Spin(7)-manifolds of cohomogeneity one

DISSERTATION

zur Erlangung des Grades eines Doktors der Naturwissenschaften

Der Fakultät für Mathematik der Technischen Universität Dortmund

vorgelegt von

Frank Reidegeld

Dortmund, Juni 2008

2

Acknowledgments

In the first place I am grateful to my thesis advisor Prof. Dr. Lorenz Schwachhöfer for providing me this interesting topic as well as for the time he spent on discussions and his valuable suggestions for improvement. I also want to thank Dr. Yaroslav Bazaikin, Prof. Dr. Jost-Hinrich Eschenburg, Dr. Mario Listing, M.Sc. Eugene Malkovich, Dr. Martin Panák, Prof. Dr. Uwe Semmelmann, Prof. Dr. Wilderich Tuschmann, and Prof. Dr. Wolfgang Ziller for helpful discussions and advice. Moreover, I would like to gratefully acknowledge my colleague Dr. Frank Klinker for debating many mathematical and other questions during my research on this thesis. I am also grateful to the other staff members of the Lehrstuhl VII (Differentialgeometrie), including Prof. em. Dr. Rolf Walter, Dr. Günter Skoruppa, and Sabine Willrich. Furthermore, I want to express my thanks to my parents and sister as well as to Melanie Kramer for her love and support during the last years. Finally, it is a pleasure to thank all the other people who have helped and encouraged me while I was working on my dissertation project.

4 CONTENTS

Contents

1 Introduction 7

2 Exceptional holonomies 17 2.1 The groups G_{2} and Spin 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 G_{2}-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Spin(7)-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Spaces with a large isometry group 41 3.1 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Metrics of cohomogeneity one . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Classification of the principal orbits 83

5 Parallel Spin 7 -manifolds with irreducible principal orbits 101 5.1 The principal orbit B^{7} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 The principal orbit Q^{1,1,1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 The principal orbit M^{1,1,0} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.4 The Aloff-Wallach spaces as principal orbits . . . . . . . . . . . . . . . . . . . 149

6 Parallel Spin 7 -manifolds with reducible principal orbits 219 6.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.2 The principal orbit SU 3 U 1 ^{2} U 1 . . . . . . . . . . . . . . . . . . . . . 225

7 Conclusion and outlook 233

A Calculations for the principal orbit Q^{1,1,1} 237

B Calculations for the principal orbit M^{1,1,0} 241

C Calculations for the Aloff-Wallach spaces as principal orbits 245 C.1 The Spin 7 -structure for the Aloff-Wallach spaces except N^{1,1} as principal

orbits and the condition for its torsion-freeness . . . . . . . . . . . . . . . . . 245

C.2 The Spin 7 -structure for the principal orbit N^{1,1} and the condition for its torsion-freeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

C.3 The Einstein condition for the generic Aloff-Wallach spaces and N^{1, 1} as principal orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

C.4 The Einstein condition for the exceptional Aloff-Wallach space N^{1,1} as principal orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

D Calculations for the principal orbit SU 3 U 1 ^{2} 265

Bibliography 269

Chapter 1

Introduction

The subject of this thesis are cohomogeneity-one metrics whose holonomy is contained in Spin 7 . Our aim is to find new examples and to prove a partial classification result. The motivation for this work is the paper ”Cohomogeneity-one G_{2}-structures” of Richard Cleyton and Andrew Swann [20]. In that paper the authors construct G_{2}-structures with and without torsion which admit a cohomogeneity-one action. We apply the methods which were developed by Cleyton and Swann to the Spin 7 -case. It turns out that our case is more intricate than the G_{2}-case. We only consider Spin 7 -structures without torsion, since this task already is sufficiently extensive.

We shortly recall what Spin 7 -structures are and why they are interesting for mathematicians and physicists. Let M be an eight-dimensional manifold. A Spin 7 -structure on M is a reduction of the coframe bundle to a principal bundle with structure group Spin 7 . Equivalently, a Spin 7 -structure is determined by a four-form Ω with the property that at every p M there is a basis of T_{p}M such that Ω can be identified via that basis with a special four-form on ℝ^{8}. We call a pair of an eight-dimensional manifold and a Spin 7 -structure a Spin 7 -manifold. On any Spin 7 -manifold, there exists a canonical orientation, which is induced by Ω Ω, and a canonical metric g, which depends non-linearly on Ω. With respect to g and the volume form Ω Ω, the four-form Ω is self-dual. A Spin 7 -structure is called parallel or torsion-free if ∇^{g}Ω 0 or equivalently dΩ 0. In this situation, g is Ricci-flat and the holonomy of the Levi-Civita connection is a subgroup of Spin 7 . The most interesting case is where the holonomy is all of Spin 7 . If this is the case, then there exists up to constant multiples a unique parallel spinor on M. This fact makes manifolds with holonomy Spin 7 interesting for physicists. In recent years, manifolds with holonomy G_{2} and Spin 7 have been studied in the context of M-theory (see Atiyah, Witten [3] and Gukov, Sparks [39]).

The equation dΩ 0 should be considered as a non-linear partial differential equation. Although the operator d is linear, the condition that Ω determines a Spin 7 -structure is a non-linear restriction. For this reason, it is difficult to construct examples of metrics with holonomy Spin 7 . That there are indeed local metrics of that kind has first been proven by Bryant [14]. The next breakthrough was the construction of complete metrics with holonomy G_{2} and Spin 7 by Bryant and Salamon [15]. The authors give an explicit description of their examples and it turns out that they are of cohomogeneity one. In particular, the metric with holonomy Spin 7 has the sphere S^{7} as principal orbit. The first compact manifolds with exceptional holonomy have been constructed by Joyce [47]. Since the techniques of Bryant [14] and Joyce [47] yield no explicit description of their metrics, it would be nice to have further explicit examples, apart from those which were constructed in [15].

The problem of finding new examples can be simplified by assuming that M admits an action of a Lie group G which preserves Ω such that the principal orbits of that action have codimension one. If this is the case, the G-action is called a cohomogeneity-one action and M, Ω is called a Spin 7 -manifold of cohomogeneity one. In the literature, there are many examples of such manifolds. Calabi [18] has constructed a complete metric with holonomy Sp 2 on T^{}ℂℙ^{2}. This metric is left invariant by a cohomogeneity-one action of SU 3 whose principal orbit is the exceptional Aloff-Wallach space N^{1,1} (see Cvetič, Gibbons, Lü, Pope [22] and Kanno, Yasui [49]). Explicit cohomogeneity-one metrics with holonomy SU 4 can be found in Cvetič et al. [21], [25] and in Herzog, Klebanov [41]. The work of these authors is based on earlier papers by Berard-Bergery [8], Page, Pope [58], and Stenzel [64]. Cohomogeneity-one metrics with holonomy Spin 7 have been described in a series of papers by Cvetič et al. [21], [22], [23], [24], [25], [26], [27] and by Kanno, Yasui [48], [49]. The results of these papers are often based on power series expansions up to a finite order and on numeric arguments. The reason behind this is that in the context of these papers dΩ 0 has an explicit solution in special cases only. Recently, Bazaikin [5] has rigorously proven the existence of two one-parameter families of non-homothetic cohomogeneity-one metrics with holonomy Spin 7 , which are asymptotically locally conical. The principal orbit is N^{1,1} ℤ_{2} for the first and S^{7} ℤ_{4} for the second family.

In the cohomogeneity-one case, the equation dΩ 0 is equivalent to a system of ordinary differential equations. There are several methods to deduce these equations. In [48], a superpotential for the equations for the Ricci-flatness is constructed, which yields a sufficient condition for the holonomy reduction. Bazaikin [5] obtained the differential equations which he studied by deforming a cone with holonomy Sp 2 . In a forthcoming paper [6], Bazaikin and Malkovich will deform cones with holonomy SU 4 . This deformation will yield a more general system of differential equations. In this thesis, we will work with a third method, which is motivated by a paper of Hitchin [42] and has also been applied in [48]. If M, Ω is of cohomogeneity one, the principal orbits are a one-parameter family of equidistant hypersurfaces. The union of all principal orbits is diffeomorphic to a product N I, where I is an interval. On any hypersurface inside a Spin 7 -manifold, there exists a canonical G_{2}-structure. The G_{2}-structure is, analogously to the Spin 7 -structure, determined by a three-form, which we denote by ω. Let \frac{}{t} be the Lie derivative in the direction of the coordinate t which parameterizes I. With this notation, the equation dΩ 0 is equivalent to:

\frac{}{t} ω dω and d ω 0 .

If we have d ω 0 for a single t, the equation holds for all t and \frac{}{t} ω dω has a unique short-time solution. The first of the above two equations is equivalent to a system of non-linear ordinary differential equations, since the principal orbit is a homogeneous space. The second one is equivalent to a system of polynomial equations, which is in many cases automatically satisfied. If we are able to solve the differential equations, we have found a metric with holonomy Spin 7 .

Before we can study the equation \frac{}{t} ω dω in a concrete situation, we first have to choose a principal orbit G H. After that, we have to find a sufficiently large space of coclosed homogeneous G_{2}-structures on G H which is preserved by \frac{}{t} ω dω. By parameterizing that space, we are finally able to deduce a system of ordinary differential equations which is equivalent to \frac{}{t} ω dω.

The seven-dimensional coset spaces G H which admit a G-invariant G_{2}-structure can be classified by algebraic methods. The group H acts on the tangent space of G H by its isotropy representation. G_{2} also acts on the tangent space as the stabilizer of the three-form ω. In this situation, H G_{2} is necessary and sufficient for G H to admit a G-invariant G_{2}-structure. Therefore, we have to classify all connected Lie subgroups H G_{2} and then all Lie groups G with H G and dim G dim H 7. With these methods, we are able to classify all possible principal orbits under some mild restrictions:

Theorem 1.1. Let G H be a compact, connected coset space which admits a G-invariant G_{2}-structure. We assume that G is compact and connected, too, and acts almost effectively on G H. If G H is not covered by a Cartesian product of lower-dimensional homogeneous spaces, G, H, and G H are up to finite coverings one of the following:

\begin{matrix}G & H & G H \\ SU 3 & U 1 & N^{k,l} with k, l ℤ \\ SO 5 & SO 3 & V^{5,2} \\ Sp 2 & Sp 1 & S^{7} \\ SO 5 & SO 3 & B^{7} \\ SU 2 ^{3} & U 1 ^{2} & Q^{1,1,1} \\ SU 3 SU 2 & SU 2 U 1 & M^{1,1,0} \end{matrix}

If G H is covered by a product, at least one of the factors is a circle.

In the above table, the indices of N^{k,l} (Q^{1,1,1}, M^{1,1,0}) denote a special embedding of U 1 (U 1 ^{2}, SU 2 U 1 ) into SU 3 (SU 2 ^{3}, SU 3 SU 2 ). Those embeddings are described explicitly in Section 5.4 (5.2, 5.3). V^{5,2} denotes the Stiefel-manifold of all orthonormal pairs in ℝ^{5} and B^{7} is the Berger space, which is described in more detail in Section 5.1. Some of the above spaces admit a transitive action by a bigger group, which also preserves the G_{2}- structure. These group actions are not listed in the table of the theorem, but are described in Chapter 4. The principal orbits which are not a product of lower-dimensional homogeneous spaces we call irreducible and the other ones we call reducible. The irreducible principal orbits coincide with the homogeneous spaces which admit a nearly parallel G_{2}-structure. Those spaces are classified in a paper of Friedrich, Kath, Moroianu, and Semmelmann [37]. Since any nearly parallel G_{2}-structure ω also satisfies d ω 0, all of those spaces admit a coclosed G_{2}-structure. If G H is a reducible principal orbit, the group G splits into G^{}U 1 and G H splits into G^{}H U 1 up to a finite cover. The six-dimensional space G^{}H admits a homogeneous SU 3 -structure. Spaces of that kind are possible principal orbits for cohomogeneity-one G_{2}-structures. For this reason, six-dimensional spaces admitting a homogeneous SU 3 -structure are studied by Cleyton and Swann [20]. Their list coincides with our list with the single exception of a space of type SU 2 ^{2} U 1 U 1 , which is included in our list, but seems to be missing in [20]. Many of the reducible principal orbits admit a homogeneous coclosed G_{2}-structure, but we will not prove this fact for all cases.

Any Riemannian product of a parallel G_{2}-structure and a circle is also a parallel Spin 7 - structure. We will shortly consider the question if there are any further examples of parallel cohomogeneity-one Spin 7 -structures with a reducible principal orbit, for example warped products. In the general case, there seems to be nothing which obstructs the existence of such examples. We investigate one particular principal orbit, namely SU 3 U 1 ^{2} U 1 , in detail. In that case, we obtain a negative result only: Any parallel cohomogeneity-one Spin 7 -manifold with that principal orbit is a Riemannian product of a circle and a parallel cohomogeneity-one G_{2}-manifold with principal orbit SU 3 U 1 ^{2}. Since that kind of manifolds has been studied by Cleyton and Swann [20], we will not consider this issue further and focus on the irreducible principal orbits.

As we have mentioned above, we have to find a sufficiently large space of coclosed G_{2}- structures on any of the orbits G H which we consider. We split this problem into two subproblems. First, we classify all G-invariant metrics on G H and then we classify all invariant G_{2}-structures which have a fixed associated metric and orientation. The classification of the metrics can be done with help of representation theory. We fix a normal background metric q on G H. Any other G-invariant metric on G H can be identified by q with an Hequivariant endomorphism of a fixed tangent space. The set of those endomorphisms can be described with help of Schur’s lemma. If the tangent space splits into pairwise inequivalent H-submodules, any G-invariant metric is diagonal with respect to any basis which is adapted to the splitting. This condition will be satisfied in many of the cases which we consider. In the other cases, we will also assume that the metric is diagonal in order to keep our calculations simple. The space of all G-invariant G_{2}-structures which have a fixed associated metric and orientation can be described by Norm_{SO 7}H Norm_{G_{2}}H. Since that space is independent of the choice of the metric and orientation, we also call it the space of all G_{2}-structures with the same associated metric and orientation. In many cases, we are able to describe that space explicitly. In the other cases, we restrict ourselves to a submanifold such that the set of all G_{2}-structures which we consider is preserved by \frac{}{t} ω dω. We are now able to make an ansatz for ω. After having checked if d ω 0, we can finally deduce a system of ordinary differential equations which is sufficient for the holonomy reduction.

The space M of cohomogeneity one on which we want to define the Spin 7 -structure has to have at least one singular orbit G K, where H K G. Otherwise, the metric would be either non-complete or a product metric. The quotient K H has to be a sphere in order to make M a manifold. In some cases, K H is a quotient of a sphere by a discrete group. We investigate these cases, too, since M is an orbifold bundle over G K in that situation.

There are several problems which have to be solved before we finally obtain a cohomogeneityone metric with reduced holonomy. The first problem is that not necessarily all solutions of our differential equations correspond to metrics which can be smoothly extended to the singular orbit. There are certain smoothness conditions, which have to be satisfied. In the case where the metric is analytic, these conditions translate into conditions on the coefficients of the Taylor expansion of the metric. Since any metric whose holonomy is contained in Spin 7 is also Ricci-flat, it follows by a well-known theorem of DeTurck and Kazdan [30] that the metric is analytic. We therefore may assume that the metric is described by a power series. In a paper of Eschenburg and Wang [32], the smoothness conditions for analytic cohomogeneity-one metrics are deduced with help of representation theoretical arguments. With help of the methods from that paper, we are able to deduce the smoothness conditions for our situation. These conditions usually state that some of the functions which describe the metric are even and some are odd. Sometimes there are also relations between different functions or other restrictions on the metric, which could not be easily seen without applying the methods of [32]. After having deduced the smoothness conditions, we can check them with help of symmetry arguments. If the metric is smooth, its holonomy is contained in Spin 7 . Therefore, there has to exist one, or if the holonomy is not all of Spin 7 , a certain family of smooth parallel four-forms. With help of this argument, the smoothness of Ω follows.

The second problem is that the differential equations degenerate at the singular orbit. In general, we have to divide by zero on the right hand side of our equations. This may cause singularities, but in all except one case we always find metrics on the singular orbit such that the zero in the denominator is compensated by another zero in the numerator. There are two subcases which we have to discuss: In the first case, there are no restrictions which we have to impose on the initial values at the singular orbit and we are in the situation of the Picard-Lindelöf theorem. The existence and uniqueness of the solutions of our initial value problem thus immediately follows. Moreover, we will see that the differential equations which we obtain in this case are explicitly solvable. In the second case, we have to impose certain extra conditions on the initial values in order to compensate the zeroes in the denominator. If this is the case, the right hand side of the differential equations is defined on a non-open set only and we cannot apply the Picard-Lindelöf theorem. In particular, we do neither know if there exists a solution of our initial value problem nor if this solution is unique. There are indeed cases where the solution depends on initial conditions of higher order which we can freely prescribe. For the cohomogeneity-one Einstein condition, the number and order of those free parameters can be calculated by means of representation theory. This result is proven in the paper of Eschenburg and Wang [32]. We therefore make the following plan: Let m be the maximal order of the free parameters for the cohomogeneity-one Einstein condition. Any of the free parameters for the equations for the holonomy reduction also has to be one of the parameters of Eschenburg and Wang. We make a power series expansion up to m^{th} order and check which of those degrees of freedom actually remain in our situation. Then we prove by an explicit calculation that for any choice of the free parameters there exists a formal power series solution of our initial value problem. The convergence of the power series follows by another result of Eschenburg and Wang [32]. After that, we finally have classified all solutions of our initial value problem.

In the first of the above two cases, we have to consider certain spaces of K-equivariant maps in order to deduce the smoothness conditions. This information is already sufficient to apply the theorem of Eschenburg and Wang [32]. As a by-product of our calculations, we thus can prove in both cases the existence of cohomogeneity-one Einstein metrics on a neighborhood of the singular orbit.

We remark that most results of Eschenburg and Wang hold only if the tangent and the normal space of the singular orbit contain no equivalent H-submodules. This is a general assumption which was made in [32]. Eschenburg and Wang [32] suppose that all of their results can be carried over to the general case. In some of the cases which we consider the above assumption is violated. Nevertheless, we show by a modification of the proof of the theorem of Eschenburg and Wang that their theorem applies to our cases as well.

It is already a demanding task to carry out the program which we have outlined above for a single principal orbit. Therefore, we will not consider all of the reducible principal orbits in this thesis. More concretely, we will not investigate cohomogeneity-one metrics whose principal orbit is the Stiefel-manifold V^{5,2} or covered by the sphere S^{7}. One of the reasons why we omit these principal orbits is that they are intensively studied in the literature. Metrics with principal orbit V^{5,2} and holonomy SU 4 are constructed by Stenzel [64] and Cvetič et al. [25]. On parallel cohomogeneity-one Spin 7 -structures whose principal orbit is covered by S^{7} there exists much literature. The most simple example of such a structure is the flat Spin 7 -structure on ℝ^{8}. Spin 7 acts by its eight-dimensional representation on ℝ^{8}. The orbits of that action are spheres and the action thus is of cohomogeneity one. As we have mentioned above, the parallel Spin 7 -structure of Bryant and Salamon [15] is also of cohomogeneity one and has S^{7} as principal orbit. Furthermore, Cvetič et. al. investigate parallel Spin 7 -structures of that kind in [23], [24], [26], and [27]. Among their examples are manifolds with principal orbit S^{7} and singular orbit S^{4} as well as manifolds with principal orbit S^{7} ℤ_{4} and singular orbit ℂℙ^{3}. Finally, the cohomogeneity-one metrics of Bazaikin [5] with principal orbit S^{7} ℤ_{4} fit in this context, too. Those metrics are a special case of the numerical examples of Cvetič et al. and also have ℂℙ^{3} as singular orbit. All of the above metrics are diagonal. On both V^{5,2} and S^{7} there exist non-diagonal homogeneous metrics. Since we can transform any of the non-diagonal metrics into a diagonal one by the action of the normalizer, it seems unlikely that there are many further examples apart from those in the literature. This argument will be explained in more detail in Section 7.

The first principal orbit which we consider in detail is the Berger space B^{7} SO 5 SO 3 . The embedding of SO 3 into SO 5 is given by the five-dimensional irreducible representation of SO 3 . Since B^{7} is isotropy-irreducible, there is up to constant multiples only one SO 5 -invariant metric and one SO 5 -invariant G_{2}-structure on B^{7}. With help of this fact, we can conclude that the only parallel cohomogeneity-one Spin 7 -structures with principal orbit B^{7} are cones over that space. Next, we consider principal orbits of type Q^{k,l,m} : SU 2 ^{3} U 1 _{k,l,m}^{2}, where the three indices describe the embedding of U 1 _{k,l,m}^{2} into SU 2 ^{3}. We will see that only Q^{1,1,1} admits a homogeneous G_{2}-structure. Since the tangent space of Q^{1,1,1} splits into pairwise inequivalent U 1 _{1,1,1}^{2}-modules, any homogeneous metric on that space is diagonal. For any fixed metric and orientation on Q^{1,1,1}, there exists a oneparameter family of SU 2 ^{3}-invariant G_{2}-structures whose associated metric and orientation coincides with the chosen ones. Moreover, there are no further G_{2}-structures of that kind and the one-parameter family can be generated by an action of U 1 by isometries. We fix one of those isometries and let it act on all of the principal orbits simultaneously. This construction yields an isometry of the whole cohomogeneity-one manifold. If we find a single parallel Spin 7 -structure, we obtain by the U 1 -action a whole family of parallel Spin 7 -structures, which all have the same associated metric. From this it follows that the holonomy is SU 4 . In the situation which we study at the moment, the equations for the holonomy reduction are explicitly solvable. There are solutions with S^{2} S^{2} and with S^{2} S^{2} S^{2} as singular orbit. The metrics of the first kind are smooth at singular orbit. In the second case, the length of the collapsing circle K U 1 ^{2} is 4πt O t^{2} for small t. If the metric was smooth, this length would have to be 2πt O t^{2} . Our metric therefore has a singularity at the singular orbit. We nevertheless suppose that it is possible to make the metric smooth by replacing the principal orbit Q^{1,1,1} by Q^{1,1,1} ℤ_{2} for a suitable ℤ_{2}-action. We will not carry out this construction explicitly, since later on there will be another principal orbit for which this will be explained in detail. All in all, we have proven the following theorem:

Theorem 1.2. Let M, Ω be a parallel cohomogeneity-one Spin 7 -manifold whose principal orbits are of type Q^{k,l,m} : SU 2 ^{3} U 1 _{k,l,m}^{2}. In this situation, the following statements are true:

1. The principal orbits are SU 2 ^{3}-equivariantly diffeomorphic to Q^{1,1,1}.

2. The metric g which is associated to Ω has holonomy SU 4 .

3. If M has a singular orbit, it has to be S^{2} S^{2} or S^{2} S^{2} S^{2}. In the first case, g can be extended to a smooth, complete metric and M is non-compact. In the second case, M is non-compact, too, but g cannot be smooth at the singular orbit. However, g can be extended such that any geodesic which does not intersect the singular orbit is defined on all of ℝ.

The above metrics have already been described in another context (see Cvetič et al. [21],

[25] and Herzog, Klebanov [41]). Our proof that there are no further Spin 7 -structures on a cohomogeneity-one manifold with principal orbit Q^{k,l,m} is new. Moreover, we have introduced a new point of view on these metrics.

The next kind of principal orbits which we consider are spaces of type SU 3 SU 2 SU 2 U 1 where the first factor of SU 2 U 1 is embedded into SU 3 . In this case, we obtain similar results as in the previous one. The embedding of U 1 into SU 3 SU 2 has to be special in order to make SU 3 SU 2 SU 2 U 1 a space which admits a SU 3 SU 2 -invariant G_{2}-structure. On a cohomogeneity-one manifold (or orbifold) with that space, which we denote by M^{1,1,0}, as principal orbit, the differential equations for the holonomy reduction are explicitly solvable. There are three possible singular orbits, namely S^{2}, ℂℙ^{2}, and S^{2} ℂℙ^{2}. If the singular orbit is S^{2}, our space of cohomogeneity one is an orbifold but not a manifold, since M^{1,1,0} is a S^{5} ℤ_{3}-bundle over the singular orbit. In Section 5.3, we will prove the following theorem:

Theorem 1.3. Let M, Ω be a parallel cohomogeneity-one Spin 7 -manifold whose principal orbit is of type M^{k,l,m} : SU 3 SU 2 SU 2 U 1 _{k,l,m}. The indices k, l, and m describe the embedding of the abelian factor of SU 2 U 1 _{k,l,m} into SU 3 SU 2 and the embedding of the semisimple factor shall be as above. In this situation, the following statements are true:

1. The principal orbit is SU 3 SU 2 -equivariantly diffeomorphic to M^{1,1,0}.

2. The metric g which is associated to Ω has holonomy SU 4 .

3. If M has a singular orbit, it has to be S^{2}, ℂℙ^{2}, or S^{2} ℂℙ^{2}. In all three cases, M is non-compact and g can be extended such that any geodesic which does not intersect the singular orbit is defined on all of ℝ. This means in particular that if g is smooth, M, g is complete. If the singular orbit is S^{2}, g cannot be a smooth orbifold metric. In the second case, the metric is smooth and in the third case the metric cannot be smooth at the singular orbit.

Although the metric is not smooth if the singular orbit is S^{2} ℂℙ^{2}, it may be possible to replace the principal orbit by M^{1,1,0} ℤ_{2} in such a way that the metric becomes a smooth one. As in the previous case, all of the above metrics are mentioned in [21], [25], and [41]. Nevertheless, our classification result is new.

Finally, we consider the Aloff-Wallach spaces as principal orbits. An Aloff-Wallach space is a coset space of type N^{k,l} : SU 3 U 1 _{k,l}, where k, l ℤ and U 1 _{k,l} is the subgroup of SU 3 which is generated by diag ik, il, i k l . The Aloff-Wallach spaces which are SU 3 -equivariantly diffeomorphic to N^{1,0} or N^{1,1} are called exceptional and the other ones are called generic. If N^{k,l} is an exceptional Aloff-Wallach space, the tangent space T_{eU 1 _{k,l}}N^{k,l} contains a pair of equivalent U 1 _{k,l}-submodules. Therefore, there exist non-diagonal SU 3 - invariant metrics on those spaces no matter how we fix the basis of the tangent space. As we have announced above, we nevertheless assume that the metric is diagonal in order to keep our considerations simple. For any diagonal metric g on an Aloff-Wallach space N^{k,l}, there exists a special coclosed G_{2}-structure such that its associated metric is g. As in the Q^{1,1,1}- and M^{1,1,0}-case, there is a U 1 -action on this G_{2}-structure which generates a whole one-parameter family of SU 3 -invariant G_{2}-structures with the same associated metric and orientation. However, there are two differences to the previous cases. First, the U 1 -action is not induced by a subgroup of the normalizer Norm_{SU 3}U 1 _{k,l}. Second, the set of all coclosed G_{2}-structures in the above family is usually discrete. Only in the case where N^{k,l} is SU 3 -equivariantly diffeomorphic to N^{1,1} and the initial G_{2}-structure is of a special type all G_{2}-structures in the family are coclosed. In the other cases, it is even possible to restrict ourselves without loss of generality to the initial coclosed G_{2}-structure. Unfortunately, we cannot exclude that the space of all SU 3 -invariant G_{2}-structures with the same associated metric and orientation is larger than the U 1 -orbit. If N^{k,l} is generic or SU 3 -equivariantly diffeomorphic to N^{1,0}, it may have further connected components, which are diffeomorphic to a circle. If N^{k,l} is SU 3 -equivariantly diffeomorphic to N^{1,1}, the space is three-dimensional and its connected components are generated by a SO 3 -action. Since we suppose that there are not many coclosed G_{2}-structures on the Aloff-Wallach spaces, we restrict ourselves to G_{2}-structures which belong to the U 1 -orbit. With help of the above facts, we can prove that the holonomy of a parallel cohomogeneity-one Spin 7 -manifold whose principal orbit is a generic Aloff-Wallach space is all of Spin 7 . If the principal orbit is SU 3 -equivariantly to N^{1,0}, the holonomy is the same. In the N^{1,1}-case, we do not investigate all G_{2}-structures in the SO 3 -orbit. Therefore, we can conclude only that the holonomy is contained in Spin 7 . If the G_{2}-structure on the principal orbit is of the special type which we have mentioned above, we can moreover prove that the holonomy is contained in SU 4 . In Kanno, Yasui [49], further results on the holonomy of cohomogeneity-one metrics with principal orbit N^{1,1} are proven. By classifying all closed connected groups K with U 1 _{k,l} K SU 3 , we see that there are four possible singular orbits, namely SU 3 U 1 ^{2}, S^{5}, SU 3 SO 3 , and ℂℙ^{2}. If the singular orbit is ℂℙ^{2}, the space on which SU 3 is acting on is usually an orbifold but not a manifold, since K U 1 _{k,l} is a lens space of type S^{3} ℤ_{k l}. If furthermore k l 0, the space has a singularity which is not an orbifold singularity. S^{5} or SU 3 SO 3 as a singular orbit is possible only if the principal orbit is of type N^{1,0}.

After having classified the possible singular orbits, we consider solutions of the equations for the holonomy reduction which are defined on a tubular neighborhood of one of those orbits. These equations have an explicit solution in special cases only. We therefore have to work with the power series methods which we have described earlier in this introduction. Since these methods yield local results only, we do not check the completeness of our metrics. The first singular orbit which we investigate is SU 3 U 1 ^{2}. If the principal orbit is a generic Aloff-Wallach space or N^{1,0}, there exist no solutions of our differential equations near the singular orbit. If the principal orbit is N^{1,1}, there exists a short-time solution, which depends on no parameters except the metric on SU 3 U 1 ^{2}. Unfortunately, the metric cannot be smoothly extended to the singular orbit. Nevertheless, this becomes possible if we replace the principal orbit by a suitable quotient N^{1,1} ℤ_{2}. This time, we explicitly carry out that procedure and obtain another cohomogeneity-one manifold with singular orbit SU 3 U 1 ^{2}. The next singular orbit which we consider is S^{5}. We will prove that any SU 3 -invariant metric on S^{5} SU 3 SU 2 can be uniquely extended to a smooth cohomogeneity-one metric which solves the equations for the holonomy reduction. If the singular orbit is SU 3 SO 3 , we find no parallel SU 3 -invariant Spin 7 -structures. In the cases where the singular orbit is ℂℙ^{2}, the metrics are no longer unique, but depend on additional free parameters of second or third order. We will explain these parameters by the results of Eschenburg and Wang [32]. To sum up, we have proven the following result:

Theorem 1.4. Let M, Ω be a parallel cohomogeneity-one Spin 7 -manifold (or -orbifold) whose principal orbit is an Aloff-Wallach space. We assume that the metric associated to Ω is diagonal with respect to a natural basis and that the G_{2}-structure on the principal orbit is always contained in the one-parameter family which we have mentioned above. In this situation, the following statements are true:

1. If the principal orbit is N^{1,1} and the coefficients of Ω satisfy certain relations, the holonomy of the metric associated to Ω is contained in SU 4 . In the other cases, it is contained in Spin 7 . If the principal orbit is a generic Aloff-Wallach space or N^{1,0}, the holonomy is all of Spin 7 .

2. If the principal orbit is N^{1,1} and the singular orbit is SU 3 U 1 ^{2}, the metric associated to Ω can never be smoothly extended to the singular orbit. If we divide the principal orbit by a suitable ℤ_{2}-action which leaves Ω invariant, the metric can always be smoothly extended. Moreover, it is uniquely determined by the metric on the singular orbit.

3. If the singular orbit is S^{5}, any SU 3 -invariant metric on the singular orbit can be uniquely extended to a smooth cohomogeneity-one metric with holonomy Spin 7 such that the principal orbit is N^{1,0} and the G_{2}-structure on the principal orbit satisfies our restrictions.

4. If the singular orbit is ℂℙ^{2}, there are initial conditions of higher order which we can prescribe. Their number and order depend on further details of the principal orbit and the initial values of 0^{th} order. There are several subcases, in which there are two initial conditions of third order, one initial condition of third order, or one initial condition of second order.

5. Under some mild restrictions, the above examples are all parallel cohomogeneity-one Spin 7 -manifolds which satisfy the assumptions on the metric and the G_{2}-structure which we have made.

The metrics with singular orbit ℂℙ^{2} which depend on two parameters of third order are new contributions of the author. The other metrics from the above theorem have already been mentioned in the literature [5], [24], [39], [48], [49]. Since those results are often numerical, many of our proofs on the local existence and smoothness of our metrics are new. The same applies to the fact that there are no further free parameters apart from those in the theorem.

With help of the theorem of Eschenburg and Wang [32], we find many examples of cohomogeneity-one Einstein metrics with principal orbit Q^{1,1,1}, M^{1,1,0}, or N^{k,l}. Earlier in this introduction, we have remarked that there is a technical assumption which is necessary for the theorem. If the principal orbit is an exceptional Aloff-Wallach space, that assumption is violated in some cases. As we have announced above, we can nevertheless apply the theorem of Eschenburg and Wang to those cases. We state the results which we obtain this way as a separate theorem. On the quaternionic projective space ℍℙ^{2} there is an Einstein metric of cohomogeneity one with principal orbit N^{1,0} (see Püttmann, Rigas [59]). The examples of cohomogeneity-one Einstein metrics which are neither ℍℙ^{2} nor have reduced holonomy are further contributions of the author.

This thesis is organized as follows: In Chapter 2, we recall some facts about G_{2}- and Spin 7 - structures and about the two groups themselves. In the course of our considerations, we need many results on manifolds with a large isometry group, which we collect in Chapter 3. In Section 3.1, we introduce some methods which simplify our calculations on homogeneous spaces. Section 3.2 deals with manifolds of cohomogeneity one. In particular, we resume the paper of Eschenburg and Wang [32] and introduce the flow equation of Hitchin [42]. In the fourth chapter, we classify the spaces which admit a homogeneous G_{2}-structure. In order to do this, we need a list of all connected Lie subgroups of G_{2}. Therefore, we classify them, too. Chapter 5 is the most extensive one of this thesis. In that chapter, we investigate the irreducible principal orbits which we have chosen in detail. We start with the Berger space B^{7}, which is studied in Section 5.1. In Section 5.2 (5.3), we explore the issue of cohomogeneityone metrics with special holonomy whose principal orbit is Q^{1,1,1} (M^{1,1,0}). Parallel Spin 7 - manifolds of cohomogeneity one with an Aloff-Wallach space as principal orbit are investigated in Section 5.4. This section is the most comprehensive section of Chapter 5. The reasons for this are that we have to consider the generic and the two exceptional Aloff-Wallach spaces separately, that the calculations for the exceptional Aloff-Wallach spaces are more difficult than those for the other principal orbits, and that the equations for the holonomy reduction do not have a general explicit solution. The sixth chapter is devoted to cohomogeneity-one Spin 7 -manifolds with a reducible principal orbit. In Section 6.1, we deal with the equation dΩ 0 in the general situation. The case where the principal orbit is SU 3 U 1 ^{2} U 1 is treated in the second section of that chapter. The seventh chapter is called ”Conclusion and outlook”. It begins with a short summary of our results. Furthermore, we motivate why we have not considered S^{7}, S^{7} ℤ_{4}, or the Stiefel-manifold V^{5,2} as a principal orbit. The calculations which we were necessary to derive the equations for the holonomy reduction and for the Einstein condition are included in the appendix.

Exceptional holonomies

2.1 The groups G_{2} and Spin 7

Before we introduce the most important facts on metrics with exceptional holonomy, we first have to take a closer look at the groups G_{2} and Spin 7 . These groups can be described with help of the octonions. Therefore, we have to give a short introduction to normed division algebras, too. This introduction is in part based on the paper of John Baez [4] on the octonions. Since most of the results of this section are well-known, we usually will omit their proofs and refer to the literature instead.

Definition 2.1.1. A normed division algebra is a (not necessarily associative) real algebra A with a unit element and a scalar product ., . satisfying

xy, xy x, x y, y x, y A .

Let A be a normed division algebra:

1. A non-zero linear map Φ : A A satisfying

Φ xy Φ x Φ y x, y A

is called an automorphism of A.

2. A linear map φ : A A with

φ xy xφ y φ x y x, y A

is called a derivation of A.

3. Let x be an element of A. The map L_{x} : A A with L_{x} y xy is called the left multiplication by x. Analogously, the map R_{x} : A A with R_{x} y yx is called the right multiplication by x.

Example 2.1.2. ℝ and ℂ together with the scalar products x, y : xy x, y ℝ and x, y : Re xȳ x, y ℂ are normed division algebras. On the quaternions ℍ there exists a scalar product, too, which makes ℍ a normed division algebra and 1, i, j, k an orthonormal basis of ℍ.

The real numbers ℝ can be embedded as ℝ 1 into any normed division algebra A. This fact allows us to define the notion of imaginary numbers in A:

Definition 2.1.3. Let A be a normed division algebra. The orthogonal complement of ℝ A is called the imaginary space Im A of A. Its elements are the imaginary numbers of A. Let x be an element of A. The projection of x onto ℝ is called the real part Re x of x and its projection onto Im A is called the imaginary part Im x of x. Re x Im x we call the conjugate x̄ of x.

We collect some useful facts on normed division algebras:

Lemma 2.1.4. Let A be a normed division algebra. In this situation, the following statements are true:

1. Let x, y Im A . Then xy yx 2 x, y and, in particular, x^{2} x ^{2}.

2. Any x A 0 has an inverse x^{1} \frac{x̄}{x ^{2}}. This fact justifies the name ”division algebra” and shows that the maps L_{x} and R_{x} are bijective.

3. Let x 1. Then, we have yx, zx xy, xz y, z . In this situation, L_{x} and R_{x} are therefore both orthogonal.

4. Since A is not necessarily associative, we do not have L_{x} L_{y} L_{xy}. The set of all L_{x} (or R_{x}) with x 1 therefore is not necessarily a group, but there is still a group generated by all L_{x} (or R_{x}). That group turns out to be a Lie subgroup of O dim A .

Remark 2.1.5. Let A be an algebra with a unit element. If for all x A 0 the maps R_{x} and L_{x} are bijective, A is called a division algebra. Any normed division algebra is a division algebra. The converse is not necessarily true.

The automorphisms of a normed division algebra have the following properties:

Lemma 2.1.6. Let A be a normed division algebra and Φ be an automorphism of A. Then, we have:

1. Φ 1 1.

2. If A is finite-dimensional, Φ is bijective. This fact simply follows from the equation Φ x Φ x^{1} 1. Now, we have justified why we call Φ an automorphism.

3. Let x be in ℝ. Then we have Φ x x. An automorphism therefore is determined by its restriction to Im A .

4. Φ is an orthogonal map, i.e., Φ x , Φ y x, y x, y A.

5. Let x be in Im A . Then Φ x is in Im A , too. We therefore can identify any automorphism of A with an element of O Im A .

2.1. THE GROUPS G_{2} AND SPIN 7 19

6. The automorphisms of A form a group, the so called automorphism group Aut(A). It is possible to show that Aut A is a Lie subgroup of O dim A 1 .

We can describe the Lie algebra of this Lie group explicitly:

Lemma 2.1.7. Let A be a normed division algebra. Furthermore, let φ and ψ be derivations of A. Then, we have:

1. For any x ℝ, we have φ x 0. Therefore, φ is determined by its restriction to Im A .

2. Let x be in Im A . Then φ x is in Im A , too. We therefore can identify any derivation with a linear map Im A Im A .

3. φ is skew-symmetric, i.e., φ x , y x, φ y 0 for all x, y A.

4. The commutator φ ψ ψ φ is a derivation, too. The set of all derivations of A therefore is a Lie algebra 𝔡𝔢𝔯 A which is a subalgebra of 𝔰𝔬 n 1 .

5. 𝔡𝔢𝔯 A is the Lie algebra of the Lie group Aut A .

On any normed division algebra, there is a canonical 3-form and a canonical 4-form:

Definition 2.1.8. Let A be a normed division algebra. The map

\begin{aligned}., . & : A A A \\ x, y & : xy yx \end{aligned}

is called the commutator of A. The map

\begin{aligned}., ., . & : A A A A \\ x, y, z & : xy z x yz \end{aligned}

is called the associator of A.

Lemma 2.1.9. Let A be a normed division algebra and let x, y, z Im A . Then x, y and x, y, z are in Im A , too.

We can use the above maps to construct our forms:

Lemma 2.1.10. Let A be a normed division algebra.

1. The map

\begin{aligned}ω & : A A A ℝ \\ ω x, y, z & : \frac{1}{2} x, y , z \end{aligned}

is a three-form, the canonical three-form of A.

2. The map

\begin{aligned}ω^{} & : A A A A ℝ \\ ω^{}x, y, z, w & : \frac{1}{2} x, y, z , w \end{aligned}

is a four-form, the canonical four-form of A.

Remark 2.1.11. 1. We have inserted the minus in the definition of the canonical four-form in order to make it consistent with later conventions for the signs.

2. Since we have xy yx 2 x, y for imaginary x and y, the restriction of ω to the imaginary space satisfies ω _{Im A ^{3}} x, y, z xy, z .

3. If x, y or z, (or w) is real, we have ω x, y, z 0 (ω^{}x, y, z, w 0).

4. From the antisymmetry of ω^{}, it follows that x^{2} y x xy , xy x x yx , and yx x y x^{2} for all x, y A. Therefore, any subalgebra of A which is generated by two elements is associative. An algebra with this property is called alternative.

5. It is easily possible to determine the objects we have defined above for the quaternions. The automorphism group of ℍ is isomorphic to SO 3 . The group generated by the leftmultiplication and the group generated by the right-multiplication with unit quaternions are both isomorphic to SU 2 . Together they generate all of SO 4 . The canonical three-form ω restricted to Im ℍ is the volume form which satisfies ω i, j, k 1. The canonical four-form of ℍ vanishes, since ℍ is associative.

The normed division algebras have been classified long ago:

Theorem 2.1.12. (See Hurwitz [44].) There are exactly four normed division algebras, namely ℝ, ℂ, ℍ, and the octonions 𝕆.

The octonions are determined up to an isomorphism by the above theorem. 𝕆 has a subalgebra which is isomorphic to ℍ. Let ϵ be an element of 𝕆 with ϵ 1 and ϵ ℍ 𝕆. Then 1, i, j, k, ϵ, iϵ, jϵ, kϵ is an orthonormal basis of 𝕆, which we call the standard basis of 𝕆. For this basis, we obtain the following multiplication table:

i j k ϵ iϵ jϵ kϵ
i -1 k -j iϵ -ϵ -kϵ jϵ
j -k -1 i jϵ kϵ -ϵ -iϵ
k j -i -1 kϵ -jϵ iϵ -ϵ
ϵ -iϵ -jϵ -kϵ -1 i j k
iϵ ϵ -kϵ jϵ -i -1 -k j
jϵ kϵ ϵ -iϵ -j k -1 -i
kϵ -jϵ iϵ ϵ -k -j i -1

In the above table, we multiply a x from the left column with a y from the upper row and obtain x y. The multiplication table is independent of ϵ and the subalgebra which isomorphic to ℍ.

Remark 2.1.13. The octonions are neither commutative nor associative, since, for example, ij k ji and ij ϵ kϵ i jϵ .

The derivations of 𝕆 are isomorphic to one of the exceptional Lie algebras:

Proposition 2.1.14. The Lie algebra 𝔡𝔢𝔯 𝕆 is isomorphic to the compact real form of the complex simple Lie algebra 𝔤_{2}.

Proof : First, we determine the dimension of 𝔡𝔢𝔯 𝕆 . Each element of the standard basis can be written in the form i^{α_{1}}j^{α_{2}} ϵ^{α_{3}} with α_{1}, α_{2}, α_{3} 0, 1 . Let Φ be an automorphism of 𝕆. Since Φ xy is determined by Φ x and Φ y , Φ is determined by Φ i , Φ j , and Φ ϵ . Let Ψ : 𝕆 𝕆 be a ℝ-linear map. The following conditions are necessary for Ψ to be an automorphism:

• Ψ i , Ψ j , Ψ ϵ Im 𝕆

• Ψ i 1, Ψ j 1, Ψ ϵ 1

• Ψ j Ψ i

• Ψ ϵ span Ψ i , Ψ j , Ψ k

A triple which satisfies the same conditions as Ψ i , Ψ j , Ψ ϵ is called a basis triple. It is possible to show that for all basis triples x, y, z there exists a unique automorphism Φ Aut 𝕆 with Φ i x, Φ j y, Φ ϵ z. We can choose the value of Φ i freely out of S^{6} Im 𝕆 . Since Φ i Φ j , we can choose for any Φ i the value of Φ j out of S^{5} span 1, Φ i ^{}. Analogously, we can choose for any Φ i , Φ j the value of Φ ϵ out of S^{3} Φ ℍ ^{}. Therefore, Aut 𝕆 is a S^{3}-bundle over the space of all orthonormal 2-frames in Im 𝕆 . That space is called the Stiefel-manifold V_{2,7} and is a S^{5}-bundle over S^{6}. Therefore, dim V_{2,7} dim S^{6} dim S^{5} 11 and dim 𝔡𝔢𝔯 𝕆 dim Aut 𝕆 dim V_{2,7} dim S^{3} 14.

Next, we determine the rank of 𝔡𝔢𝔯 𝕆 . In order to do this, we consider the following Cartan subalgebra of 𝔰𝔬 7 :

𝔱 : ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 & λ_{1} \\ & λ_{1} & 0 \\ & & & 0 & λ_{2} \\ & & & λ_{2} & 0 \\ & & & & & 0 & λ_{3} \\ & & & & & λ_{3} & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; λ_{1}, λ_{2}, λ_{3} ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

The above matrices should be considered as matrix representations of ℝ-linear maps with respect to the standard basis of Im 𝕆 . 𝔡𝔢𝔯 𝕆 𝔱 is a Cartan subalgebra of 𝔡𝔢𝔯 𝕆 . We therefore have to check which of the above matrices are derivations of 𝕆. After a short calculation, we see that

𝔡𝔢𝔯 𝕆 𝔱  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & λ_{1} \\ & λ_{1} & 0 \\ & & & 0 & λ_{2} \\ & & & λ_{2} & 0 \\ & & & & & 0 & λ_{1} λ_{2} \\ & & & & & λ_{1} λ_{2} & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; λ_{1}, λ_{2} ℝ ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; (2.1)

Since this is a two-dimensional Lie algebra, the rank of 𝔡𝔢𝔯 𝕆 is 2. Aut 𝕆 O 7 is a compact Lie group. It is a well known fact that the Lie algebra of a compact Lie group has to be the direct sum of an abelian and a semisimple Lie algebra. With help of the classification of the complex simple Lie algebras we see that there is only one 14-dimensional complex Lie algebra of rank 2 which is of this type, namely 𝔤_{2}. Since Aut 𝕆 is compact, 𝔡𝔢𝔯 𝕆 has to be the compact real form of 𝔤_{2}. We will denote that real form shortly by 𝔤_{2}, too.

Remark 2.1.15. Any φ 𝔤_{2} is determined by its values on i, j, ϵ for the same reasons as automorphisms of 𝕆.

Before we proceed, we will mention a few facts on the structure of 𝔤_{2}. The Dynkin diagram of 𝔤_{2} consists of two nodes and a threefold directed arrow connecting the two nodes:

.

Its root system is generated by a short and a long root and has 12 Weyl chambers. The Weyl group of 𝔤_{2} is the Dieder group D_{6} (see Humphreys [43]). We will now prove some properties of Aut 𝕆 :

Lemma 2.1.16. Aut 𝕆 is a simply connected Lie subgroup of SO 7 . Moreover, it is isomorphic to the simply connected Lie group with Lie algebra 𝔤_{2}, which is denoted by G_{2}.

Proof : V_{2,7} is connected, since it is a S^{5}-bundle over S^{6}. The S^{3}-bundle Aut 𝕆 over V_{2,7} is connected for the same reasons. Aut 𝕆 therefore is not only a Lie subgroup of O 7 , but of SO 7 , too.

It is known that the center of the Lie group G_{2} is trivial. Therefore, there are no connected Lie groups covered by G_{2}, except G_{2} itself. Since G_{2} is the only connected Lie group with Lie algebra 𝔤_{2}, we have shown that Aut 𝕆 is isomorphic to G_{2} and therefore simply connected.

We now consider the group which is generated by all L_{x} with the property that x 𝕆 is of unit length. Until we have found an explicit description of this group, we denote it by K. The following result will be useful in order to find that description:

Lemma 2.1.17. (See Cacciatori, Cerchiai, Della Vedova, Ortenzi, and Scotti [17].) The automorphism group G_{2} of 𝕆 is a Lie subgroup of K. Moreover, it consists of precisely those elements of K which leave 1 𝕆 invariant.

Corollary 2.1.18. Let 𝔨 be the Lie algebra of K. 𝔤_{2} consists of exactly those ψ 𝔨 with ψ 1 0.

Since K acts transitively on the unit sphere in 𝕆, we obtain a second corollary from the above lemma:

Corollary 2.1.19. The coset space K G_{2} is diffeomorphic to the 7-sphere. Moreover, K is connected and a Lie subgroup of SO 8 .

The last part of the corollary can be shown by similar arguments as in the proof of Lemma 2.1.16. We are now able to describe K explicitly:

Proposition 2.1.20. The group generated by the left multiplication with unit octonions is isomorphic to Spin 7 and acts on 𝕆 ℝ^{8} by the spinor representation.

Proof : Since K G_{2} S^{7}, K is a G_{2}-bundle over S^{7}. We have the following exact sequence:

… π_{1} G_{2} π_{1} K π_{1} S^{7} ….

G_{2} and S^{7} are both simply connected. The above sequence therefore becomes:

… 0 π_{1} K 0 …

and we have shown that K is simply connected. Let 𝔨 be the Lie algebra of K and q be a biinvariant metric on K. The q-orthogonal complement 𝔪 of 𝔤_{2} 𝔨 is a 𝔤_{2}-module. On page 25, we will show that the non-trivial 𝔤_{2}-module with the lowest dimension is sevendimensional. Therefore, 𝔪 is either a trivial module or equivalent to ℝ^{7} with 𝔤_{2} 𝔤𝔩 7 acting by matrix multiplication. In the first case, G_{2} would be a normal subgroup of K. If this was the case, L_{i}G_{2}L_{i} would be contained in G_{2}. Let Φ be an arbitrary automorphism of 𝕆. If our assumption was true, Ψ : 𝕆 𝕆 with Ψ x : iΦ ix would have to fix 1. Since there are Φ with Φ i i, G_{2} is not a normal subgroup of K. Since K is compact, 𝔨 is the direct sum of an abelian and a semisimple Lie algebra. dim 𝔨 21 and the only possibilities for 𝔨 therefore are:

1. 𝔰𝔬 6 𝔥, where 𝔥 is a six-dimensional Lie algebra,

2. 𝔰𝔬 7 ,

3. 𝔰𝔭 3 .

In the first case, we had shown the existence of a six-dimensional faithful representation of 𝔤_{2}, since 𝔤_{2} 𝔰𝔬 6 𝔤𝔩 6 . Since such a representation does not exist, we can exclude that case. In the third case, we had 𝔤_{2} 𝔰𝔭 3 𝔤𝔩 6, ℂ . Therefore, that case can be excluded for the same reasons as the first one. Since K is simply connected, we have shown that K Spin 7 . The orbits of the action of K on 𝕆 are seven-spheres. Therefore, K acts irreducibly on 𝕆 and the action of K has to be given by the spinor representation. Remark 2.1.21. We can describe G_{2} as Spin 7 SO 7 , where the first group acts by the spinor representation on 𝕆 and the second one acts on Im 𝕆 .

Convention 2.1.22. We will sometimes denote the Lie algebra of Spin 7 by 𝔰𝔭𝔦𝔫 7 , when we want to emphasize the fact that it acts on 𝕆 ℝ^{8} by the spinor representation.

We will now construct certain forms with stabilizer G_{2} or Spin 7 . Later on, we will see that we can describe any G_{2}- or Spin 7 -structure with help of those forms. We identify the basis i, j, k, ϵ, iϵ, jϵ, kϵ of Im 𝕆 with the standard basis e_{1}, …, e_{7} of ℝ^{7}. The canonical three-form ω ^{3} Im 𝕆 ^{}can also be identified with a three-form on ℝ^{7}, which we denote by ω, too. We obtain:

ω dx^{123} dx^{145} dx^{167} dx^{246} dx^{257} dx^{347} dx^{356} .  (2.2)

The dx^{i} are as usual defined by dx^{i} e_{j} : δ_{j}^{i} and we further have defined dx^{ijk} : dx^{i} dx^{j} dx^{k}. We can describe the canonical four-form on 𝕆 in the same way and obtain:

ω^{}dx^{1247} dx^{1256} dx^{1346} dx^{1357} dx^{2345} dx^{2367} dx^{4567}  (2.3)

where dx^{ijkl} : dx^{i} dx^{j} dx^{k} dx^{l}. We equip ℝ^{7} with the standard scalar product and the volume form vol with vol e_{1}, e_{2}, …, e_{7} 1. It is easy to see that

ω^{}ω

where : ^{3} ℝ^{7} ^{}^{4} ℝ^{7} ^{}is the Hodge star operator. We denote the standard basis of ℝ^{8} by e_{0}, …, e_{7} . By mapping this basis to 1, i, j, k, ϵ, iϵ, jϵ, kϵ , we can identify 𝕆 and ℝ^{8}. We choose an orientation on 𝕆 such that 1, i, j, k, ϵ, iϵ, jϵ, kϵ is positive oriented. The four-form

Ω : ω dx^{0} ω

is self-dual. We can write down Ω explicitly:

\begin{aligned}Ω & dx^{0123} dx^{0145} dx^{0167} dx^{0246} dx^{0257} dx^{0347} dx^{0356} \\ & dx^{1247} dx^{1256} dx^{1346} dx^{1357} dx^{2345} dx^{2367} dx^{4567} . \end{aligned}  (2.4)

Convention 2.1.23. From now on, we will often identify ℝ^{7} with Im 𝕆 and ℝ^{8} with 𝕆 by the above identifications of the bases.

The forms ω, ω^{}, and Ω are clearly G_{2}-invariant. In the following, we will determine their stabilizer and annihilator exactly:

Definition 2.1.24. Let V be a vector space and α : V^{n} ℝ a multilinear map. The stabilizer Stab α of α is the set of all linear maps Φ : V V with Φ^{}α α, i.e.,

α Φ v_{1} , …, Φ v_{n} α v_{1}, …, v_{n} v_{1}, …, v_{n} V .

The stabilizer obviously is a closed group, hence:

Lemma 2.1.25. The stabilizer of a multilinear map α : V^{n} ℝ is a Lie subgroup of GL V .

Definition 2.1.26. Let V be a vector space and α : V^{n} ℝ a multilinear map. The annihilator Ann α of α is the set of all linear maps ϕ : V V with

α ϕ x_{1} , x_{2}, …, x_{n} α x_{1}, ϕ x_{2} , …, x_{n} … α x_{1}, x_{2}, …, ϕ x_{n} 0 .

Lemma 2.1.27. The annihilator of a multilinear map together with the bracket ϕ, ψ : ϕ ψ ψ ϕ is a Lie algebra. Moreover, it is the Lie algebra of the stabilizer group.

We are now able to formulate our theorem:

Theorem 2.1.28. Let ω  ^{3} Im 𝕆 ^{}, ω^{}^{4} Im 𝕆 ^{}, and Ω  ^{4} 𝕆^{}be the forms we

have constructed above. For these forms the following statements are true:

1. The stabilizer of ω is G_{2}.

2. The stabilizer of ω^{}is G_{2} id, id .

3. The stabilizer of Ω is Spin 7 .

4. The annihilator of ω and ω^{}is 𝔤_{2}.

5. The annihilator of Ω is 𝔰𝔭𝔦𝔫 7 acting on 𝕆 by the spinor representation.

Proof : The last two statements directly follow from the first three ones. A proof of the first statement can be found in a paper of Robert Bryant [14]. The stabilizer of Ω has been determined in [14], too. In the diploma thesis of the author [61], it has been shown that Stab ω^{}G_{2} id, id .

Remark 2.1.29. The orbit of ω ^{3} ℝ^{7} ^{}with respect to the canonical action of GL 7 is an open set. This can be easily shown by the following calculation:

dim GL ℝ^{7} dim Stab ω 49 14 35 dim ^{3} ℝ^{7} ^{}.

Moreover: On ℝ^{7} there is another three-form ω^{}, which is stabilized by the split real form of the complex Lie group G_{2}. The union of the orbits of ω and ω^{}is an open, dense subset of 3 ℝ^{7} ^{}(see Reichel [60] and Schouten [62]).

At the end of this section, we will briefly investigate the representations of 𝔤_{2} and 𝔰𝔭𝔦𝔫 7 . Furthermore, we will decompose some modules, which will become interesting for us, into irreducible submodules. We start with the representations of 𝔤_{2}. Since 𝔤_{2} is a simple Lie algebra, all representations of 𝔤_{2} are either faithful or trivial. The center of G_{2} contains only the neutral element. Therefore, the representations of G_{2} are either faithful or trivial, too. We denote the irreducible representation of 𝔤_{2} with highest weight a, b by 𝕍_{a,b}. The fundamental representations 𝕍_{1,0} and 𝕍_{0,1} of 𝔤_{2} have the following description:

1. 𝕍_{1,0} is 𝔤_{2}-equivariantly isomorphic to ℝ^{7}, where 𝔤_{2} 𝔤𝔩 7 acts on ℝ^{7} by matrix multiplication.

2. 𝕍_{0,1} is the adjoint representation of 𝔤_{2}.

The dimensions of the smallest representations of 𝔤_{2} are:

\begin{matrix}dim 𝕍_{a,b} & b=0 & 1 & 2 \\ a= 0 & 1 & 14 & 77 \\ 1 & 7 & 64 & 286 \\ 2 & 27 & 189 & 729 \\ 3 & 77 & 448 & 1547 \end{matrix}

Before we consider the representations of 𝔰𝔭𝔦𝔫 7 , we will decompose certain 𝔤_{2}-modules into submodules. The 𝔤_{2}-module 𝔤𝔩 7 ℝ^{7} ℝ^{7} ^{}decomposes into four irreducible 𝔤_{2}- submodules:

𝔤𝔩 7 𝕍_{0,0} 𝕍_{1,0} 𝕍_{0,1} 𝕍_{2,0} ,  (2.5)

where

• 𝕍_{0,0} consists of the identity map on ℝ^{7} and its multiples,

• 𝕍_{1,0} consists of the maps L_{x} R_{x}, where x Im 𝕆 ℝ^{7},

• 𝕍_{0,1} is 𝔤_{2} 𝔤𝔩 7 , and

• 𝕍_{2,0} are the trace-free, symmetric 7 7-matrices.

The decomposition of the spaces ^{k} ℝ^{7} ^{}is useful to know in the context of G_{2}-structures. There are the following splittings which also can be found in Bryant [14]:

\begin{matrix}^{0} ℝ^{7} ^{} & 𝕍_{0,0} \\ ^{1} ℝ^{7} ^{} & 𝕍_{1,0} \\ ^{2} ℝ^{7} ^{} & 𝔰𝔬 7 𝕍_{1,0} 𝕍_{0,1} \\ ^{3} ℝ^{7} ^{} & 𝕍_{0,0} 𝕍_{1,0} 𝕍_{2,0} \end{matrix}

The trivial summand of ^{3} ℝ^{7} ^{}simply is span ω . Since the Hodge star operator is 𝔰𝔬 7 - and therefore 𝔤_{2}-equivariant, the spaces ^{k} ℝ^{7} ^{}and ^{7 k} ℝ^{7} ^{}are 𝔤_{2}-equivariantly isomorphic. For this reason, the decomposition of ^{}ℝ^{7} ^{}is determined by the above splittings. In the following chapter, we need to know the normalizer of G_{2} in SO 7 , i.e., the group:

Norm_{SO 7}G_{2} : g SO 7 gG_{2}g^{1} G_{2} .

Let g SO 7 be arbitrary. We want to check if ghg^{1} G_{2} for all h G_{2}. This statement is equivalent to:

ω ghg^{1} x , ghg^{1} y , ghg^{1} z ω x, y, z x, y, z ℝ^{7} , h G_{2} .

We define u : g^{1} x , v : g^{1} y , and w : g^{1} z . The above equation now becomes:

h^{}g^{}ω g^{}ω .

Therefore, we have to check for which g SO 7 the pull-back g^{}ω is G_{2}-invariant. Since the maximal trivial 𝔤_{2}-submodule of ^{3} ℝ^{7} ^{}is one-dimensional, the only G_{2}-invariant threeforms are the multiples of ω. For this reason, there has to exist a λ ℝ with g^{}ω λω. We conclude that g is an element of the group ^{3} \overline{λ} G_{2}. Since g SO 7 , λ has to equal 1. We obviously have G_{2} Norm_{SO 7}G_{2} and therefore have proven the following lemma:

Lemma 2.1.30. The normalizer group Norm_{SO 7}G_{2} of G_{2} in SO 7 is G_{2}.

Before we proceed to the representations of 𝔰𝔭𝔦𝔫 7 , we decompose 𝔰𝔭𝔦𝔫 7 into irreducible 𝔤_{2}-submodules:

Lemma 2.1.31. Let 𝔰𝔭𝔦𝔫 7 be the Lie algebra of Spin 7 , which acts on 𝕆 by the spinor representation. In this situation, we have:

𝔰𝔭𝔦𝔫 7 𝔤_{2} L_{Im 𝕆} ,

where L_{Im 𝕆} is the vector space of all left multiplications by imaginary octonions. Moreover, L_{Im 𝕆} is a 𝔤_{2}-module, which is equivalent to ℝ^{7}.

Proof : In Corollary 2.1.19 we have shown that Spin 7 G_{2} S^{7}, where S^{7} can be identified with the unit sphere in 𝕆. More precisely, any x S^{7} corresponds to the coset L_{x}G_{2} Spin 7 . We consider Spin 7 as a G_{2}-bundle over S^{7} and split the tangent space T_{e}Spin 7 𝔰𝔭𝔦𝔫 7 into its horizontal and vertical part. The lemma easily follows from the observation that the tangent space of S^{7} at 1 𝕆 coincides with Im 𝕆 .

Finally, we briefly consider the representations of the Lie algebra 𝔰𝔭𝔦𝔫 7 , which is isomorphic to 𝔰𝔬 7 . Since 𝔰𝔬 7 is a simple Lie algebra, the representations of 𝔰𝔬 7 are either faithful or trivial. The center of Spin 7 is isomorphic to ℤ_{2}. Therefore, the non-trivial representations of Spin 7 either are faithful or have ℤ_{2} as kernel. The latter can be considered as representations of SO 7 . The rank of 𝔰𝔬 7 is 3 and its Dynkin diagram is B_{3}. We denote the irreducible representation of 𝔰𝔬 7 with highest weight a, b, c by 𝕍_{a,b,c}. The fundamental representations of 𝔰𝔬 7 can be described as follows:

1. 𝕍_{1,0,0} is the vector representation of 𝔰𝔬 7 .

2. 𝕍_{0,1,0} is the representation of 𝔰𝔬 7 on ^{2} ℝ^{7} ^{}or equivalently the adjoint representation.

3. 𝕍_{0,0,1} is the spinor representation of 𝔰𝔬 7 .

As in the 𝔤_{2}-case, it is possible to decompose the spaces ^{k} ℝ^{8} ^{}^{k} 𝕍_{0,0,1} into irreducible 𝔰𝔬 7 -modules. This work has been done in [14] and the following decompositions have been obtained:

\begin{matrix}^{0} ℝ^{8} ^{} & 𝕍_{0,0,0} \\ ^{1} ℝ^{8} ^{} & 𝕍_{0,0,1} \\ ^{2} ℝ^{8} ^{} & 𝕍_{0,1,0} 𝕍_{1,0,0} \\ ^{3} ℝ^{8} ^{} & 𝕍_{1,0,1} 𝕍_{0,0,1} \\ ^{4} ℝ^{8} ^{} & 𝕍_{2,0,0} 𝕍_{0,0,2} 𝕍_{1,0,0} 𝕍_{0,0,0} \end{matrix}

Since the Hodge star operator on ^{}ℝ^{8} ^{}is 𝔰𝔭𝔦𝔫 7 -equivariant, we are able to decompose all of ^{}ℝ^{8} ^{}. The irreducible submodules of that space can be described explicitly (see Bryant [14]). For example, we have:

1. ^{3} ℝ^{8} ^{}𝕍_{0,0,1} Ω α α ℝ^{8} ^{}

2. ^{3} ℝ^{8} ^{}𝕍_{1,0,1} β ^{3} ℝ^{8} ^{}Ω β 0

3. ^{4} ℝ^{8} ^{}𝕍_{0,0,0} c Ω c ℝ

4. ^{4} ℝ^{8} ^{}𝕍_{0,0,2} γ ^{4} ℝ^{8} ^{}γ γ

The dimensions of the irreducible submodules of ^{}ℝ^{8} ^{}, which are no fundamental representations of 𝔰𝔬 7 , are:

1. dim 𝕍_{1,0,1} 48

2. dim 𝕍_{2,0,0} 27

3. dim 𝕍_{0,0,2} 35

2.2. G_{2}-STRUCTURES 29

2.2 G_{2}-structures

In this section, we will introduce the most elementary facts on G_{2}-structures. Before we start, we first have to make some definitions:

Definition 2.2.1. 1. Let M be a n-dimensional smooth manifold. The coframe bundle of M is the set of all bijective linear maps u : T_{p}M ℝ^{n}, where p M is arbitrary. This set can be given the structure of a GL n -principal bundle over M.

2. Let G be a Lie subgroup of GL n . A G-structure on M is a reduction of the coframe bundle to a principal bundle with structure group G.

3. Whenever we write about G_{2}-structures, we consider G_{2} as a Lie subgroup of GL 7 . The underlying manifold M therefore has to be seven-dimensional.

4. A seven-dimensional manifold equipped with a fixed G_{2}-structure we will call a G_{2}- manifold.

In Theorem 2.1.28, we have seen that the stabilizer of the three-form ω ^{3} ℝ^{7} ^{}coincides with G_{2}. Therefore, we can describe any G_{2}-structure by a three-form:

Lemma 2.2.2. Let M be a seven-dimensional manifold and let ω be a three-form on M with the following property: For each p M, there exist an open neighborhood U of p and linearly independent vector fields X_{1}, …, X_{7} on U with

ω _{U} X^{123} X^{145} X^{167} X^{246} X^{257} X^{347} X^{356} .

In the above formula, X^{i} is defined by X^{i} X_{j} : δ_{j}^{i} and X^{ijk} by X^{i} X^{j} X^{k}. If this is the case

\begin{aligned}𝔊_{G_{2}} : u : T_{p}M ℝ^{7} & u linear, p M, { \atop ω_{p}} X, Y, Z ω u X , u Y , u Z \\ & X, Y, Z T_{p}M , \end{aligned}

where ω ^{3} ℝ^{7} ^{}denotes the three-form (2.2), is a G_{2}-structure on M. Conversely, let 𝔊_{G_{2}} be a G_{2}-structure on M. Then

{ \atop ω_{p}} X, Y, Z : ω u X , u Y , u Z , where 𝔊_{G_{2}} u : T_{p}M ℝ^{7}

determines a well-defined, smooth three-form on M with the above properties.

In the following, we will often denote G_{2}-manifolds as pairs of a manifold and a three-form. With help of the decompositions we have found in Section 2.1, we are able to construct G_{2}-invariant objects on a G_{2}-manifold:

Lemma 2.2.3. Let M be a G_{2}-manifold with a G_{2}-structure 𝔊_{G_{2}} and let ϕ : G_{2} GL V be a representation. We denote the maximal trivial submodule of V by W. The G_{2}-invariant sections of the vector bundle 𝔊_{G_{2}} _{G_{2}} V are exactly the sections of 𝔊_{G_{2}} _{G_{2}} W.

In particular, we find the following invariant objects:

Lemma 2.2.4. Let M, ω be a G_{2}-manifold. The following objects on M are G_{2}-invariant:

1. The three-form ω.

2. The metric g and the volume form vol which are defined by

g X, Y vol : \frac{1}{6} X ω Y ω ω ,

where X ω : ω X, , denotes the interior product of X and ω. Let u : T_{p}M ℝ^{7} be an element of the G_{2}-structure. We can obtain the value of g_{p} X, Y by inserting u^{1} e_{1} , …, u^{1} e_{7} in the right hand side of the above equation. Analogously, we can obtain vol_{p} Z_{1}, …, Z_{7} by setting X_{p} : u^{1} e_{1} and Y_{p} : u^{1} e_{2} .

3. The Hodge star operator associated to g and vol is obviously G_{2}-equivariant. Therefore, the four-form ω is G_{2}-invariant, too. For any p M, ω_{p} can be identified by an element of the G_{2}-structure with ω^{}^{4} ℝ^{7} ^{}.

4. The spinor representation of 𝔰𝔬 7 is eight-dimensional and can be identified with 𝕆. Since G_{2} Spin 7 , it is a representation of G_{2}, too. The associated vector bundle of this representation is the spinor bundle of M. Since 𝕆 splits into ℝ Im 𝕆 with respect to the action of G_{2}, the spinor bundle splits into a seven-dimensional and a one-dimensional G_{2}-invariant subbundle. Therefore, there exist G_{2}-invariant spinors on M.

From this lemma, we immediately obtain the following important corollary:

Corollary 2.2.5. Let M, ω be a G_{2}-manifold. Then M is orientable and admits a canonical spin structure.

Convention 2.2.6. Let M, ω be a G_{2}-manifold. We call the tensor field ω ( ω, g, vol) the canonical 3-form (4-form, metric, volume form) on M. The orientation of M induced by vol we will call the canonical orientation of M. Analogously, the spin-structure induced by the inclusion G_{2} Spin 7 will be called the canonical spin structure. Alternatively, we often call these objects the objects associated to the G_{2}-structure.

The converse of Corollary 2.2.5 is also true:

Lemma 2.2.7. (See Lawson, Michelsohn [53], pp. 348f.) Let M be a seven-dimensional orientable manifold which admits a spin structure. Then M admits a G_{2}-structure, too.

Proof : Let N be a manifold. It is known that if the dimension of the real spinor bundle is greater than dim N, there exists a nowhere vanishing spinor on N (see Isham, Pope, Warner [45]). Let η be a nowhere vanishing spinor on M, which has to exist, since 8 7. A G_{2}- structure on M can be defined by:

ω^{ijk} : η^{}γ^{ijk}η

where the γ^{ijk} are the gamma matrices.

We call a connection ∇ on the tangent bundle compatible with the G_{2}-structure if ∇ω 0. The G_{2}-invariant objects which we have defined above are parallel with respect to any compatible connection. Moreover:

Lemma 2.2.8. Let M be a seven-dimensional manifold with a G_{2}-structure 𝔊_{G_{2}} and let ∇ be a connection on T M which is compatible with the G_{2}-structure. Furthermore, let V be a G_{2}-module. We can extend ∇ to a connection on the vector bundle 𝔊_{G_{2}} _{G_{2}} V. The dimension of the maximal trivial submodule of V we denote by n. In this situation, the set of all G_{2}-invariant parallel sections of 𝔊_{G_{2}} _{G_{2}} V is a n-dimensional vector space.

In particular, the tensor fields ω, g, vol, ω are parallel and G_{2}-invariant. Moreover, they are up to constant multiples the only elements of ^{3} T^{}M, S^{2} T^{}M , ^{7} T^{}M, and ^{4} T^{}M with these properties. Finally, we have up to constant multiples exactly one G_{2}-invariant spinor on M which is parallel with respect to ∇.

Remark 2.2.9. 1. From the decomposition of ^{}ℝ^{7} ^{}we have found in Section 2.1, we can conclude that the constant functions, ω, ω, and vol are up to constant multiples the only G_{2}-invariant parallel forms on M.

2. If the holonomy of ∇ is all of G_{2}, the above sections are the only parallel sections of the considered bundles. If the holonomy is smaller, more parallel sections, which are not necessarily G_{2}-invariant, may exist. As an example we consider the canonical G_{2}- structure on the flat ℝ^{7}. The space of all parallel three-forms on ℝ^{7} has dimension 35. Since ^{3} ℝ^{7} ^{}contains exactly one trivial summand, the space of all G_{2}-invariant parallel three-forms on ℝ^{7} is only one-dimensional.

Since 𝔤_{2} 𝔰𝔬 7 , a torsion-free compatible connection on a given G_{2}-manifold does not necessarily exist. The obstruction for the existence of such a connection is the intrinsic torsion:

Definition 2.2.10. Let G be a Lie subgroup of O n and 𝔊_{G} a G-structure on a n-dimensional manifold M. We denote the Lie algebra of G by 𝔤.

1. The vector bundle over M associated to the adjoint representation of G, is called the adjoint vector bundle 𝔤M.

2. A connection ∇ on T M is called compatible with the G-structure 𝔊_{G} if ∇_{X} is a section of 𝔤M for all vector fields X.

3. We define the map

\begin{aligned}δ & : 𝔤M T^{}M T M ^{2}T^{}M \\ δ α X, Y & : α X Y α Y X \end{aligned}

and the G-invariant vector bundle

H^{0,2}M : T M ^{2}T^{}M δ 𝔤M T^{}M .

4. Let Θ be the torsion of an arbitrary compatible connection. The intrinsic torsion of the G-structure 𝔊_{G} is defined as Θ^{int} : Θ δ 𝔤M T^{}M and is a section of H^{0,2}M.

5. 𝔊_{G} is called parallel or torsion-free if Θ^{int} 0 on all of M.

Remark 2.2.11. 1. The adjoint vector bundle is a subbundle of the endomorphism bundle T M T^{}M. This fact explains the notation in the definition of δ.

2. Both notions of a compatible connection ∇ on a G_{2}-manifold coincide, i.e., ∇ω 0 is equivalent to ∇_{X} 𝔤_{2}M.

3. Let V M be the vector bundle associated to the orthogonal complement of 𝔤 𝔰𝔬 n with respect to a G-invariant inner product on 𝔰𝔬 7 . H^{0,2}M is G-equivariantly isomorphic to V M T^{}M.

4. Θ^{int} is independent of the choice of the compatible connection and therefore well-defined.

We return to the case, where G coincides with G_{2} O 7 . It is possible to show that the intrinsic torsion is determined by dω and d ω:

Lemma 2.2.12. Let M, ω be a G_{2}-manifold with intrinsic torsion Θ^{int}. There is an injective, linear, and G_{2}-equivariant map

φ : H^{0,2}M  ^{4}T^{}M  ^{5}T^{}M

such that φ Θ^{int} dω, d ω .

In our situation, H^{0,2}M can be identified with T M T^{}M. The fibers of H^{0,2}M therefore decompose into 𝕍_{0,0} 𝕍_{1,0} 𝕍_{0,1} 𝕍_{2,0}. Depending on the subbundle of which the intrinsic torsion is a section there are 16 types of G_{2}-structures. Because of Lemma 2.2.12, each of the 16 types can be described by an equation which dω and d ω have to satisfy. Those equations have been determined by Fernandez and Gray [34]. For our considerations, we are only interested in the following types of G_{2}-structures:

Definition 2.2.13. A G_{2}-manifold M, ω with intrinsic torsion Θ^{int} is called

1. symplectic or closed if dω 0. This condition is equivalent to Θ^{int} 𝕍_{0,1}.

2. cosymplectic or coclosed if d ω 0. This condition is equivalent to Θ^{int} 𝕍_{0,0} 𝕍_{2,0}.

3. nearly parallel if dω λ ω and d ω 0 for a λ : M ℝ. This condition is equivalent to Θ^{int} 𝕍_{0,0}.

4. parallel or torsion-free if the intrinsic torsion vanishes.

Remark 2.2.14. We can conclude from dω λ ω the equation d ω 0 only if λ does not vanish. Since there are G_{2}-manifolds with dω 0 but d ω 0, we have to include the second condition in the definition of nearly parallel.

We now collect some properties of nearly parallel G_{2}-manifolds:

Proposition 2.2.15. (See Friedrich, Kath, Moroianu, and Semmelmann [37].) Let M, ω be a nearly parallel G_{2}-manifold. The canonical metric g on M is an Einstein metric. Its Einstein constant is equal to \frac{3}{8}λ^{2}, where λ is taken from the definition of ”nearly parallel”. In particular, λ has to be constant. Moreover, there exists a G_{2}-invariant Killing spinor η on M, i.e.:

∇_{X}^{g}η μX η vector fields X ,

where the dot denotes the Clifford multiplication, ∇^{g} the Levi-Civita connection, and μ \frac{1}{8}λ is a constant.

There are the following equivalent conditions for the torsion-freeness of a G_{2}-manifold:

Theorem 2.2.16. (See Fernandez, Gray [34].) Let M, ω be a G_{2}-manifold. We denote the canonical metric on M by g and the Levi-Civita connection of g by ∇^{g}. In this situation, the following statements are equivalent:

1. The intrinsic torsion of the G_{2}-structure vanishes.

2. The holonomy of g is a subgroup of G_{2}.

3. ∇^{g}ω 0.

4. dω 0 and d ω 0.

In this context, the following fact should be mentioned:

Lemma 2.2.17. Let M, g be a seven-dimensional oriented Riemannian manifold whose holonomy is contained in G_{2}. Then, there exists a parallel G_{2}-structure on M whose associated metric is g and whose associated orientation is the same as of M. If the holonomy is all of G_{2}, the G_{2}-structure is moreover unique.

We finally state the following facts about parallel G_{2}-manifolds:

Proposition 2.2.18. (See Bonan [12].) The canonical metric on a parallel G_{2}-manifold is Ricci-flat.

Proposition 2.2.19. (See Joyce [47], p. 245.) The restricted holonomy group of the Levi-Civita connection on a parallel G_{2}-manifold is one of the following:

1. G_{2},

2. SU 3 , which acts irreducibly on a six-dimensional subspace of the tangent space and trivially on its orthogonal complement,

3. SU 2 , which acts irreducibly on a four-dimensional subspace and trivially on its orthogonal complement,

4. the trivial group 1 .

Proposition 2.2.20. (See Wang [66].) Let M, g be a seven-dimensional Riemannian manifold with holonomy G_{2}. The space of all spinors which are parallel with respect to the Levi-Civita connection is one-dimensional.

2.3 Spin(7)-structures

In this section, we will present some facts about Spin 7 -structures. Many of this facts will be similar to the corresponding results in Section 2.2. Analogously to that case, we call an eightdimensional manifold with a fixed Spin 7 -structure a Spin 7 -manifold. Since the stabilizer of the four-form (2.4) is Spin 7 , we can describe any Spin 7 -structure by a four-form:

Lemma 2.3.1. Let M be an eight-dimensional manifold and let Ω be a four-form on M with the following property: For each p M there exists an open neighborhood U of p and linearly independent vector fields X_{0}, …, X_{7} on U with

\begin{aligned}Ω _{U} & X^{0123} X^{0145} X^{0167} X^{0246} X^{0257} X^{0347} X^{0356} \\ & X^{1247} X^{1256} X^{1346} X^{1357} X^{2345} X^{2367} X^{4567} . \end{aligned}

In the above formula, X^{i} is defined by X^{i} X_{j} : δ_{j}^{i} and X^{ijkl} by X^{i} X^{j} X^{k} X^{l}. If this is the case,

𝔊_{Spin 7} : u : T_{p}M ℝ^{8} u linear, p M, { \atop Ω_{p}} W, X, Y, Z Ω u W , u X , u Y , u Z
W, X, Y, Z T_{p}M ,

where Ω ^{4} ℝ^{8} ^{}is the four-form (2.4), is a Spin 7 -structure on M. Conversely, let 𝔊_{Spin 7} be a Spin 7 -structure on M. Then

{ \atop Ω_{p}} W, X, Y, Z : Ω u W , u X , u Y , u Z , where 𝔊_{Spin(7)} u : T_{p}M ℝ^{8}

determines a well-defined, smooth four-form on M with the above properties.

Because of the above lemma, we will denote Spin 7 -manifolds as pairs of a manifold and a four-form. The Spin 7 -invariant objects on a Spin 7 -manifold can be obtained by a construction analogous to Lemma 2.2.3. The most important of them are given by the following lemma:

Lemma 2.3.2. Let M, Ω be a Spin 7 -manifold. The following objects on M are Spin 7 - invariant:

1. Ω Ω, which is a volume form on M. We call Ω Ω the canonical volume form vol.

2. There is a canonical metric g on M, Ω which is determined by the following relation (see Karigiannis [50]):

6 X Y ^{2} vol X Y Ω X Y Ω Ω .

In the above formula, . denotes the norm on the fibers of ^{2} T M which is induced by g.

Corollary 2.3.3. Any Spin 7 -manifold is orientable.

Remark 2.3.4. 1. As in Section 2.2, we often call the canonical volume form and metric the volume form and metric associated to the Spin 7 -structure.

2. It is easy to prove that Ω is self-dual with respect to the metric g and the orientation induced by Ω Ω.

3. With help of the decompositions we have found in Section 2.1, it could be easily proved that the constant function 1, Ω, and Ω Ω are a basis of the space of all Spin 7 -invariant forms on M.

4. We can lift the inclusion Spin 7 SO 8 to an inclusion of Spin 7 into Spin 8 . Therefore, any Spin 7 -manifold admits a spin-structure. Since the dimension of a Spin 7 -manifold is even, the spinor bundle splits into a positive and a negative part. The positive spinor bundle splits into a one-dimensional and an irreducible part and the negative spinor bundle is irreducible (see Joyce [47] p.256 for details). Therefore, we have up to a multiplication by a scalar function exactly one Spin 7 -invariant spinor on any Spin 7 -manifold (see Wang [66]).

The existence of a spin-structure on an eight-dimensional manifold is not sufficient for the existence of a Spin 7 -structure on that manifold. Moreover, the Euler characteristic of the positive or the negative part of the spinor bundle has to vanish (see Lawson, Michelsohn [53]).

A connection ∇ on the tangent bundle of a Spin 7 -manifold M, Ω is compatible with the Spin 7 -structure if and only if ∇Ω 0. Analogously to Section 2.2, we have the following lemma:

Lemma 2.3.5. Let M, Ω be a Spin 7 -manifold and let ∇ be a connection on T M which is compatible with the Spin 7 -structure. In this situation, there is up to a constant multiple exactly one positive, parallel, and Spin 7 -invariant spinor and no negative, non-zero, and Spin 7 -invariant spinor on M.

Our next theme is the intrinsic torsion of a Spin 7 -structure. It is possible to prove by means of representation theory that the fiber H_{p}^{0,2}M is isomorphic to 𝕍_{1,0,1} 𝕍_{0,0,1}. This is exactly the decomposition of ^{5} T_{p}^{}M into irreducible Spin 7 -modules. Therefore, it is not surprising that the intrinsic torsion is measured by dΩ and that we have an analogon to Lemma 2.2.12:

Lemma 2.3.6. Let M, Ω be a Spin 7 -manifold with intrinsic torsion Θ^{int}. There is a bijective, linear, and Spin 7 -equivariant map

φ : H^{0,2}M ^{5}T^{}M ,

such that φ Θ^{int} dΩ.

There are four types of Spin 7 -structures, which are distinguished by the invariant subbundle of H^{0,2}M of which Θ^{int} is a section. In order to give simple descriptions of each of this types, we define the Lee-form:

Definition 2.3.7. Let M, Ω be a Spin 7 -manifold. The one-form

θ : dΩ Ω

is called the Lee-form of M, Ω .

We are now able to describe the four types:

Definition 2.3.8. A Spin 7 -manifold M, Ω , whose intrinsic torsion and Lee-form we denote by Θ^{int} and θ, is called

1. parallel or torsion-free if the intrinsic torsion vanishes,

2. locally conformally parallel if Θ^{int} 𝕍_{0,0,1} or equivalently if dΩ θ Ω,

3. balanced if Θ^{int} 𝕍_{1,0,1} or equivalently if θ 0,

4. generic if none of the above is the case.

Remark 2.3.9. If Θ^{int} 𝕍_{0,0,1}, then there exists for each p M an open neighborhood U of p and a conformal change of the Spin 7 -structure on U such that the new Spin 7 -structure is parallel (see Cabrera [16]). This fact justifies the name ”locally conformal parallel”.

We are first of all interested in parallel Spin 7 -structures. The most important criteria for the torsion-freeness of a Spin 7 -structure are the following:

Theorem 2.3.10. (See Bryant [14].) Let M, Ω be a Spin 7 -manifold. We denote the canonical metric on M by g and its Levi-Civita connection by ∇^{g}. In this situation, the following statements are equivalent:

1. The intrinsic torsion vanishes.

2. The holonomy of g is a subgroup of Spin 7 .

3. ∇^{g}Ω 0.

4. dΩ 0.

Analogously to Section 2.2, the following two statements can be proven:

Lemma 2.3.11. Let M, g be an eight-dimensional oriented Riemannian manifold whose holonomy is contained in Spin 7 . Then, there exists a parallel Spin 7 -structure on M whose associated metric is g and whose associated orientation is the same as of M. If the holonomy is all of Spin 7 , the Spin 7 -structure is moreover unique.

Proposition 2.3.12. (See Bonan [12].) The canonical metric of a parallel Spin(7)-manifold is Ricci-flat.

If Ω is parallel, then the holonomy of the canonical metric is contained in Spin 7 , but not necessarily equal to Spin 7 . According to Joyce [47], there are the following possibilities for the holonomy group:

Proposition 2.3.13. Let M, Ω be a parallel Spin 7 -manifold. The possible restricted holonomy groups of the Levi-Civita connection and their inclusions are given by the diagram below:

[figure: 1; SU 2; SU 2; SU 3; G_{2}; SU 2 SU 2; Sp 2; SU 4; Spin 7]

The last three groups in the lower row of this diagram act irreducibly on the tangent space. If the restricted holonomy is SU 2 SU 2 , the first factor acts irreducibly on a fourdimensional subspace of the tangent space and the second one acts irreducibly on its orthogonal complement. The other groups act irreducibly on a certain subspace of the tangent space and trivially on its orthogonal complement, which in each case has a positive dimension. In particular, we have:

1. SU 2 acts irreducibly on a four-dimensional subspace.

2. SU 3 acts irreducibly on a six-dimensional subspace.

3. G_{2} acts irreducibly on a seven-dimensional subspace.

Remark 2.3.14. The inclusions SU 3 G_{2} and SU 4 Spin 7 from the above diagram are not given by the identity, but by conjugation by an element of SO 8 . As in the G_{2}-case, we have:

Proposition 2.3.15. (See Wang [66].) Let M, g be an eight-dimensional Riemannian manifold with holonomy Spin 7 . The space of all spinors which are parallel with respect to the Levi-Civita connection is one-dimensional.

There are several methods to determine the holonomy of a parallel Spin 7 -manifold M, Ω . We first show how it can be checked if the holonomy is a subset of G_{2}. If this is the case, the holonomy acts trivially on a non-trivial subspace of the tangent space. Therefore, there exists a parallel vector field on the manifold. It is usually easy to decide if such a vector field exists. If the metric associated to the Spin 7 -structure is complete, we can show by the de Rham decomposition theorem that M is covered by a product with a flat factor. If M even is compact, M has to be finitely covered by a Riemannian product of a flat torus and another manifold. The converse of this statement is also true:

Lemma 2.3.16. Let M, Ω be a compact parallel Spin 7 -manifold whose associated metric we denote by g. We assume that the restricted holonomy of g is one of SU 2 SU 2 , Sp 2 , SU 4 , or Spin 7 . Then π_{1} M is finite.

Proof : We assume that π_{1} M is infinite. Since g is Ricci-flat, it follows from the Cheeger-Gromoll splitting theorem that M has a finite cover, which is globally isometric to N T^{k}, where N is a simply connected, compact manifold and T^{k} is a flat k-dimensional torus. Since π_{1} M , we have k 1. In this situation, the restricted holonomy acts trivially on a at least one-dimensional subspace of the tangent space. Therefore, the restricted holonomy cannot be one of the above groups.

The following lemma will help us decide if the holonomy is a subgroup of SU 4 :

Lemma 2.3.17. Let M, Ω be a parallel Spin 7 -manifold. There exists a unique SO 8 - structure on M, i.e. a Riemannian metric and an orientation, which is the extension of the Spin 7 -structure. We assume that the holonomy of the associated metric is a subgroup of SU 4 . The holonomy bundle or its extension defines a parallel SU 4 -structure on M. Let S be the space of all parallel Spin 7 -structures on M, which are extensions of the SU 4 - structure and induce the same Riemannian metric and orientation. In this situation, any connected component of S is diffeomorphic to a circle.

Conversely, let M be an eight-dimensional manifold with a one-parameter family of parallel Spin 7 -structures which is diffeomorphic to a circle and whose members all induce the same metric and orientation. Then, there exists a parallel SU 4 -structure on M.

Proof : First, we consider the situation on ℝ^{8}. This restriction is justified, since we will see below that a change of the parallel Spin 7 -structure can be described by conjugation of Spin 7 by an h GL 8 .

Since we want the SO 8 -structure to be preserved, h has to leave SO 8 invariant. The normalizer Norm_{GL 8}SO 8 is SO 8 ℝ 0 , where the second factor is given by λ Id_{ℝ^{8}} λ ℝ 0 . Since conjugation by λ Id_{ℝ^{8}} leaves any matrix invariant, we will assume that h SO 8 . In the statement of the lemma, we assume that the SU 4 -structure is fixed. Therefore, SU 4 has to be preserved by h, too, and we have h Norm_{SO 8}SU 4 . It is known that Norm_{SO 8}SU 4 U 4 ℤ_{2}.

Next, we search for the group K of all h Norm_{SO 8}SU 4 which preserve Spin 7 , i.e., K Norm_{SO 8}SU 4 Norm_{SO 8}Spin 7 . Since SU 4 Spin 7 Norm_{SO 8}Spin 7 , we have SU 4 K U 4 ℤ_{2}. Thus, the identity component of K is either SU 4 or U 4 . In order to determine the identity component, we consider the group

 ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}e^{iφ} \\ & e^{iφ} \\ & & e^{iφ} \\ & & & e^{iφ} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; φ ℝ ;  ;  ;  ;  ;  ;

which we shortly denote by U 1 . U 4 obviously is finitely covered by the direct product of SU 4 and U 1 . We have to answer the question, if U 1 Spin 7 U 1 ^{1} Spin 7 . Let k U 1 be a matrix with kSpin 7 k^{1} Spin 7 . Since the stabilizer of the four-form Ω is exactly Spin 7 , we have to consider the following equation:

kgk^{1} ^{}Ω Ω g Spin 7 .

As usual, the asterisk denotes the pull-back. The above equation is equivalent to:

g^{}k^{}Ω k^{}Ω g Spin 7 .

Since Ω and its multiples are the only Spin 7 -invariant four-forms on ℝ^{8}, we have:

k^{}Ω λΩ for a λ ℝ .

Therefore, k has to be a multiple of an element of Spin 7 . Since det k 1, we even know that k Spin 7 ℤ_{2}, where ℤ_{2} is generated by Id_{ℝ^{8}}. By a straightforward calculation, we can show that 𝔲 1 𝔰𝔭𝔦𝔫 7 0 . Therefore, U 1 Spin 7 is discrete. We now have shown that U 1 is not contained in Norm_{SO 8}Spin(7) and that therefore the identity component of K is SU 4 . We can conclude that U 4 K U 1 ℤ_{k} U 1 for a k ℕ.

Let h Norm_{SO 8}SU 4 be arbitrary. We consider the given Spin 7 -structure as a principal bundle 𝔊_{Spin 7}. The following set defines a h Spin 7 h^{1} - or equivalently a new Spin 7 - structure on M:

h u : T_{p}M ℝ^{8} u 𝔊_{Spin(7)} .

This construction describes an action of Norm_{SO 8}SU 4 on the set of all Spin 7 -structures on M. The considerations we have made prove that its isotropy group is K. The action of Norm_{SO 8}SU 4 leaves the SU 4 -structure invariant, but its action on the Spin 7 -structure generates a set of new Spin 7 -structures. Since all of them are extensions of the parallel SU 4 -structure, they are parallel, too. The Spin 7 -structures can be extended to the same SO 8 -structure, since Norm_{SO 8}SU 4 SO 8 . We have shown above that Norm_{SO 8}SU 4 K is a homogeneous space whose connected components are all diffeomorphic to a circle. Therefore, S has the same property, too.

We finally have to prove the second part of the lemma. Since any parallel object is determined by its value at a single point, it suffices to prove our statement at a fixed p M. We denote the members of the one-parameter family of Spin 7 -structures by Ω_{θ} ^{4} T_{p}^{}M, where θ S^{1}. Let Stab Ω_{θ} be the stabilizer group of the four-form Ω_{θ} and let

G _{θ S^{1}} Stab Ω_{θ} .

It is easy to see that there exists a parallel G-structure on M. If G was Spin 7 , all Ω_{θ} would be stabilized by Spin 7 . Since our one-parameter family is diffeomorphic to a circle, the Ω_{θ} are not contained in a one-dimensional subspace of ^{4} T_{p}^{}M. Therefore, the space of all Spin 7 -invariant four-forms has to be at least two-dimensional. Since we have proven that this space is in fact one-dimensional, we have obtained a contradiction and the holonomy is Spin 7 .

If the holonomy was not contained in SU 4 , it would have to be G_{2}. The action of G_{2} splits the tangent space into a seven-dimensional and a one-dimensional irreducible submodule. There is up to a sign only one unit vector X which is orthogonal to the space on which G_{2} acts non-trivially. Let X^{}be the dual of X. The only parallel Spin 7 -structures which are an extension of the G_{2}-structure ω and induce the same metric as Ω are ω X^{}ω. Our assumption on the one-parameter family of parallel Spin 7 -structures is thus not satisfied and we can exclude this case.

Remark 2.3.18. 1. The above lemma describes the connected components of S. It makes no statement on the number of those components. Since we will not need that number, the above lemma is sufficient for our considerations.

2. In the situation of the lemma, there are two possibilities: Either, there exists a parallel vector field on M. In this case, the holonomy is contained in SU 3 . Or, a parallel vector field does not exist. If this is the case, the restricted holonomy is one of the groups SU 4 , Sp 2 , or SU 2 SU 2 .

For the metrics we consider in Chapter 5, we determine the set of all parallel Spin 7 -structures which have the same associated metric and orientation. If we do not find a one-parameter family in this set, the holonomy is not a subgroup of SU 4 . Conversely, we know that the holonomy is contained in SU 4 if there exists such a family.

If we want to discuss the holonomy group itself, we have to search for certain parallel forms on the manifold. Since we will consider spaces with a large isometry group, we have geometric and algebraic methods at hand to decide if such forms exist. Those methods will be explained in more detail when we need them.

Spaces with a large isometry group

3.1 Homogeneous spaces

Many equations in differential geometry, for example, the Einstein equation Ric λg and the equation dΩ 0 on a Spin 7 -manifold become simpler if we assume that the metric has a large isometry group. The most symmetric case is that the isometry group acts transitively on the manifold. In that case, we have a so called homogeneous space. It will turn out that there are no homogeneous spaces with exceptional holonomy. For that reason we will assume later on that the metric is of cohomogeneity one, i.e., most of the orbits have codimension one. Nevertheless, we will first give a short introduction to the issue of homogeneous spaces. The reason for this is that in the non-homogeneous case the orbits of an isometric action are still homogeneous. If we want to make calculations on manifolds with a large isometry group, it will therefore often be useful to first consider the geometry on the orbit spaces. We start with some basic definitions:

Definition 3.1.1. Let M, g be a Riemannian manifold. The set Isom M, g of all isometries of M, g is called the symmetry or isometry group of M, g .

The isometry group is clearly a group. Moreover, it is a Lie group:

Proposition 3.1.2. Let M, g be a Riemannian manifold. Isom M, g has a canonical differentiable structure such that Isom M, g becomes a Lie group and the action Isom M, g M M becomes differentiable.

We are now able to define the concept of a homogeneous space:

Definition 3.1.3. Let M, g be a Riemannian manifold and let G Isom M, g be a closed subgroup of the isometry group. If G acts transitively on the manifold M, we call M, g a (G-)homogeneous space and g a (G-)homogeneous metric.

Remark 3.1.4. 1. The idea behind the concept of a homogeneous space is that there is no way to distinguish one point of the homogeneous space from another.

2. A Riemannian manifold can be G-homogeneous with respect to different groups G. For example, let S^{2n 1} ℝ^{2n} ℂ^{n} be the 2n 1 -dimensional sphere with the round metric. O 2n and U n act both transitively on S^{2n 1} , although U n O 2n . As long as G is known from the context, it will not be mentioned.

Homogeneous spaces have the following useful property:

Proposition 3.1.5. Any homogeneous space is complete.

It is possible to describe homogeneous spaces as coset spaces:

Proposition 3.1.6. Let M, g be a G-homogeneous space, where G Isom M, g . We fix an arbitrary p M and denote the isotropy group of the G-action at p by H_{p}. The map

\begin{aligned}φ & : M G H_{p} \\ φ q & : gH_{p} , \end{aligned}

where g is an isometry in G mapping p to q, is a well-defined G-equivariant diffeomorphism. Furthermore, there exists a unique G-invariant metric on G H_{p} such that φ becomes an isometry.

Conversely, if we equip any coset space G H with a G-invariant metric g, (G H,g) is a homogeneous space.

Because of the above proposition, we can restrict ourselves to coset spaces G H while studying homogeneous spaces. It is useful to make some restrictions on the pair G, H :

Convention 3.1.7. A homogeneous space G H can also be written as G K H K . H K shall be embedded in such a way into G K that the second factor of H K coincides with the second factor of G K. Since we want to keep the list of possible groups acting on G H short, we will assume from now on that G acts almost effectively on G H, i.e., the elements of G which act trivially are a discrete set. We also require that H is a closed subgroup of G, which is equivalent to G H being a Hausdorff space.

Remark 3.1.8. 1. For the rest of this thesis, we define G H as the space of all left cosets gH. When we write ”group action” (or ”invariant object”), we therefore mean ”left group action” (or ”left-invariant object”). If we consider a right action, it will explicitly mentioned.

2. Let g be an element of the normalizer Norm_{G}H. Since we have kH g kg H, the conjugation map kH gkg^{1} H is well-defined and fixes eH. Those kind of maps will be important later on.

3. Let H_{p} be the isotropy group of an isometric action at p and let g be an isometry which maps p to q. The isotropy group at q is gH_{p}g^{1} . It therefore makes sense to speak of the isotropy group of a homogeneous space without specifying a point.

4. Let H be a closed subgroup of G and let H^{}gHg^{1} be a conjugate of H. Then there exists a G-equivariant diffeomorphism ϕ : G H G H^{}which is described by ϕ kH gkg^{1} H^{}. We can therefore identify embeddings of H into G which are equal up to a conjugation by an element of G.

5. Nonconjugate embeddings of H into G will often give topologically different spaces G H. There are, for example, infinitely many non-homotopic spaces of type SU 3 U 1 . (See Section 5.4.) Moreover, if we embed H K in a non-canonical way into G K we do not necessarily have G K H K G H. (See Lemma 3.1.12.) We will therefore specify the embedding of H into G if it is not clear from the context.

The following lemma gives another useful restriction for the shape of G and H:

Lemma 3.1.9. Let G be a (not necessarily connected) Lie group and H be a (not necessarily connected) closed subgroup of G. Furthermore, let g be a G-invariant metric on G H. Then, there exists a simply connected Lie group G^{}, a closed connected subgroup H^{}of G^{}, and a G^{}-invariant metric g^{}on G^{}H^{}such that G^{}H^{}is the universal cover of each connected component of G H and the covering map is a local isometry. In particular, G^{}H^{}is connected and simply connected.

Proof : Let G_{e} be the identity component of G, and H_{e} the identity component of H. We choose G^{}as the universal cover of G_{e}. Let π : G^{}G_{e} be the covering map. H^{}can be chosen as the identity component of π^{1} H_{e} and g^{}as the lift of the metric on G_{e} H G_{e} .

Corollary 3.1.10. Let G H be an arbitrary coset space, K GL dim G dim H a closed subgroup, and 𝔎 a G-invariant K-structure on G H. Furthermore, let G^{}H^{}be the simply connected coset space from the lemma above. Then G^{}H^{}carries a G^{}-invariant K-structure which is the lift of 𝔎

Remark 3.1.11. Because of the above lemma and its corollary we will restrict ourselves from now on to the case where G is simply connected and H is connected.

The next result will be needed for the detailed description of the orbifold singularities which we have to consider in Section 5.3 and 5.4.

Lemma 3.1.12. Let G be a Lie group and H be a closed subgroup of G. We consider the space G U 1 H U 1 ^{}where U 1 ^{}G U 1 is not a subgroup of G and does not necessarily coincide with the abelian factor of G U 1 . In this situation, G H G-equivariantly covers G U 1 H U 1 ^{}and the preimage of e H U 1 ^{}in G H is given by the discrete group G U 1 ^{}.

Proof : U 1 ^{}is a group of the following type:

U 1 ^{}ψ e^{imφ} , e^{inφ} G U 1 φ ℝ ,

where m, n ℤ, n 0 and ψ : U 1 G is an injective group homomorphism. Since the group by which we divide G U 1 is isomorphic to H U 1 , we have hψ e^{imφ} ψ e^{imφ} h. We define the following map:

\begin{matrix}Φ : G H & G U 1 H U 1 ^{} \\ gH & g, e hψ e^{imφ} , e^{inφ} h H, φ ℝ . \end{matrix}

This map is a G-equivariant covering map of G U 1 H U 1 ^{}:

1. \underline{Φ is well-defined:} Let k H be arbitrary. We have:

Φ gkH gk, e hψ e^{imφ} , e^{inφ} h H, φ ℝ
g, e khψ e^{imφ} , e^{inφ} h H, φ ℝ
g, e hψ e^{imφ} , e^{inφ} h H, φ ℝ
Φ gH

2. \underline{Φ is G-equivariant:} We choose an arbitrary g^{}G and obtain:

Φ g^{}gH g^{}g, e hψ e^{imφ} , e^{inφ} h H, φ ℝ
g^{}, e g, e hψ e^{imφ} , e^{inφ} h H, φ ℝ
g^{}, e Φ gH

3. \underline{Φ is surjective:} Let g, e^{iϕ} G U 1 be arbitrary. We have:

Φ gψ e^{i \frac{m}{n} ϕ} H gψ e^{i \frac{m}{n} ϕ} , e hψ e^{imφ} , e^{inφ} h H, φ ℝ
g, e ψ e^{i \frac{m}{n} ϕ} hψ e^{imφ} , e^{inφ} h H, φ ℝ
g, e hψ e^{im φ \frac{ϕ}{n}} , e^{inφ} h H, φ ℝ
g, e hψ e^{imφ} , e^{in φ \frac{ϕ}{n}} h H, φ ℝ
g, e^{iϕ} hψ e^{imφ} , e^{inφ} h H, φ ℝ

4. \underline{The preimage of e H U 1 ^{}is discrete:} An element gH G H is mapped by Φ to e H U 1 ^{}G U 1 H U 1 ^{}if and only if

g, e hψ e^{imφ} , e^{inφ} h H, φ ℝ .

Since n 0, the preimage of H U 1 ^{}is ΓH, where

Γ G U 1 ^{}ψ e^{2πik \frac{m}{n}} k ℤ ℤ_{\frac{n}{gcd n,m}} .

This group is clearly discrete.

Remark 3.1.13. In the situation of the above lemma, we can rewrite G U 1 H U 1 ^{} as G H Γ or G H Γ respectively. The action of Γ we divide out is defined by the rightmultiplication of cosets gH with elements of Γ. This action is well-defined, since the groups H and Γ commute. Since Γ is abelian, it is possible to consider this action as a left action.

There is a theorem on the topology of coset spaces, which will not explicitly needed in this thesis, but nevertheless is useful to know:

Theorem 3.1.14. Let G be a compact Lie group and let G H be a Riemannian homogeneous space with an infinite fundamental group. Then G H is covered by G^{}H T where T is a flat torus and G^{}H is a homogeneous space with a finite fundamental group.

Proof : We assume without loss of generality that G and H are both connected. The Lie algebra of G we denote by 𝔤 and the Lie algebra of H by 𝔥. If there is a x 𝔤 which is not contained in 𝔥 and satisfies x, y 0 for all y 𝔤, the Lie algebra 𝔤 is a direct sum of type 𝔤^{}𝔲 1 . Since G is compact, we have G H G^{}H S^{1}. By repeating this construction as often as possible, we obtain G H K H T, where T is a flat torus. Since it is not possible to repeat our construction again, the Lie algebra 𝔨 of K contains no x with x Z 𝔨 and x 𝔥. It remains to show that in this situation the fundamental group of K H is finite.

We equip K with the negative of the Killing form as a metric. As we will state below, there exists a unique K-invariant metric q on K H such that the projection π : K K H becomes a Riemannian submersion. The fact that a x 𝔨 with the properties described above does not exist guarantees that q is positive definite. It is possible to show by a short calculation that the Ricci-curvature of q is \frac{1}{4}q. It follows from the Bonnet-Myers theorem that the fundamental group of K H is finite.

Our next aim is to introduce certain techniques which we will need in order to carry out explicit calculations on homogeneous spaces. Let G H be a homogeneous space and let X T_{e}G be arbitrary. For any t ℝ and any G-invariant metric on G H the map

gH exp tX g H

is an isometry of G H. Therefore, the vector field

gH \frac{}{t} _{t 0} exp tX g H

is a Killing vector field on G H, which we will shortly denote by X^{}. The following simple lemmas will be useful later on:

Lemma 3.1.15. Let G H be a coset space and X, Y 𝔤 : T_{e}G be arbitrary. Then the following identities hold:

1. X Y ^{}X^{}Y^{},

2. λX ^{}λX^{}for all λ ℝ.

In the following, the right-invariant vector fields on Lie groups will become important. We therefore introduce the following notation:

Convention 3.1.16. Let G be a Lie group and X 𝔤. We denote the right-invariant vector field on G which coincides with X at e by X^{}.

The projection of X^{}simply is the Killing vector field X^{}:

Lemma 3.1.17. Let G H be a coset space and π : G G H be the projection map. For any X 𝔤 we have:

dπ X_{g}^{}X_{gH}^{}.

We want to know how the Killing vector field X^{}behaves under the action of G on G H. The following lemma answers this question:

Lemma 3.1.18. Let G be a Lie group and H be a closed subgroup of G. Furthermore, let g, h G and X 𝔤 be arbitrary. The Killing vector field X^{}on G H obeys the following equation:

dL_{h} X_{gH}^{}Ad_{h} X _{hgH}^{},

where L_{h} : G H G H denotes the left-multiplication by h and Ad is the adjoint action of G on 𝔤.

The fact that X^{}, Y^{}_{e} X, Y for all X, Y 𝔤 has the following consequence:

Lemma 3.1.19. For any X, Y 𝔤, the Lie bracket of the Killing vector fields X^{}and Y^{}on G H is given by the following formula:

X^{}, Y^{}X, Y ^{}.

The group H acts from the left on the tangent space T_{eH}G H by h, X dL_{h} X . Therefore, T_{eH}G H is not only a vector space but also a H-module. The H-module structure on T_{eH}G H can be described by the following proposition, which follows directly from the above lemmas:

Proposition 3.1.20. Let G be a Lie group and H be a closed subgroup of G. We define the following map:

\begin{matrix}φ : 𝔤 & T_{eH}G H \\ X & X_{eH}^{}. \end{matrix}

In this situation, the following statements are true:

1. φ is linear and surjective.

2. ker φ T_{e}H : 𝔥.

3. T_{eH}G H is H-equivariantly isomorphic to the quotient 𝔤 𝔥, where the action of H on 𝔤 𝔥 is induced by the restriction of the adjoint action Ad : G GL 𝔤 to H.

Remark 3.1.21. Let g G be arbitrary. The isotropy group of the G-action at gH G H is gHg^{1} . Therefore, T_{gH}G H together with the action h, X dL_{ghg^{1}} X is a H-module, too. If we replace φ by the more general map

\begin{matrix}φ_{g} : 𝔤 & T_{gH}G H \\ X & Ad_{g} X _{gH}^{}, \end{matrix}

we obtain the same result on the H-module structure of T_{gH}G H as on the structure of T_{eH}G H.

Convention 3.1.22. From now on, we will assume that G is a compact Lie group. In this case, there exists a biinvariant metric on G. We will fix one of those metrics and denote it by q. In the following, we will often have to consider the q-orthogonal complement of 𝔥 in 𝔤, which we will denote by 𝔪.

Remark 3.1.23. Let Ad be the adjoint action of G. From the biinvariance of q, it follows that Ad_{h} 𝔪 𝔪 for all h H. 𝔪 therefore is a H-module. It is easy to see that 𝔪 is H-equivariantly isomorphic to 𝔤 𝔥 and therefore to any tangent space T_{gH}G H.

When we do explicit calculations on a homogeneous space, we need to fix a basis of the tangent space. The choice of that basis should in a certain way respect the splitting of the tangent space into H-submodules:

Definition 3.1.24. Let G H be a n-dimensional homogeneous space. As above, we identify the tangent space at a point p G H with the H-module 𝔪. Let

𝔪 𝔪_{1} … 𝔪_{k}

be a splitting of 𝔪 into (not necessarily irreducible) H-submodules. We denote the dimensions of 𝔪_{1}, …, 𝔪_{k} by m_{1}, …, m_{k}. A basis e_{i} _{i 1...n} of 𝔪 is called adapted to the splitting 𝔪_{1} … 𝔪_{k} if

𝔪_{1} span e_{1}, …, e_{m_{1}}
𝔪_{2} span e_{m_{1} 1} , …, e_{m_{1} m_{2}}
…
𝔪_{k} span e_{m_{1} ... m_{k 1} 1} , …, e_{n}

Remark 3.1.25. 1. Later on, when we consider concrete homogeneous spaces, we will mostly choose a basis of 𝔪 which is adapted to a fixed splitting. Furthermore, the splitting 𝔪_{1} … 𝔪_{k} and the basis e_{i} _{i 1...n} will usually be q-orthogonal and all vectors of a family e_{m_{1} ... m_{l} 1} , …, e_{m_{1} ... m_{l 1}} will be of the same length with respect to q. It turns out that our calculations will become more simple if we describe our objects with respect to such a basis.

2. Since it often will be necessary to describe the action of 𝔥 on 𝔪 explicitly, we will often extend our basis of 𝔪 by a q-orthogonal basis of 𝔥 to a basis of all of 𝔤.

Our next aim is to describe the set of G-invariant metrics on a homogeneous space G H. In order to describe an arbitrary G-invariant metric, we first need a background metric on G H:

Proposition 3.1.26. Let G be a compact Lie group equipped with a biinvariant metric q and let H be a closed subgroup of G. There is a unique G-invariant metric on G H such that the projection map π : G G H becomes a Riemannian submersion.

Convention 3.1.27. The metric on G H which is induced by q we will denote by q, too. In the literature, metrics of this kind are often called normal metrics.

Let g be an arbitrary G-invariant metric on G H. Since G H is a homogeneous space, g is determined by its values on the tangent space T_{eH}G H. Because of Remark 3.1.23, g can be considered as an element of S^{2} 𝔪 . In order to have a well-defined G-invariant extension to all of G H, g S^{2} 𝔪 has to be H-invariant. These considerations motivate the following lemma:

Lemma 3.1.28. Let g be a G-invariant metric on a homogeneous space G H. As above, we identify g with an element of S^{2} 𝔪 . The endomorphism φ : 𝔪 𝔪 defined by:

q φ X , Y : g X, Y X, Y 𝔪

is H-equivariant, q-symmetric and positive definite with respect to q. Conversely, any linear φ : 𝔪 𝔪 with these three properties defines by the above formula a G-invariant metric on G H.

The possible linear maps φ : 𝔪 𝔪 describing a G-invariant metric on G H can be classified with help of Schur’s lemma. Therefore, the above result is useful for practical purposes. It will often be convenient to restrict ourselves to diagonal metrics, since this will simplify many calculations:

Definition 3.1.29. Let G H be a n-dimensional homogeneous space and let e_{i} _{i 1,...,n} be a q-orthogonal basis of 𝔪. A G-invariant metric g on G H is called diagonal with respect to e_{i} _{i 1,...,n} if the matrix representation of g S^{2} 𝔪 with respect to e_{i} _{i 1,...,n} is a diagonal matrix.

Since e_{i} _{i 1,...,n} is q-orthogonal, g is diagonal if and only if φ is represented by a diagonal matrix.

Remark 3.1.30. Let G H be a n-dimensional homogeneous space with the property that 𝔪 splits into pairwise inequivalent irreducible H-submodules. Furthermore, let e_{i} _{i 1,...,n} be a q-orthogonal basis of 𝔪 which is adapted to the splitting of 𝔪. In this situation, it follows from Schur’s lemma that any G-invariant metric on G H is diagonal with respect to e_{i} _{i 1,...,n} . Therefore, the restriction to diagonal metrics is in many cases not a real one. In the other cases, we will carefully discuss what we will loose by this restriction.

Next, we will present formulas for the sectional and the Ricci-curvature of a homogeneous space. These formulas will be needed later on when we express the Einstein condition Ric λg for a cohomogeneity-one metric as an explicit system of ordinary differential equations. The following results, except the formula from Corollary 3.1.37, can also be found in Besse [10], pp. 183ff. Before we are able to determine the curvature tensor of a homogeneous space, we first have to consider the Levi-Civita connection:

Proposition 3.1.31. Let g be a G-invariant metric on a homogeneous space G H, whose Levi-Civita connection we denote by ∇ and let X, Y, and Z be in 𝔤. Then, we have:

2g ∇_{X^{}}Y^{}, Z^{}g X^{}, Y^{}, Z^{}g X^{}, Z^{}, Y^{}g Y^{}, Z^{}, X^{}.

Proof : Since X^{}is a Killing vector field, the Lie derivative L_{X^{}}g vanishes. Therefore, we have the following equation, which is the key to the proof:

X^{}g Y^{}, Z^{}L_{X^{}}g Y^{}, Z^{}
g L_{X^{}}Y^{}, Z^{}g Y^{}, L_{X^{}}Z^{}
g X^{}, Y^{}, Z^{}g X^{}, Z^{}, Y^{}.

The statement of the proposition can now be obtained from the equations for the torsionfreeness of ∇ and its compatibility with the metric.

Our next step is to express ∇_{X^{}}Y^{}_{eH} in terms of the Lie bracket of 𝔤. This can be done by the following corollary:

Corollary 3.1.32. Let G H be a homogeneous space equipped with a G-invariant metric g. Furthermore, let X, Y 𝔪 be arbitrary. Then, the Levi-Civita connection ∇ of g at the point eH is given by:

∇_{X^{}}Y^{}_{eH} \frac{1}{2} X, Y _{𝔪} U X, Y ,

where U : 𝔪 𝔪 𝔪 is defined by:

2g_{eH} U X, Y , Z : g_{eH} Z, X _{𝔪}, Y g_{eH} Z, Y _{𝔪}, X

and ., . _{𝔪} denotes the orthogonal projection of the Lie bracket of 𝔤 onto 𝔪.

Remark 3.1.33. ∇_{X^{}}Y^{}is not necessarily a Killing vector field. Therefore, we have to be careful if we want to consider ∇_{X^{}}Y^{}at another point as eH or to determine the second derivatives of a vector field.

We are now able to prove a formula for the sectional curvature: Let X, Y T_{eH}G H be an orthonormal pair of tangent vectors. In this case, Sec_{eH} X, Y g_{eH} R_{eH} X, Y Y, X . We extend X and Y to Killing vector fields X^{}and Y^{}on G H and obtain after a straightforward calculation:

Proposition 3.1.34. Let G H be a homogeneous space and g be a G-invariant metric on G H. Furthermore, let X, Y be a g-orthonormal pair in 𝔪 𝔪. Then, the sectional curvature of g satisfies:

sec_{p} X, Y g_{p} X^{}, Y^{}, X^{}, Y^{}g_{p} ∇_{Y^{}}Y^{}, ∇_{X^{}}X^{}∇_{X^{}}Y^{}_{p}^{2} X^{}, Y^{}_{p}^{2} ,

where p G H can be chosen arbitrarily.

We want to have a formula for sec X, Y in terms of the Lie bracket of 𝔤. This can be done by applying Corollary 3.1.32. Since that corollary is only valid at eH G H, the formula we obtain is only valid at eH, too:

Corollary 3.1.35. In the situation of the proposition above, the sectional curvature of the metric g at the point eH is given by:

\begin{aligned}sec_{eH} X, Y & \frac{1}{2}g_{eH} X, Y _{𝔤}, X _{𝔪}, Y \frac{1}{2}g_{eH} X, Y _{𝔤}, Y _{𝔪}, X g_{eH} U X, X , U Y, Y \\ & U X, Y _{eH}^{2} \frac{3}{4} X, Y _{𝔪} _{eH}^{2} \end{aligned}

Since the Ricci curvature is the average of certain sectional curvatures, we can use the above expression to derive a formula for the Ricci-curvature of a homogeneous space:

Proposition 3.1.36. Let G H be a n-dimensional homogeneous space. Furthermore, let g be a G-invariant metric on G H. Then we have for any tangent vector X T_{eH}G H 𝔪 and any g-orthonormal basis X_{i} _{1 i n} of T_{eH}G H:

\begin{aligned}Ric_{eH} X, X & \frac{1}{2} _{i 1}^{n} g_{eH} X, X_{i} _{𝔪}, X, X_{i} _{𝔪} \frac{1}{2}κ X, X \frac{1}{4} _{i,j 1}^{n} g_{eH} X_{i}, X_{j} _{𝔪}, X ^{2} \\ & g_{eH} _{i 1}^{n} U X_{i}, X_{i} , X _{𝔪}, X , \end{aligned}

where κ denotes the Killing form of the Lie algebra 𝔤 of G.

We now prove a formula for the Ricci-curvature which is suitable for explicit calculations. Before we start, we have to fix some notations. As usual, we consider a G-invariant metric g on a n-dimensional homogeneous space G H and fix a g-orthonormal basis X_{i} _{1 i n} of the tangent space 𝔪. For the following considerations, we have to extend g to an H-invariant inner product on 𝔤 and X_{i} _{1 i n} to a g-orthonormal basis X_{i} _{1 i dim 𝔤} of 𝔤. Since g is H-invariant, the spaces 𝔥 and 𝔪 are orthogonal. The structure constants of the Lie algebra 𝔤 we define by:

X_{i}, X_{j} _{k 1}^{dim 𝔤}c_{ij}^{k}X_{k} .

With these notations U _{i,j,k 1}^{n} U_{ij}^{k}X^{i} X^{j} X_{k} can be written as:

U_{ij}^{k} \frac{1}{2} c_{ki}^{j} c_{kj}^{i} .

In particular, we have U_{ii}^{j} c_{ji}^{i}. Now we are able to compute the Ricci tensor:

Ric_{eH} X_{i}, X_{i}  \frac{1}{2} _{j 1}^{n} g X_{i}, X_{j} _{𝔪}, X_{i}, X_{j} _{𝔪}  \frac{1}{2}κ X_{i}, X_{i}  \frac{1}{4} _{j,k 1}^{n} g X_{j}, X_{k} _{𝔪}, X_{i} ^{2}

g _{j 1}^{n} U X_{j}, X_{j} , X_{i} _{𝔪}, X_{i}

\frac{1}{2} _{j,k 1}^{n} g c_{ij}^{k}X_{k}, c_{ij}^{k}X_{k}  \frac{1}{2} _{j 1}^{dim 𝔤}g X_{i}, X_{i}, X_{j} , X_{j}  \frac{1}{4} _{j,k,l 1}^{n} g c_{jk}^{l}X_{l}, X_{i} ^{2}

g _{j,k 1}^{n} c_{kj}^{j}X_{k}, X_{i} _{𝔪}, X_{i}

\frac{1}{2} _{j,k 1}^{n} c_{ij}^{k} ^{2} \frac{1}{2} _{j,k 1}^{dim 𝔤} g X_{i}, c_{ij}^{k}X_{k} , X_{j} \frac{1}{4} _{j,k 1}^{n} c_{jk}^{i} ^{2}

g _{j,k,l 1}^{n} c_{kj}^{j}c_{ki}^{l}X_{l}, X_{i}

\frac{1}{2} _{j,k 1}^{n} c_{ij}^{k} ^{2} \frac{1}{2} _{j,k,l 1}^{dim 𝔤} g c_{ij}^{k}c_{ik}^{l}X_{l}, X_{j} \frac{1}{4} _{j,k 1}^{n} c_{jk}^{i} ^{2}

_{j,k 1}^{n} c_{kj}^{j}c_{ki}^{i}

_{j,k 1}^{n}  \frac{1}{2} c_{ij}^{k} ^{2} \frac{1}{4} c_{jk}^{i} ^{2} c_{kj}^{j}c_{ki}^{i}  \frac{1}{2} _{j,k 1}^{dim 𝔤} c_{ij}^{k}c_{ik}^{j}

(3.1)

As a by-product of the above calculation, we have shown the following formula for the Killing form:

κ X_{i}, X_{i}  _{j,k 1}^{dim 𝔤} c_{ij}^{k}c_{ik}^{j} .  (3.2)

If we assume that the basis X_{i} _{1 i dim 𝔤} is not only orthogonal with respect to g but also with respect to κ, we can simplify the above equation. In this situation, we have:

0 κ X_{i}, X_{j}, X_{j} κ X_{i}, X_{j} , X_{j} _{k 1}^{dim 𝔤}c_{ij}^{k}κ X_{k}, X_{j} c_{ij}^{j}κ X_{j}, X_{j}

and therefore c_{ij}^{j} 0. By polarizing the equation (3.1) we finally have shown the following corollary:

Corollary 3.1.37. Let G H be a n-dimensional homogeneous space equipped with a Ginvariant metric g. Furthermore, let X_{i} _{1 i n} be a g-orthonormal basis of 𝔪 which is orthogonal with respect to the Killing form κ of 𝔤. We extend g to an H-invariant inner product on 𝔤 and X_{i} _{1 i n} to a g-orthonormal basis X_{i} _{1 i dim 𝔤} , such that the extended basis is still κ-orthogonal. In this situation, the Ricci-curvature of g is determined by:

Ric_{eH} X_{i}, X_{j}  _{k,l 1}^{n}  \frac{1}{2}c_{ik}^{l}c_{jk}^{l} \frac{1}{4}c_{kl}^{i}c_{kl}^{j}  \frac{1}{4} _{k,l 1}^{m} c_{ik}^{l}c_{jl}^{k} c_{jk}^{l}c_{il}^{k} ,

where the coefficients c_{ij}^{k} are defined by X_{i}, X_{j} _{k 1}^{dim 𝔤}c_{ij}^{k}X_{k} and i, j can chosen arbitrarily

from 1, …, n .

Remark 3.1.38. 1. Since the Ricci-curvature is invariant under isometries, the Ricci-tensor of a G-invariant metric on a homogeneous space G H is G-invariant, too. Unfortunately, the Ricci-tensor of a diagonal homogeneous metric is not necessarily diagonal.

2. In practice, the condition that X_{i} _{1 i dim 𝔤} is κ-orthogonal will often be satisfied. The bases of 𝔤 we choose in Chapter 5 are orthogonal with respect to the biinvariant metric q we have fixed. Since the restriction of the biinvariant metric to a simple factor of a semisimple Lie group is a multiple of the Killing form, our bases therefore are orthogonal with respect to the Killing form, too.

In Chapter 5, we will often have to compute the exterior derivatives of G-invariant differential forms on a homogeneous space G H. Therefore, we will briefly describe how this can be done explicitly. Since we have to consider many different geometric objects on the next pages, we will work with a fixed basis e_{i} _{1 i dim 𝔤} of 𝔤. This allows us to distinguish the different objects by the position of their indices. We also fix a biinvariant metric q on G and denote the corresponding normal metric on G H by q, too. We remark that the following considerations could also be carried out without referring to a basis or a normal metric. Since in Chapter 5 we have to work with a fixed basis and normal metric anyway, we will not do this. We start our considerations by introducing certain right-invariant one-forms on G:

Definition 3.1.39. Let G be a Lie group, whose Lie algebra we denote by 𝔤, and e_{i} _{1 i dim 𝔤} be a basis of G. As above, we denote the right-invariant vector field on G associated to e_{i} by e_{i}^{}. We define e^{i}^{}as the unique right-invariant one-form satisfying:

e^{i}^{}e_{j}^{}q e_{i}^{}, e_{j}^{}i, j 1, …, dim 𝔤 .

Remark 3.1.40. The term ”right-invariant” means in this context that the following equation holds:

R_{h^{1}} ^{}e^{i}^{}_{k} e^{i}^{}_{kh} h, k G, i 1, …, dim 𝔤 .

Since the action of G by h, α R_{h^{1}} ^{}α is a right action on the cotangent bundle, it is justified to speak of right-invariance in the above definition.

On the space G H there exist certain one-forms which are dual to the Killing vector fields e_{i}^{}:

Definition 3.1.41. Let G H be a coset space and e_{i} _{1 i dim 𝔤} be a basis of the Lie algebra 𝔤. We define the one-form e^{i}^{}on G H by the equation:

e^{i}^{}e_{j}^{}q e_{i}^{}, e_{j}^{}i, j 1, …, dim 𝔤 .

Remark 3.1.42. At any point p G H, e^{i}_{p}^{}, which is an element of the dim 𝔤 dim 𝔥 - dimensional vector space T_{p}^{}G H, has to satisfy dim 𝔤 equations. Nevertheless, the one-form e^{i}^{}is uniquely defined, since the kernel of the map

\begin{aligned}φ & : 𝔤 T_{p}G H \\ φ X & : X_{p}^{} \end{aligned}

is a conjugate of 𝔥.

The one-forms e^{i}^{}and e^{i}^{}are related by a lemma analogous to Lemma 3.1.17:

Lemma 3.1.43. Let G H be a coset space and let e^{i}^{}and e^{i}^{}be defined as above. We denote the projection of G onto G H by π. Furthermore, let pr_{V} : 𝔤 𝔤 be the projection onto a subspace of 𝔤 with respect to q. In this situation, we have for all i 1, …, dim 𝔤 and k G:

π^{}e^{i}^{}_{k} e^{i} pr_{Ad_{k} 𝔥} e^{i} _{k}^{}.

Proof : We show the lemma by evaluating the above equation on an arbitrary e_{j}^{}. The point k G will be omitted from our formulas for reasons of simplicity. We have:

π^{}e^{i}^{}e_{j}^{}
e^{i}^{}dπ e_{j}^{}
e^{i}^{}e_{j}^{}
q e_{i}^{}, e_{j}^{}

Since π is a Riemannian submersion, whose fiber at kH G H is kH G, we can transform the above term into:

q e_{i} pr_{Ad_{k} 𝔥} e_{i} ^{}, e_{j} pr_{Ad_{k} 𝔥} e_{j} ^{}
q e_{i} pr_{Ad_{k} 𝔥} e_{i} ^{}, e_{j}^{}q e_{i} pr_{Ad_{k} 𝔥} e_{i} ^{}, pr_{Ad_{k} 𝔥} e_{j} ^{}
e^{i} pr_{Ad_{k}𝔥} e^{i} ^{}e_{j}^{}

The exterior derivatives of the one-forms e^{i}^{}can be directly determined with help of the definition of d:

de^{i}^{}e_{j}^{}, e_{k}^{}e_{j}^{}e^{i}^{}e_{k}^{}e_{k}^{}e^{i}^{}e_{j}^{}e^{i}^{}e_{j}^{}, e_{k}^{}

e_{j}^{}\begin{matrix}q e_{i}^{}, e_{k}^{} \\ \end{matrix}_{const.} e_{k}^{}\begin{matrix}q e_{i}^{}, e_{j}^{} \\ \end{matrix}_{const.} e^{i}^{}e_{j}^{}, e_{k}^{}

e^{i}^{}e_{j}^{}, e_{k}^{}

e^{i}^{}e_{j}, e_{k} ^{} q e_{i}, e_{j}, e_{k}

We now assume that the basis e_{i} _{1 i dim 𝔤} is q-orthogonal. In this situation, we obtain the following explicit formula for de_{i}^{}:

de^{i}^{}_{1 j k dim 𝔤} \frac{q e_{i}, e_{j}, e_{k}}{q e_{i}, e_{i}}e^{jk}^{},

where e^{jk}^{}denotes e^{j}^{}e^{k}^{}. Since the pull-back π^{}commutes with the exterior differential, we are now able to deduce a formula for de^{i}^{}. In order to make our formula easy to manage, we assume that there is a n 1, …, dim 𝔤 such that e_{i} _{1 i n} spans 𝔪 and e_{i} _{n 1,...,dim 𝔤} spans 𝔥. The pull-back of e^{i}^{}is not e_{i}^{}. Instead we have the following relations:

π^{}e^{i}^{}_{e}  e^{i}_{e}^{}i 1, …, n
π^{}e^{i}^{}_{e}  0  i n 1, …, dim 𝔤

Because of the formula in Lemma 3.1.43, which is more complicated than Lemma 3.1.17, these relations are not necessarily true at another point of G than e. With help of the above relations and the formula for de^{i}^{}, we obtain:

Lemma 3.1.44. Let G H be a coset space. We denote the Lie algebra of G by 𝔤. Let e_{i} _{1 i dim 𝔤} be a basis of 𝔤 which is orthogonal with respect to a biinvariant metric q. Furthermore, we assume that there exists a n 1, …, dim 𝔤 with 𝔪 span e_{i} _{1 i n} and 𝔥 span e_{i} _{n 1,...,dim 𝔤} . In this situation, we have:

de^{i}^{}_{eH} _{1 j k n} \frac{q_{e} e_{i}, e_{j}, e_{k}}{q_{e} e_{i}, e_{i}} e^{jk}^{}_{eH} ,

where e^{jk}^{}denotes e^{j}^{}e^{k}^{}.

By applying the anti-derivation property of d, we could compute the exterior derivatives of arbitrary k-forms of type:

c_{i_{1}...i_{k}}e^{i_{1}...i_{k}}^{}: c_{i_{1}...i_{k}}e^{i_{1}}^{}… e^{i_{k}}^{}(3.3)

at the point eH. Unfortunately, k-forms of that kind are in general not G-invariant. Nevertheless, the considerations we have made can be used for determining the exterior derivatives of G-invariant forms.

If we restrict a G-invariant k-form on G H to the point eH G H, we obtain an H-invariant tensor in ^{k} T_{eH}^{}G H. Conversely, any H-invariant α ^{k} T_{eH}^{}G H can be extended by the action of G to a well-defined G-invariant differential form on G H, which we denote by α, too.

There exists a unique k-form of type (3.3) which coincides at eH with α. That form we will denote by α^{}. Our next aim is to determine dα_{eH} and to compare it with dα_{eH}^{}. We recall that the exterior derivative of a differential form is defined by:

\begin{aligned}dα X_{1}^{}, …, X_{k 1}^{} & _{i 1}^{k 1} 1 ^{i 1} X_{i}^{}α X_{1}^{}, …, X̂_{i}^{}, …, X_{k 1}^{} \\ & _{1 i j k 1} 1 ^{i j} α X_{i}^{}, X_{j}^{}, X_{1}^{}, …, X̂_{i}^{}, …, X̂_{j}^{}, …, X_{k 1}^{}. \end{aligned}

The vector fields we have inserted into dα in the above formula could be chosen arbitrarily. For our calculations, we assume that they are Killing vector fields. We obtain for the first kind of summands:

X_{i}^{}α_{eH} X_{1}^{}, …, X̂_{i}^{}, …, X_{k 1}^{}

\frac{}{t} _{t 0} α_{exp tX_{i} H} X_{1}^{}, …, X̂_{i}^{}, …, X_{k 1}^{}

\frac{}{t} _{t 0} α_{eH} dL_{exp tX_{i}} X_{1}^{}, …, { \atop dL_{exp tX_{i}} X_{i}^{}}, …, dL_{exp tX_{i}} X_{k 1}^{}

\frac{}{t} _{t 0} α_{eH} Ad_{exp tX_{i}} X_{1} ^{}, …, { \atop Ad_{exp tX_{i}} X_{i} ^{}}, …, Ad_{exp tX_{i}} X_{k 1} ^{}

_{j 1, j i}^{k 1} α_{eH} X_{1}^{}, …, X̂_{i}^{}, …, \begin{matrix}ad_{X_{i}} X_{j} ^{} \\ \end{matrix}_{entry of X_{j}^{}}, …, X_{k 1}^{}

Therefore, we have: dα_{eH} X_{1}^{}, …, X_{k 1}^{}

_{1 i j k 1} 1 ^{i 1} α_{eH} X_{1}^{}, …, X̂_{i}^{}, …, \begin{matrix}ad_{X_{i}} X_{j} ^{} \\ \end{matrix}_{entry of X_{j}^{}}, …, X_{k 1}^{}

_{1 i j k 1} 1 ^{j 1} α_{eH} X_{1}^{}, …, \begin{matrix}ad_{X_{j}} X_{i} ^{} \\ \end{matrix}_{entry of X_{i}^{}}, …, X̂_{j}^{}, …, X_{k 1}^{}

_{1 i j k 1} 1 ^{i j} α_{eH} X_{i}^{}, X_{j}^{}, X_{1}^{}, …, X̂_{i}^{}, …, X̂_{j}^{}, …, X_{k 1}^{}

_{1 i j k 1}  1 ^{i 1 j 2}  1 ^{j 1 i 1}

α_{eH} ad_{X_{i}} X_{j} ^{}, X_{1}^{}, …, X̂_{i}^{}, …, X̂_{j}^{}, …, X_{k 1}^{}

_{1 i j k 1} 1 ^{i j 1} α_{eH} X_{i}, X_{j} ^{}, X_{1}^{}, …, X̂_{i}^{}, …, X̂_{j}^{}, …, X_{k 1}^{}

_{1 i j k 1} 1 ^{i j} α_{eH} X_{i}, X_{j} ^{}, X_{1}^{}, …, X̂_{i}^{}, …, X̂_{j}^{}, …, X_{k 1}^{}

(3.4)

We could have made a similar calculation for dα_{gH}. Since α is homogeneous, dα is determined by dα_{eH}. Therefore, there is no need to compute dα_{gH}. We now assume that α is a one-form. In this case, we have:

dα_{eH} X_{1}^{}, X_{2}^{}α_{eH} X_{1}, X_{2} ^{}.

For the one-form α^{}, which is dual to a Killing vector field, we have:

dα_{eH}^{}X_{1}^{}, X_{2}^{}α_{eH}^{}X_{1}, X_{2} ^{}.

Up to the sign, the values of dα_{eH} and dα_{eH}^{}coincide. With help of the anti-derivation property of d, this observation can be generalized to forms of higher order:

Proposition 3.1.45. Let α be a G-invariant differential form on a coset space G H. Furthermore, let α^{}be the unique differential form of type (3.3) which coincides with α at the point eH. Then, we have:

dα_{eH} dα_{eH}^{}.

Remark 3.1.46. 1. Let e_{i} _{1 i n} be a basis of 𝔪. An H-invariant α ^{k} T_{eH}^{}G H can be written as

α α_{i_{1}...i_{k}}e^{i_{1}...i_{k}} .  (3.5)

The extension of α by the action of G to all of G H will be denoted by (3.5), too, although there may not exist global coordinate vector fields e_{1}, …, e_{n}. In most cases, even linearly independent vector fields e_{1}, …, e_{n} will not exist on G H.

2. When we want to determine the exterior derivative of a G-invariant differential form α at eH, we will do this by the following procedure: We formally apply the product rule for the exterior differential to (3.5) and replace each e^{i_{j}} we have to derivate by de^{i_{j}} _{eH}^{}. After that, we multiply the result by 1 and finally have obtained dα_{eH}.

At the end of this section, we will collect some useful results on homogeneous G-structures. Let 𝔉^{K H} be the coframe bundle of a coset space K H. K acts from the right on 𝔉^{K H} by k, u u dL_{k} , where u : T_{p}M ℝ^{dim K H} and k K. It therefore makes sense to speak of K-invariant G-structures on K H. If we have a K-invariant G_{2}- (Spin 7 -)structure on K H, the associated three- (four-)form is K-invariant, too. Conversely, any K-invariant three- (four-)form stabilized by G_{2} (Spin 7 ) defines a K-invariant G_{2}- (Spin 7 -)structure.

In Chapter 4, we will classify the possible principal orbits for cohomogeneity-one Spin 7 - structures. We will show in Section 3.2, that there exists a canonical homogeneous G_{2}- structure on those orbits. Conversely, any G-invariant G_{2}-structure on a coset space G H can be extended to a cohomogeneity-one Spin 7 -structure on G H I, where I is an interval. We therefore need a necessary and sufficient criterion for the existence of a G-invariant G_{2}- structure on G H:

Lemma 3.1.47. Let G H be a seven-dimensional coset space admitting a G-invariant G_{2}- structure 𝔊. H acts on the tangent space T_{eH}G H by h, v dL_{h} v . Therefore, we can consider H as a subgroup of GL T_{eH}G H . G_{2} acts on T_{eH}G H, too, where G_{2} GL T_{eH}G H is given by:

u^{1} v 𝔊 u, v : T_{eH}G H ℝ^{7} .

In this situation, we have H G_{2}.

Proof : Let h H and 𝔊 u : T_{eH}G H ℝ^{7} be arbitrary. Since 𝔊 is G-invariant, u dL_{h} : T_{eH}G H ℝ^{7} is an element of the G_{2}-structure, too. The endomorphism u^{1} v is for any 𝔊 v : T_{eH}G H ℝ^{7} an element of G_{2}. By setting v u dL_{h} we obtain the statement of the lemma.

Remark 3.1.48. 1. In the above lemma, we could define G_{2} GL T_{eH}G H also as the stabilizer of the associated three-form.

2. It is possible to prove a similar lemma for G-invariant K-structures on G H, where K GL dim G H is an arbitrary Lie subgroup.

3. On a homogeneous space, there may exist more than one transitive group action. It is possible that a G_{2}-structure is invariant under one of those actions, but not under another one. The flat G_{2}-structure on ℝ^{7}, for example, is invariant under translations but not under the whole Euclidean group.

Lemma 3.1.49. Let G H be a seven-dimensional coset space. As above, we consider H as a subgroup of GL T_{eH}G H . If there exists a linear φ : T_{eH}G H ℝ^{7} with H φ^{1} G_{2}φ, G H admits a G-invariant G_{2}-structure.

Proof : The set

φ dL_{h} dL_{g^{1}} : T_{gH}G H ℝ^{7} h H, g G

defines a G-invariant φHφ^{1} -structure on G H. Since the extension of this set to a G_{2}- structure is G-invariant, too, the statement of the lemma is true.

We assume that we have found a fixed G-invariant G_{2}-structure on a coset space G H. Furthermore, we assume without loss of generality that G and H are both connected. The G_{2}-structure is given by a principal bundle with fiber G_{2}, which we denote as above by 𝔊. 𝔊 can be extended to a SO 7 -structure, which induces an invariant metric and an orientation on G H. Our next aim is to classify the invariant G_{2}-structures on G H which have the same extension to a SO 7 -structure as 𝔊.

First, we will make some general observations. The projection map of 𝔊 we denote by π : 𝔊 G H. We fix a point p G H. Since 𝔊 is G-invariant, any element of the fiber π^{1} p determines the whole G_{2}-structure. Let π^{1} p u : T_{p}G H ℝ^{7}. We have:

π^{1} p h u h G_{2} .

Furthermore, we have:

𝔊 h u dL_{k^{1}} _{L_{k} p} h G_{2}, k G .

In Chapter 5, where we have to construct invariant G_{2}-structures, it will therefore suffice to find a u π^{1} p or equivalently a frame u^{1} e_{1} , …, u^{1} e_{7} .

We choose another G_{2}-structure ℌ on G H with the same properties as 𝔊. Its projection map is denoted by π^{}: ℌ G H. Let v π^{1} p . Since v determines ℌ, and 𝔊 is fixed, the map ϕ : v u^{1} : ℝ^{7} ℝ^{7} determines ℌ. We can therefore solve our problem by classifying the possible endomorphisms ϕ.

Our next step is to describe the set of all ϕ. The maps u^{1} : ℝ^{7} T_{p}G H and v : T_{p}G H ℝ^{7} preserve the orientation and the scalar product. Therefore, we have ϕ SO 7 . We remark that

t u^{1} t π^{1} p G_{2} .

Since H acts on T_{p}G H, we consider H as a subgroup of GL T_{p}G H . We can identify T_{p}G H by u with ℝ^{7}. Therefore, we can take H as a subgroup of GL 7 . We have shown in Lemma 3.1.47 that with this identifications H G_{2}. The set

w v^{1} w π^{1} p

is given by G_{2}, too. As above, we identify T_{p}G H by v with ℝ^{7} and use this identification to define an action of H on ℝ^{7}. It is easy to see that h H acts by matrix multiplication with ϕhϕ^{1} . For the same reasons as above, we have ϕHϕ^{1} G_{2}. All in all, we have shown that ϕ is an element of the set

M : ϕ SO 7 ϕHϕ^{1} G_{2} .

Conversely, each ϕ M describes a G_{2}-structure with the desired properties.

If ϕHϕ^{1} G_{2} and ψHψ^{1} G_{2}, we do not necessarily have ϕψ H ϕψ ^{1} G_{2}. Therefore, M is not necessarily a group, but an invariant set with respect to the adjoint action of H. H and ϕHϕ^{1} are both connected subgroups of G_{2}. In Lemma 4.1, we will show that connected subgroups of G_{2} which act with the same weights on ℝ^{7} are conjugate to each other not only by an element of SO 7 but by a further φ G_{2}, too. Therefore, we have:

φ^{1} ϕHϕ^{1} φ H G_{2} .

Since both ϕ and φ^{1} ϕ describe the same G_{2}-structure, we can consider instead of M the normalizer of H in SO 7 :

Norm_{SO 7}H ϕ SO 7 ϕHϕ^{1} H .

We remark that the space of all G-invariant G_{2}-structures with a given associated metric and orientation does not depend on the choice of the SO 7 -structure. Let 𝔎 be the extension of 𝔊 to a SO 7 -structure and let 𝔎^{}be a further invariant SO 7 -structure on G H. We choose an orientation-preserving isometry ψ : T_{p}G H T_{p}G H where the former G H is equipped with 𝔎 and the latter with 𝔎^{}. The set

h u ψ dL_{k^{1}} _{L_{k} p} h G_{2}, k G

is an invariant G_{2}-structure which can be extended to 𝔎^{}. Conversely, let 𝔊^{}be an invariant G_{2}-structure with 𝔎^{}as its extension and 𝔊^{}v : T_{p}G H ℝ^{7}. Then

h v ψ^{1} dL_{k^{1}} _{L_{k} p} h G_{2}, k G

is a G_{2}-structure with 𝔎 as its extension. In order to describe the set of all G_{2}-structures with the desired properties we have to divide the normalizer by the action of those elements of SO 7 which leave G_{2} invariant. We obtain:

Norm_{SO 7}H Norm_{SO 7}H Norm_{SO 7}G_{2} .

In Lemma 2.1.30, we have shown that Norm_{SO 7}G_{2} G_{2}. Furthermore, we have: Norm_{SO 7}H G_{2} Norm_{G_{2}}H. All in all, we have proven the following lemma:

Lemma 3.1.50. Let G H, where H is connected, be a seven-dimensional homogeneous space admitting a G-invariant G_{2}-structure. The space of all G-invariant G_{2}-structures which have a fixed associated metric and orientation is H-equivariantly diffeomorphic to:

Norm_{SO 7}H Norm_{G_{2}}H .

In particular, this space does not depend on the choice of the metric and orientation.

If one searches for metrics with exceptional holonomy, homogeneous metrics seem to be a good starting point. Unfortunately, there are no non-trivial homogeneous parallel G_{2}- or Spin 7 -structures. The reason for this is the following theorem:

Theorem 3.1.51. (See Besse [10] p.191 or Alekseevskii, Kimelfeld [1].) Any homogeneous Ricci-flat metric is necessary flat.

Proof : Let G H be a homogeneous space equipped with a Ricci-flat metric g. Since the Ricci curvature of G H is non-negative, we can apply the Cheeger-Gromoll splitting theorem. The universal cover of G H therefore is the Riemannian product of a simply connected homogeneous space G^{}H^{}and a flat ℝ^{d}. If we choose d maximal, G^{}H^{}contains no line. It is known that any non-compact homogeneous space contains a line. Therefore G^{}H^{}has to be compact. A theorem of Bochner [11] states that if the Ricci curvature of a compact Riemannian manifold is non-positive, the identity component of its isometry group is a torus. Since G^{}H^{}is Ricci-flat, G^{}therefore is a Lie subgroup of a Lie group whose identity component is a torus. Since, we can assume that G^{}and H^{}are connected, G^{}is a torus, too. If dim G^{}0, π_{1} G^{}ℤ^{dim G^{}}. Since we want G^{}H^{}to be simply connected, π_{1} H^{}has to be isomorphic to π_{1} G^{}. This is only possible, if G^{}H^{}. Therefore G^{}H^{}is a point and we have shown that the universal cover of G H is a flat Euclidean space.

Since there are no non-flat homogeneous metrics of exceptional holonomy, it is natural to search for metrics of that kind on spaces whose isometry group is large but not acting transitively on the manifold. In the next section, we will therefore introduce the most important results on metrics of cohomogeneity one.

3.2 Metrics of cohomogeneity one

Since there are no homogeneous metrics with exceptional holonomy, we have to consider metrics whose isometry group is smaller. First, we make the following assumption:

Convention 3.2.1. For reasons of simplicity, we only consider group actions by compact, connected Lie groups. This assumption guarantees that all orbits are closed subsets of the manifold M on which the group acts on. If M is a Riemannian manifold (a Spin 7 -manifold), we assume that the group action preserves the metric (the Spin 7 -structure) on M. Furthermore, we assume that M is connected. All of the above assumptions are maintained throughout this thesis unless otherwise stated.

In the following, we will often have to consider the orbit space M G of an isometric group action:

Lemma 3.2.2. Let M, g be a connected Riemannian manifold with an isometric action of a Lie group G. Furthermore, let π : M M G be the projection map and let d : M M 0, be the distance function on M induced by g. The quotient M G together with the function

\begin{aligned}d^{} & : M G M G 0, \\ d^{}x^{}, y^{} & : inf d x, y π x x^{}, π y y^{} \end{aligned}

is a metric space.

Example 3.2.3. The orbit space M G is not necessarily a manifold. We consider the Euclidean space ℝ^{n} with SO n acting on it by matrix multiplication. The orbits of this action are the spheres around the origin and the origin itself. The space ℝ^{n} SO n is isometric to the interval 0, with the standard metric, which is not a manifold.

By considering the above example, we see that an isometric action can have different types of orbits. The orbits whose neighborhood in M G can be identified with an open set in ℝ^{n} are of special interest for us:

Definition 3.2.4. Let M, g be a connected Riemannian manifold with an isometric action by a Lie group G. An orbit O of this action is called a principal orbit if there is an open subset U of M with the following properties: O U and U is G-equivariantly diffeomorphic to O V, where V ℝ^{n} is an open subset.

Since we are interested in spaces with a large isometry group, we have to find a measure for the size of G:

Definition 3.2.5. In the situation of the above definition, dim V (or equivalently dim M dim O) is called the cohomogeneity of the G-action on M, g .

If M is connected, any two principal orbits are G-equivariantly diffeomorphic. Therefore, the above definition is justified. Furthermore, we can identify any principal orbit with a coset space G H, where H G is fixed.

Convention 3.2.6. From now on, we assume that we have a Lie group G acting isometrically on a Riemannian manifold M, g , such that the cohomogeneity of the G-action is 1. In this situation, the subset U from Definition 3.2.4 is G-equivariantly diffeomorphic to O I, where I ℝ is an open interval. The isotropy group of the G-action on the principal orbit we denote by H.

It is possible to show the following lemma:

Lemma 3.2.7. In the situation of Convention 3.2.6, the following statements are true:

1. The union of all principal orbits of the G-action is an open, dense subset of M.

2. Any orbit of the cohomogeneity-one action is G-equivariantly diffeomorphic to a coset space G K, where K is a Lie subgroup of G. We can conjugate K in such a way that H K G.

The union of all principal orbits will often be denoted by M^{o}. We now describe the nonprincipal orbits in more detail. There are two kinds of these orbits, which are distinguished by the dimension of K H:

Definition 3.2.8. In the situation of the above lemma, let G K be a non-principal orbit. G K is called a

• singular orbit if dim K dim H.

• exceptional orbit if dim K dim H, i.e., the quotient K H is discrete.

The Lie group K is not arbitrary, but has to satisfy a certain condition:

Proposition 3.2.9. (See Mostert [55].) Let M, g be a Riemannian manifold with a cohomogeneity-one action by a Lie group G. Furthermore, let the principal orbits of this action be G-equivariantly diffeomorphic to G H and let G K be a non-principal orbit. In this situation, the quotient K H is diffeomorphic to a sphere.

Remark 3.2.10. • If we do not require M to be a smooth manifold, but still an orbifold, K H is diffeomorphic to S^{k} Γ, where k : dim K dim H and Γ is a discrete subgroup of O k 1 acting on S^{k}.

• Any non-principal orbit has a neighborhood in M G which consists solely of principal orbits. The above proposition therefore guarantees that there is a neighborhood of the singular orbit which is a disc bundle over that orbit. We call such a neighborhood a tubular neighborhood. In the orbifold case, we have a S^{k} Γ-bundle instead of a disc bundle. We will speak in that situation of a tubular neighborhood, too.

Convention 3.2.11. • In the following, we assume for reasons of simplicity that M is a manifold. It is easily possible to modify the results of this section in such a way that they are true for orbifolds, too.

• Motivated by the above observation, we will often speak of a collapse of the sphere S^{k} at the singular orbit or simply of a S^{k}-collapse when we approach a singular orbit.

For our next theorem we need the following lemma:

Lemma 3.2.12. On a connected Riemannian manifold M, g with a G-action of cohomogeneity one there exists a geodesic γ : I M which intersects all orbits perpendicularly. Let π : M M G be the projection map. In this situation, we obviously have π γ I M G.

With help of this lemma, it is possible to show the following result on the topology of M G:

Theorem 3.2.13. (See Mostert [55].) Let G be a group acting with cohomogeneity one on a manifold M. The quotient M G is homeomorphic to one of the following:

1. M G ℝ,

2. M G S^{1},

3. M G 0, ,

4. M G 0, 1 .

All points in the interior of M G correspond to principal orbits. The points on the boundary, namely 0 in the third and 0, 1 in the fourth case correspond to non-principal orbits. In the situation of the fourth case, we denote the isotropy group of the G-action at 0 by K^{}and the isotropy group at 1 by K^{}. If we have two non-principal orbits, the following diagram has to commute:

\xymatrix{ & G \\ K^{} \ar[ur] &  & K^{} \ar[ul] \\  & H \ar[ul] \ar[ur]}

Let K G be the isotropy group of a non-principal orbit. The action of K on the sphere S^{k} K H is transitive. Since we can fill ℝ^{k 1} by spheres of nonnegative radii, we can extend this action to an action on ℝ^{k 1} . It is known that any transitive action by a Lie group on a sphere is the restriction of a linear action (see Montgomery, Samelson [54]). Therefore, the action on ℝ^{k 1} we consider is linear, too. This action is the same as the action of K on the normal space of a p G K. It should be noted that this action of K on ℝ^{k 1} is solely determined by the choice of G, H, and K. These considerations together with the above theorem make it possible to show the following result on the global shape of M:

Corollary 3.2.14. Let M, g be a Riemannian manifold with an isometric cohomogeneityone action of a Lie group G. Furthermore, let H be the isotropy group of the G-action on a principal orbit. If the G-action has exactly one non-principal orbit, we denote it by K. If there are two non-principal orbits, we denote them by K^{}and K^{}. In this situation, M is, depending on the topology of M G, G-equivariantly diffeomorphic to one of the following:

1. If M G ℝ, then M G H ℝ.

2. If M G S^{1}, then M G H _{ℤ} ℝ. The action of ℤ on G H ℝ we divide out is the following: ℤ acts on ℝ by translation and the action of ℤ on G H is generated by the canonical right action of an element of Norm_{G}H.

3. If M G 0, , then M is G-equivariantly diffeomorphic to the disc bundle G _{K} D over G K, where D is the unit disc of dimension dim K dim H 1. Since D is diffeomorphic to ℝ^{dim D}, we can also describe M as a vector bundle over the nonprincipal orbit. The projection of this bundle onto G K is a Riemannian submersion.

4. If M G 0, 1 , then M G _{K^{}}D^{}G _{K^{}}D^{}where G _{K^{}}D^{}and G _{K^{}}D^{} have the same description as above. The two spaces are glued along a copy of G H by a G-equivariant diffeomorphism of G H.

In particular, the shape of M G together with the groups H and K^{}determines the global shape of M.

We are able to exclude some cases from our considerations. First, we will show that it is possible to restrict ourselves to manifolds without an exceptional orbit. We assume that M is a cohomogeneity-one manifold with at least one exceptional orbit. Since the quotient K H has to be a sphere, we have K H ℤ_{2}. There are three possible cases:

1. If M has only one non-principal orbit, which of course has to be exceptional, it is possible to construct a two-fold cover of M which is G-equivariantly diffeomorphic to G H ℝ.

2. Next, we assume that M has one exceptional and one singular orbit. We assume without loss of generality that G K^{}is exceptional. In this case, M is covered by a manifold of type G _{K^{}}D^{}G _{K^{}}D^{}which has two singular orbits of type G K^{}. This cover is two-fold, too. In the course of our considerations, we will not consider manifolds of this and the previous type anymore, since we can avoid them by passing to a cover.

3. We finally assume that M has two exceptional orbits. In this case, we can repeat the construction from the second case infinitely often. M is therefore a compact manifold with infinite fundamental group. Since we are interested in parallel Spin 7 -manifolds of cohomogeneity one, we assume that M carries a parallel Spin 7 -structure. It follows from Lemma 2.3.16 that the holonomy group acts trivially on a non-trivial subspace of the tangent space and thus is a subgroup of G_{2}.

Since we are only interested in examples with an irreducible holonomy representation, we will assume from now on that M has no exceptional orbits. The case where M has no singular orbit can be excluded, too: If M G S^{1}, then M is the total space of a homogeneous bundle over S^{1} and hence has infinite fundamental group. Furthermore, M is compact. We can therefore apply the same arguments as above and exclude that case. Next, we consider the case where M, g is complete and M G ℝ. In this situation, the geodesic which intersects all orbits perpendicularly is a line. By applying the Cheeger-Gromoll splitting theorem, we can show that M, g is a Riemannian product of ℝ and G H. Since we are first of all interested in complete examples which are not locally a Riemannian product, we will assume from now on that M has at least one singular orbit. In this situation, it is very easy to decide if M is complete or compact:

Lemma 3.2.15. Let M, g be a cohomogeneity-one manifold with a singular orbit. In this situation, we have:

1. M is compact if and only if it has two singular orbits.

2. M is complete and non-compact if and only if M has exactly one singular orbit, and there exists a geodesic γ which intersects all orbits perpendicularly and has infinite length.

3. M is non-complete if and only if M has exactly one singular orbit, and there exists a γ which cannot infinitely be extended.

Proof : We will prove that in the situation of the second case the manifold is complete. The other statements of the lemma then are obviously true. Let p_{n} _{n ℕ} be a Cauchy sequence in M. The projection map M M G we denote as usual by π. Since

d π x , π y d x, y x, y M ,

π p_{n} _{n ℕ} is a Cauchy sequence, too. We equip 0, with the standard metric. γ : 0, M is an injective map. Therefore, M G can isometrically be identified with 0, , which is complete. We define x 0, as lim_{n} π p_{n} . Since all orbits of the G-action are compact, π is a proper map, i.e., all compact subsets N 0, have a compact preimage π^{1} N . The set π p_{n} n ℕ x a compact one. Therefore, π^{1} p_{n} n ℕ x is compact, too. p_{n} is a sequence in that set. There exists a subsequence of p_{n} which converges against a p M. The Cauchy sequence p_{n} _{n ℕ} therefore has to converge against p, too.

The subject of the following pages are cohomogeneity-one Einstein metrics. This issue is important for us, since metrics with holonomy Spin 7 are Ricci-flat and therefore Einstein. Our considerations will be in large part a synopsis of the paper of Eschenburg and Wang [32]. Before we start, we fix some notation, which we will maintain throughout this Section:

Convention 3.2.16. In the following, let M, g be a Riemannian manifold with an isometric cohomogeneity-one action of a Lie group G. As usual, we assume that G is compact and connected. Furthermore, let G H be the principal orbit of this action and G K be a singular orbit. The Lie algebra of G we denote by 𝔤 and the Lie algebra of H by 𝔥. As in Section 3.1, we fix a biinvariant metric q on 𝔤. The q-orthogonal complement of 𝔥 𝔤 is called 𝔪. Furthermore, we denote the Lie algebra of 𝔨 by K and the q-orthogonal complement of 𝔨 𝔤 by 𝔭. The normal space of G K M will be denoted by 𝔭^{}. Finally, let the dimension of the sphere K H be k.

The notation ”𝔭^{}” is motivated by the following lemma which follows from our earlier considerations:

Lemma 3.2.17. The tangent space of the singular orbit G K is K-equivariantly isomorphic to 𝔭. The normal space of G K at p is the orthogonal complement of 𝔭 T_{p}M and a Kmodule, too. K acts on 𝔭^{}by cohomogeneity one. Furthermore, its action on 𝔭^{}is determined by the choice of G, H, and K.

Let β be a G-invariant tensor field defined on a tubular neighborhood of a singular orbit G K. As we have seen, the tubular neighborhood is a disc bundle over G K, which we will shortly denote by D. For the theorem on the cohomogeneity-one Einstein equation which we will introduce, we have to consider the question how we can describe β by a power series and which power series of that type can smoothly be extended to the singular orbit.

The vector bundle over D of which β is a section we denote by B. Since the tangent space of p G K D splits into the K-modules 𝔭 and 𝔭^{}, we can consider the fiber B_{p} as a K-module, too. If p is outside the singular orbit, B_{p} should be considered as an H-module.

The fiber of the disc bundle D at a p G K will be denoted by D_{p}. Since β is G-invariant, it is determined by its restriction to a fixed D_{p}. We choose a vertical unit vector v 𝔭^{}. Since the following considerations hold for any choice of v, we assume without loss of generality that we have chosen v in such a way that it is stabilized by H. Let γ : 0, r D_{p}, where r is the radius of D_{p}, be the geodesic with γ 0 p and γ^{}0 v. Since K acts transitively on the sphere K H, β is determined by β γ.

We can identify the open disc in 𝔭^{}of radius r by the exponential map with D_{p}. Therefore, we can identify β γ with a map b : vt t 0, r B. By letting K act on vt, we can extend b vt to a map S_{t}^{k} B, where S_{t}^{k} denotes the sphere of radius t in 𝔭^{}. Moreover, we can extend b vt by linearity to a map 𝔭^{}B. In the following, we will make use of both points of view and sometimes consider b vt as an element of B and sometimes as a map 𝔭^{}B. We finally define a map b on 0, r by b t : b vt . b satisfies:

b : 0, r Hom_{K} 𝔭^{}, B ,  (3.6)

where Hom_{K} V, W denotes the space of K-equivariant homomorphisms from V to W. Alternatively, we have:

b : 0, r B ,  (3.7)

where b t has to be invariant with respect to the isotropy group H.

The point t 0 plays a special role. At that point the isotropy group of the G-action is K. Therefore, b t has to turn into a K-invariant tensor when t approaches 0. If we consider the b described by (3.6), t 0 does not play a special role, since b t always has to be a K-equivariant homomorphism. This observation will be the key for the Theorem 3.2.18 we will introduce below. In the following, we restrict b to 0, r and consider the question if we can extend b to 0.

We assume that b is an analytic function. This assumption is justified, since Einstein metrics of dimension 3 are real analytic in geodesic normal coordinates (see Besse [10], Section 5.F. and DeTurck, Kazdan [30]). In this situation, we can express the b from (3.7) as a power series

b t _{m 0}^{}b_{m}t^{m} .

Since we can extend b t to an element of Hom_{K} 𝔭^{}, B , the summand b_{m}t^{m} has to be the restriction of a homogeneous polynomial P_{m} to span v . More precisely, P_{m} is a m^{th}-order polynomial defined on 𝔭^{}with values in the fiber B_{p}. Furthermore, P_{m} has to be K-invariant. It is possible to show that any choice of K-invariant homogeneous polynomials P_{m} defines a tensor field β, which can be smoothly extended to the singular orbit. This is the statement of our theorem. In order to properly state the theorem, we define the following spaces of homogeneous polynomials:

W_{m} : P : S^{m} 𝔭^{}B_{p} P is linear and K-equivariant .

Since K acts orthogonally on 𝔭^{}, we can identify 𝔭^{}K-equivariantly with its dual space. Next, we define the evaluation map:

\begin{aligned}ϵ_{m} & : W_{m} B_{p} \\ ϵ_{m} P & : P v . \end{aligned}

We remark that ϵ_{m} P is an H-invariant polynomial, since H stabilizes v. Now, we are able to state our theorem:

Theorem 3.2.18. (See Eschenburg and Wang [32].) Let M, g be Riemannian manifold with an isometric action of cohomogeneity one by a Lie group G. We assume that this action has a singular orbit. The isotropy group of the G-action at the singular orbit will be denoted by K. Let B ^{r} T M ^{s} T^{}M be a tensor bundle over M whose fibers at the singular orbit are K-equivariantly isomorphic to the K-module B. Let b : 0, r B, where r 0, be a real analytic map with Taylor expansion _{m 1}^{}b_{m}t^{m}. By the construction we have described above, we can identify b with a tensor field β. This tensor field is defined on a tubular neighborhood of the singular orbit, but not on the singular orbit itself. β is well-defined and has a smooth extension to the singular orbit if and only if

b_{m} ϵ_{m} W_{m} m ℕ_{0} ,

where W_{m} and ϵ are defined as above.

We are in particular interested in the case where β is a metric tensor. It is useful to make the following assumption:

Assumption 3.2.19. For the following considerations on the cohomogeneity-one Einstein equation, we assume that 𝔭 and 𝔭^{}have no equivalent H-submodules in common. This assumption will be satisfied in most cases we consider. In this situation, we have the following splitting:

S^{2} 𝔭 𝔭^{}^{H} S^{2} 𝔭 ^{H} S^{2} 𝔭^{}^{H} .  (3.8)

The space V^{H} shall be defined as the set of all H-invariant elements of the H-module V.

We consider the space

P : S^{m} 𝔭^{}S^{2} 𝔭 𝔭^{}P is linear and K-equivariant

which is a special space of type W_{m}. Any P in that space can be considered as a K-invariant m^{th}-order polynomial with values in S^{2} 𝔭 𝔭^{}. eH K H 𝔭^{}is invariant with respect to H. Therefore, P eH has to be H-invariant, too. If Assumption 3.2.19 is satisfied, we can conclude that P eH S^{2} 𝔭 ^{H} S^{2} 𝔭^{}^{H}. Since S^{2} 𝔭 S^{2} 𝔭^{}is a K-module, we have P v S^{2} 𝔭 S^{2} 𝔭^{}for any v 𝔭^{}. All in all, we have shown that if 3.2.19 is true, we have:

S^{m} 𝔭^{}S^{2} 𝔭 𝔭^{}P is linear and K-equivariant
S^{m} 𝔭^{}S^{2} 𝔭 S^{2} 𝔭^{}P is linear and K-equivariant
S^{m} 𝔭^{}S^{2} 𝔭 P is linear and K-equivariant
S^{m} 𝔭^{}S^{2} 𝔭^{}P is linear and K-equivariant

(3.9)

The existence of the splittings (3.8) and (3.9) will simplify some of the following representation theoretical arguments. If Assumption 3.2.19 is not satisfied, it is sometimes possible to prove results which are analogous to those we will introduce. Nevertheless, only the case where 𝔭 and 𝔭^{}have no H-submodules in common has been investigated in [32].

Remark 3.2.20. On the disc bundle D, we can introduce a radial coordinate t such that the cohomogeneity-one metric g becomes:

g g_{t} dt^{2} ,

where g_{t} is a t-dependent G-invariant metric on the principal orbit. In Chapter 5, where we do concrete calculations, we will often describe our metric in that way. g_{t} should be considered as an element of S^{2} 𝔪 rather than of S^{2} 𝔭 𝔭^{}. Nevertheless, both points of view on the metric are equivalent.

We now apply Theorem 3.2.18 in order to find smoothness conditions on metrics near a singular orbit. The cases m 0 and m 1 are of special interest, since the smoothness conditions we obtain have a simple geometric interpretation. First, we investigate the case m 0. In order to do this, we have to consider the space

W_{0} P : ℝ S^{2} 𝔭 S^{2} 𝔭^{}P is linear and K-equivariant ,

which describes the metric at a point of the singular orbit. Of course, W_{0} also contains elements which correspond to symmetric forms which are not positive definite. We nevertheless call W_{0} the space of possible metrics at the singular orbit, since this detail will be irrelevant for our representation theoretical considerations. The only restriction from Theorem 3.2.18 is that the metric at a point of the singular orbit has to be K-invariant. Next, we consider the case m 1. The first derivative of the metric can be considered as an element of

W_{1} P : 𝔭^{}S^{2} 𝔭 S^{2} 𝔭^{}P is linear and K-equivariant .

Eschenburg and Wang [32] have shown that

P : S^{2l 1} 𝔭^{}S^{2} 𝔭^{}P is linear and K-equivariant 0 l ℕ_{0} .

For l 0 this translates into the condition that the S^{2} 𝔭^{}-part of the metric has to approach the flat metric on D_{p}. Equivalently, the metric on K H has to approach the round metric with sectional curvature \frac{1}{t^{2}} sufficiently fast when t approaches 0. For the smoothness conditions of higher order there are similar interpretations, which we will not make explicit.

Remark 3.2.21. 1. Let g be a smooth cohomogeneity-one metric with a singular orbit. The collapsing spheres are distance spheres with respect to g. For different choices of g, the distance spheres are in general not the same. When we write g as g_{t} dt^{2}, we choose \frac{}{t} as a unit normal vector field on the collapsing spheres. This vector field of course depends on the choice of g, too. It is possible to choose different g such that the g_{t} S^{2} 𝔪 are the same but \frac{}{t} is different. In that situation, the metrics g are isometric to each other. We will therefore fix a vector in 𝔭^{}, which describes the direction of the geodesic intersecting all orbits perpendicularly, and restrict ourselves to bilinear forms in S^{2} 𝔭 𝔭^{}with the property that our vector is of unit length and orthogonal to the tangent space of G H. Since metrics which are isometric in the above sense will be identified in Chapter 5, the number of possible g_{t} which we have to consider is smaller than the number of metrics which satisfy the conditions from Theorem 3.2.18.

2. Let q be a K-invariant background metric on 𝔭^{}. Since K acts transitively on a sphere in 𝔭^{}, q is unique up to multiplication by a constant. In Chapter 5, we describe our cohomogeneity-one metrics by g_{t} dt^{2}. For t 0, the restriction of g_{t} to S^{2} 𝔭^{} describes the metric on the collapsing sphere. It is easy to see that the length of any tangent vector of G H with respect to q is ct O t^{2} for a c ℝ. Therefore, the smoothness condition of m^{th} order for the vertical part of the metric describes the m^{th} derivative of the restriction of \frac{1}{t^{2}} g_{t} to the sphere K H.

3. In the orbifold case, we can make similar considerations as in the manifold case. Nevertheless, the smoothness conditions for this case may be not the same as for manifolds. For example, let ℤ_{k} be a discrete subgroup of SO 2 . ℝ^{2} ℤ_{k} is a cone whose vertex corresponds to 0 ℝ^{2}. Let g be the metric on ℝ^{2} ℤ_{k} which is induced by the Euclidean metric on ℝ^{2}. If we multiply the part of g which is directed along the circles around the origin of ℝ^{2} ℤ_{k} by k, we again obtain a metric which is isometric to the Euclidean metric on ℝ^{2}. Therefore, we have to modify the vertical part of the metric at the singular orbit if our orbifold has such a conical singularity. If dim K H 2, the smoothness conditions for the restriction of \frac{}{t}g_{t} to K H remain the same, since K H still has to be a space whose sectional curvature is approximately \frac{1}{t^{2}} . We will deal with the further influences of the orbifold singularities on the smoothness conditions when we encounter those singularities.

We now describe the cohomogeneity-one Einstein condition explicitly. On the union of the principal orbits M^{o}, this condition can be described as a system of ordinary differential equations of second order. Since M^{o} is a dense subset of M, we only have to search for the initial values on the singular orbit such that the metric becomes smooth. We therefore obtain an initial value problem which we have to solve. Since the differential equations degenerate at the singular orbit, its solutions may depend on additional parameters apart from the initial metric and its first derivative.

We again introduce a radial coordinate t on a tubular neighborhood of a singular orbit. Any cohomogeneity-one metric g on that neighborhood can be described by:

g g_{t} dt^{2} ,

where g_{t} _{t 0,r} is a t-dependent family of G-invariant metrics on the orbits. For t 0, g_{t} becomes a degenerate bilinear form.

Before we can find an explicit description of the Einstein condition, we have to determine the shape operator L ∇\frac{}{t} of the principal orbits. As in the section before, we identify the restriction of the metric to the principal orbit with an endomorphism φ of 𝔪. It is easy to see that

L \frac{1}{2}φ^{1} \frac{}{t}φ .

Let e_{i} _{i 1,...,n} be a basis of 𝔪. We assume from now on that g_{t} is for all t a diagonal metric with respect to this basis. If the matrix representation of g_{t} with respect to e_{i} _{i 1,...,n} is diag f_{1}^{2} t , …, f_{n}^{2} t , we have:

L diag \frac{f_{1}^{}}{f_{1}}, …, \frac{f_{n}^{}}{f_{n}} .

After some calculations involving the Gauss-, the Codazzi-, and the Riccatti-equation, we finally can express the cohomogeneity-one Einstein condition as a system of ordinary differential equations of second order:

Theorem 3.2.22. (See Eschenburg and Wang [32].) Let M, g be a Riemannian manifold admitting an isometric cohomogeneity-one action by a Lie group G. The Ricci-curvature of g and its Ricci-endomorphism we denote both by Ric. We identify the union of all principal orbits by a G-equivariant diffeomorphism ϕ with G H I, where I is an interval and H is the isotropy group of the G-action on the principal orbit. We choose ϕ in such a way that eH I becomes a geodesic which intersects all orbits perpendicularly and is parameterized by arclength. In this situation, we have

g g_{t} dt^{2} ,

where t is the coordinate directed along I and g_{t} _{t I} is a family of metrics on G H. We denote the shape operator of the principal orbit by L_{t} and its Ricci-curvature by Ric_{t}. As above, we will denote the Ricci-endomorphism of the principal orbit by the same symbol. The Einstein condition Ric λg is in this situation equivalent to:

\frac{}{t}L_{t} trL_{t} L_{t} Ric_{t}  λg_{t}  (3.10)
tr \frac{}{t}L_{t} tr L_{t}^{2}  λ  (3.11)
tr ı_{X} d^{∇}L_{t}  0 vector fields X \frac{}{t}  (3.12)

If g_{t} is for all t I diagonal with respect to a basis e_{i} _{i 1,...,n} of 𝔪, we have

g_{t} diag f_{1} t ^{2}, …, f_{n} t ^{2}

for some functions f_{1}, …, f_{n} : I ℝ. Furthermore, the first two equations of the cohomogeneityone Einstein condition become:

\begin{matrix}\frac{f_{i}^{}t}{f_{i} t}  \frac{f_{i}^{}t}{f_{i} t} ^{2} \frac{f_{i}^{}t}{f_{i} t} _{j 1}^{n} \frac{f_{j}^{}t}{f_{j} t} Ric_{t} \frac{1}{f_{i} t}e_{i}, \frac{1}{f_{i} t}e_{i} &  & λ i 1, …, n \\ Ric_{t} e_{i}, e_{j} &  & 0 1 i j n \\ _{i 1}^{n 1} \frac{f_{i}^{}t}{f_{i} t} &  & λ \end{matrix}  (3.13)

Remark 3.2.23. 1. In equation (3.12), we consider L_{t} as a one-form on G H with values in T G H. d^{∇}α denotes the covariant exterior derivative of a vector-valued k-form α which is defined by:

\begin{aligned}d^{∇}α X_{1}, …, X_{k 1} & _{i 1}^{k 1} 1 ^{i 1} ∇_{X_{i}} α X_{1}, …, X̂_{i}, …, X_{k 1} \\ & _{1 i j k 1} 1 ^{i j} α X_{i}, X_{j} , X_{i}, …, X̂_{i}, …, X̂_{j}, …, X_{k 1} . \end{aligned}

In (3.12), ı denotes the interior product of a vector with a vector-valued differential form. The equation (3.12) is equivalent to Ric X, \frac{}{t} 0, which has to be satisfied for any cohomogeneity-one Einstein metric. It turns out that (3.12) is automatically satisfied if dim K H 1 (see Corollary 2.5. and 2.6. in [32]). Since all cohomogeneityone manifolds which we consider are of that kind, we only have to take care of (3.10) and (3.11).

2. We assume that g is a cohomogeneity-one metric such that (3.12) is satisfied. The system consisting of (3.10) and (3.11) contains n^{2} 1 equations for the n^{2} functions g_{t} e_{i}, e_{j} . Nevertheless, that system is not overdetermined. It has been proven in [32] that it has a unique short-time solution for any initial value of g_{t} and L_{t} on a principal orbit.

3. We consider the case that g_{t} is a diagonal metric but Ric_{t} is non-diagonal. The existence of metrics of that kind cannot be a priori ruled out. In that situation, the equation Ric e_{i}, e_{j} 0 is a rational equation for the metric functions f_{1}, …, f_{n}. We therefore have a non-trivial restriction which the metric functions have to satisfy. This seems to contradict our remark on the existence of short-time solutions. The explanation of this fact is as follows: If we have Ric e_{i}, e_{j} 0, the assumption that g_{t} is always diagonal is false and the condition Ric λg forces the metric to become non-diagonal as soon as we leave the initial principal orbit.

4. In Grove, Ziller [38] and Schwachhöfer, Tuschmann [63], similar formulas for the Riccicurvature of a cohomogeneity-one metric have been obtained. The notation we make use of in this thesis is in part motivated by those papers.

We consider the equations (3.13) for the diagonal case. When we approach a singular orbit, some of the f_{i} have to converge to zero. Since L_{t} contains the term \frac{f_{i}^{}}{f_{i}}, our equations degenerate at the singular orbit. In that situation, the existence and number of short-time solutions does not directly follow from Picard-Lindelöf’s or a similar theorem. In the nondiagonal case, we face similar difficulties. Nevertheless, Eschenburg and Wang [32] have been able to prove a theorem on the local existence of cohomogeneity-one Einstein metrics. The cohomogeneity-one Einstein-equations can be rewritten as

\frac{}{t}L_{t} \frac{1}{t^{2}} A g_{t} \frac{1}{t}B g_{t}, L_{t} C t, g_{t}, L_{t} ,

where A, B, and C are some functions which could be explicitly determined and have a smooth extension to t 0. A necessary condition for the existence of local solutions is that the right-hand side of the above equation can be smoothly extended to t 0 after we have inserted the initial values. Fortunately, this is always the case. Eschenburg and Wang [32] have made a power series ansatz for the metric g. We assume that the singular orbit is at t 0 and obtain:

g_{t} _{m 0}^{}g_{m}t^{m} .

On the previous pages, we have seen that the coefficients g_{m} can be considered as elements of S^{2} 𝔭 𝔭^{}and have to satisfy the smoothness conditions described in Theorem 3.2.18. Since the G-invariant metric g_{t} on G H has to correspond to an element of S^{2} 𝔭 𝔭^{}^{H} S^{2} 𝔭 ^{H} S^{2} 𝔭^{}^{H}, we have g_{m} S^{2} 𝔭 ^{H} S^{2} 𝔭^{}^{H}, too. It is possible to show that g_{m 2} can be chosen as an arbitrary solution of a recursion equation of type

L_{m}g_{m 2} D_{m} g_{0}, …, g_{m 1} ,  (3.14)

where L_{m} is a H-equivariant endomorphism of S^{2} 𝔭 ^{H} S^{2} 𝔭^{}^{H} and D_{m} is a function which we will not consider in detail. We have to show that there are solutions of (3.14) which satisfy the conditions of Theorem 3.2.18. That theorem states that g_{m 2} has to be in ϵ_{m 2} W_{m 2} . We can therefore restrict L_{m} to a map ϵ_{m 2} W_{m 2} S^{2} 𝔭 ^{H} S^{2} 𝔭^{}^{H}. In [32], it is shown that L_{m} ϵ_{m 2} W_{m 2} ϵ_{m} W_{m} and L_{m} : ϵ_{m 2} W_{m 2} ϵ_{m} W_{m} is surjective. Furthermore, we have D_{m} g_{0}, …, g_{m 1} ϵ_{m} W_{m} . Therefore, a formal power series solution of our system of ordinary differential equations which satisfies the smoothness conditions does exist. The number of free parameters of m 2 ^{nd} order is given by dim ker L_{m} dim ϵ_{m 2} W_{m 2} dim ϵ_{m} W_{m} . Since K acts transitively on the spheres around 0 in 𝔭^{}, any polynomial P W_{m} can be recovered from P v , where v 𝔭^{}is the vector we have chosen on page 66. Therefore, the evaluation map ϵ_{m} is injective and the number of free parameters equals dim W_{m 2} dim W_{m}. We are able to extend our considerations to m 1, 2 . The possible initial values of 0^{th} and 1^{st} order are simply the K-invariant inner products on 𝔭 𝔭^{} and the G-invariant tensors in T G H ^{}T^{}M. Therefore, we define:

{ \atop W_{m}} : P : S^{m} 𝔭^{}S^{2} 𝔭 𝔭^{}P is linear and K-equivariant m 0 0  m 0  (3.15)

With this notation, the number of free parameters of m^{th} order becomes dim { \atop W_{m}} dim { \atop W_{m 2}} , which equals dim kerL_{m 2} for m 2. On page 68, we have shown that for m 0:

{ \atop W_{m}} P : S^{m} 𝔭^{}S^{2} 𝔭 P is linear and K-equivariant
P : S^{m} 𝔭^{}S^{2} 𝔭^{}P is linear and K-equivariant .

We denote the first summand by { \atop W_{m}^{h}} and the second one by { \atop W_{m}^{v}}. There are some facts about the modules { \atop W_{m}^{v}} and { \atop W_{m}^{h}} which should be mentioned. We start with { \atop W_{m}^{v}}. As we have mentioned before, Hom_{K} S^{m} 𝔭^{}, S^{2} 𝔭^{}is 0 if m is odd. In the paper of Eschenburg and Wang [32], it has been shown that Hom_{K} S^{m} 𝔭^{}, S^{2} 𝔭^{}Hom_{K} S^{2} 𝔭^{}, S^{2} 𝔭^{}if m is positive and even. For m 0, we have dim { \atop W_{m}^{h}} 1, since the K-invariant inner product on 𝔭^{}is unique up to multiplication by a constant. The above considerations show that only in the 0^{th} and 2^{nd} order there are initial conditions for the vertical directions we can prescribe. The initial condition of 0^{th} order is omitted in [32], since we can rescale the metric on the normal space of G K by changing \frac{}{t} into λ\frac{}{t}. The number of 2^{nd} order initial conditions in the vertical direction is dim Hom_{K} S^{2} 𝔭^{}, S^{2} 𝔭^{}1.

Next, we consider { \atop W_{m}^{h}}. The function defined by q tv : t^{2} can be extended by the action of K to a K-invariant quadratic form on all of 𝔭^{}. This form we denote by q, too. It is easy to see that there is up to multiplication by a positive constant only one positive definite K-invariant quadratic form on 𝔭^{}. We have the following K-equivariant embedding

\begin{aligned}ı & : { \atop W_{m}^{h}} { \atop W_{m 2}^{h}} \\ ı P & : q P , \end{aligned}

where denotes the symmetric product. Therefore, we can consider { \atop W_{m}^{h}} as a subspace of { \atop W_{m 2}^{h}} and obtain the following chains of K-modules:

\begin{matrix}{ \atop W_{0}^{h}} & { \atop W_{2}^{h}} { \atop W_{4}^{h}} … { \atop W_{2m}^{h}} … \\ { \atop W_{1}^{h}} & { \atop W_{3}^{h}} { \atop W_{5}^{h}} … { \atop W_{2m 1}^{h}} … \end{matrix}  (3.16)

We will show that both of these chains stabilize at a sufficiently large m. Let S be the sphere in 𝔭^{}. Any polynomial P : 𝔭^{}S^{2} 𝔭 can be identified with its restriction to S. Therefore, the space of all K-invariant polynomials _{m ℕ_{0}} { \atop W_{m}^{h}} can be considered as a subspace of the greater space { \atop W^{h}} of all K-equivariant maps S S^{2} 𝔭 . Analogously to above, we define an evaluation map:

\begin{aligned}ϵ & : { \atop W^{h}} S^{2} 𝔭 \\ ϵ P & : P v . \end{aligned}

Since any P { \atop W^{h}} is determined by P v , ϵ is injective. Therefore, dim { \atop W^{h}} is finite. We conclude from the fact that the dimensions of the spaces in (3.16) are non-decreasing that they have to stabilize at a certain m. All in all, we have shown that the solutions of the cohomogeneity-one Einstein equation depend only on a finite number of parameters.

Until now, we only have found formal power series as solutions. It still has to be checked if they converge. Let

{ \atop g_{m}} : _{k 0}^{m} g_{k}t^{k}

be the metric we obtain by summing up the first m 1 summands of our power series. We take { \atop g_{m}} as the starting point of a Picard iteration. The metric we obtain coincides with { \atop g_{m}} up to the m^{th} order. Since we can choose the degree m arbitrarily high, we can prescribe all of the initial conditions we have found. Therefore, we obtain for any choice of the initial conditions a convergent solution. The above considerations are the main ideas for the proof of the following theorem:

Theorem 3.2.24. (See Eschenburg and Wang [32].) Let M be a manifold equipped with a cohomogeneity-one action by a compact Lie group G. We assume that the principal orbits of this action are G-equivariantly diffeomorphic to G H and that it has a singular orbit G K, where H K G are closed subgroups of each other.

Let 𝔭 be the tangent space of the singular orbit. Any tubular neighborhood of the singular orbit is a disc bundle. The tangent space of its fiber at a point of the singular orbit we denote by 𝔭^{}. The K-module structures of 𝔭 and 𝔭^{}are independent of the point we consider. We assume that 𝔭 and 𝔭^{}have no H-submodule 0 in common.

Let g_{0} be a G-invariant metric on the singular orbit. Furthermore, let g_{0}^{}be an arbitrary linear, K-equivariant map g_{0}^{}: 𝔭^{}S^{2} 𝔭 , which we extend by the G-action to all of G K. We finally choose an arbitrary λ ℝ. In this situation, there exists a G-invariant Einstein metric g on a sufficiently small tubular neighborhood of the singular orbit which has the following properties:

1. g has λ as Einstein constant.

2. The restriction of g to the singular orbit is g_{0}.

3. The first derivation of g at the singular orbit in the directions normal to the singular orbit is g_{0}^{}.

Furthermore, the set of all those Einstein metrics depends on additional initial conditions of higher order. The number of those initial conditions of m^{th} order is given by:

dim { \atop W_{m}^{h}} dim { \atop W_{m 2}^{h}} in the directions horizontal to G K
dim W_{2}^{v} 1 in the vertical directions if m 2
0 in the vertical directions if m 2

where the notations of these formulas are explained on the previous pages.

Remark 3.2.25. 1. In the cases we consider in Chapter 5, the number of possible initial conditions for the cohomogeneity-one Einstein equation will seem to be smaller than the theorem predicts. The reason for this is that metrics which are isometric to each other in the sense of Remark 3.2.21 will be identified.

2. If we insert m 0 or m 1 in the above formulas, we see that g_{0} and g_{0}^{}can be chosen arbitrarily.

3. The geometric meaning of the freedom in the vertical direction is that we can choose the trace-free part of g_{0}^{}freely.

4. In contrast to the rest of this section, we do not assume in the above theorem that G, H, and K are connected.

5. Our representation theoretical considerations do not depend on the value of the Einstein constant. Therefore, we can construct solutions with an arbitrary sign of the Einstein constant. Since the above theorem gives only local solutions, we do not have to be concerned with the topological restrictions for the sign of the Einstein constant.

On the next pages, we will mention a few general results on cohomogeneity-one Spin 7 - structures. We require that the action of G leaves not only the metric but also the Spin 7 - structure invariant:

Definition 3.2.26. Let M, Ω be a Spin 7 -manifold equipped with an isometric cohomogeneity-one action of a Lie group G. M, Ω is called a Spin 7 -manifold of cohomogeneity one if the action of any g G leaves Ω invariant.

On any principal orbit of a cohomogeneity-one Spin 7 -manifold, there exists a canonical G_{2}-structure:

Lemma 3.2.27. Let M, Ω be a Spin 7 -manifold with a cohomogeneity-one action by a Lie group G preserving the four-form Ω. Any principal orbit is G-equivariantly diffeomorphic to G H, where H is a Lie subgroup of G. Furthermore, let i : G H M be the inclusion map of a fixed principal orbit. The pull-back i^{}Ω is a G-invariant four-form ω^{}on G H. Its Hodge dual ω determines a G-invariant G_{2}-structure on G H.

Proof : The only thing we have to show is that ω is stabilized by a group isomorphic to G_{2}. Since ω is homogeneous, it suffices to show this for an arbitrary p G H. There exists an orthonormal basis e_{0}, …, e_{7} of T_{i p}M such that

Ω_{i p} { \atop ω^{}}_{i p} e^{0} { \atop ω_{i p}}

and the coefficients of { \atop ω^{}}_{i p} and { \atop ω_{i p}} with respect to e_{1}, …, e_{7} coincide with the coefficients of the forms ω, ω ^{}Im 𝕆 ^{}. Since Spin 7 acts transitively on the unit sphere, there is an element g Spin 7 which maps e_{0} into the tangent vector \frac{}{t} which is orthogonal to the principal orbit. We denote the pull-back of the action of g^{1} on T_{i p}M by g^{1} ^{}and obtain:

Ω_{i p} g^{1} ^{}{ \atop ω^{}}_{i p} dt g^{1} ^{}{ \atop ω_{i p}} .

Since the action of Spin 7 is orthogonal, e_{1}, …, e_{7} are mapped to tangent vectors of the principal orbit. We apply the pull-back of i to the above equation:

ω_{p}^{}i^{}Ω_{i p} i^{}g^{1} ^{}{ \atop ω^{}}_{i p} .

ω_{p}^{}can be obtained from ω ^{4} Im 𝕆 ^{}by the pull-back of an endomorphism T_{p}G H Im 𝕆 . Therefore, ω ^{3} Im 𝕆 ^{}and ω_{p}^{}ω_{p} ^{3} T_{p}^{}G H are stabilized by the same group, which is G_{2}.

The converse of the above lemma is also true:

Lemma 3.2.28. Let ω be a G-invariant G_{2}-structure on a homogeneous space G H and let I ℝ be an interval. Then, there exists a G-invariant Spin 7 -structure Ω on G H I. Furthermore, let t I be arbitrary and

\begin{aligned}i & : G H G H I \\ i p & : p, t . \end{aligned}

The above Ω can be chosen in such a way that i^{}Ω ω.

Proof : Let t be the coordinate directed along I. We simply choose:

Ω ω dt ω .

We are first of all interested in parallel Spin 7 -structures of cohomogeneity one. The following necessary condition for dΩ 0 can easily be seen:

Lemma 3.2.29. In the situation of Lemma 3.2.27, let dΩ 0. On any principal orbit, the G-invariant G_{2}-structure ω is cosymplectic, i.e., d_{G H} ω 0.

Convention 3.2.30. In the above lemma, d_{G H} denotes the exterior differential operator on the principal orbit G H. In the course of this thesis, we will often have to distinguish between the exterior differential on a manifold M and on submanifolds of M. We will therefore index d by the submanifold we consider in order to avoid confusion. Throughout this section, denotes the Hodge star operator on G H.

Proof : We choose an arbitrary principal orbit and denote the inclusion map of this orbit into M by i. Since i^{}Ω ω, we have:

dΩ 0  i^{}dΩ 0  d_{G H} i^{}Ω 0  d_{G H} ω 0

Conversely, any cosymplectic G-invariant G_{2}-structure on G H can be uniquely extended to a parallel Spin 7 -structure:

Theorem 3.2.31. Let G H be a seven-dimensional homogeneous space carrying a G-invariant G_{2}-structure ω with d_{G H} ω 0. Then there exists an ϵ 0 and a one-parameter family ω_{t} _{t ϵ,ϵ} of G-invariant three-forms on G H such that the initial value problem

\frac{}{t} ω_{t}  d_{G H}ω_{t}  (3.17)
ω_{0}  ω  (3.18)

has a unique solution on G H ϵ, ϵ with the following properties:

1. ω_{t} is a G_{2}-structure on G H.

2. d_{G H} ω_{t} 0.

3. ω_{t} is in the same cohomology class in H^{4} M, ℝ as ω.

In the above formula, \frac{}{t} denotes the Lie derivative in t-direction. Furthermore, the four-form Ω : ω dt ω is a G-invariant parallel Spin 7 -structure on G H ϵ, ϵ .

Conversely, let Ω be a parallel Spin 7 -structure preserved by a cohomogeneity-one action of a Lie group G. The isotropy group of the G-action on the principal orbit we denote by H. We identify the union of all principal orbits G-equivariantly with G H I, where the interval I is parameterized by arclength. In this situation, the G_{2}-structure on the principal orbit satisfies equation (3.17).

Proof : It is easy to see that Ω ω dt ω is a closed form on G H I if and only if:

\frac{}{t} ω  d_{G H}ω
d_{G H} ω  0

Therefore, we only have to show that ω_{t} exists and that it satisfies the three properties stated in the theorem. The short-time existence and uniqueness of ω_{t} follows directly from Picard-Lindelöf’s theorem. Since d_{G H} commutes with the action of G, ω_{t} has to be G-invariant. We now prove the three properties separately:

1. The canonical action of GL 7 on Im 𝕆 induces an action of GL 7 on ^{3} Im 𝕆 ^{}. The orbit of ω ^{3} Im 𝕆 ^{}is an open set. Therefore, there exists an ϵ^{}0, ϵ such that for all t ϵ^{}, ϵ^{}ω_{t} is the pull-back of ω with respect to a linear map T_{p}G H Im 𝕆 .

2. Since we have d_{G H} ω_{0} 0, it suffices to show the equation

\frac{}{t}d_{G H} ω_{t} 0

in order to prove d_{G H} ω_{t} 0. d_{G H} is a t-independent operator on G H. Therefore, we have:

\frac{}{t}d_{G H} ω d_{G H} \frac{}{t} ω d2_{G H}ω 0 ,

which proves the second property of ω_{t}.

3. Finally, we have to prove that ω_{t} ω_{0} is an exact form. This statement can be shown with help of the fundamental theorem of calculus:

ω_{t} ω_{0} _{0}^{t} \frac{}{τ} ω_{τ} dτ _{0}^{t} d_{G H}ω_{τ} dτ d_{G H} _{0}^{t} ω_{τ} dτ .

Convention 3.2.32. Let α be a tensor field on G H I. For the rest of this thesis, we will always denote the Lie derivative of α in the t-direction by \frac{}{t}α.

Remark 3.2.33. 1. In [42], Hitchin introduced the notion of a stable form. A form on a vector space V is called stable if its GL V -orbit is an open set. The three- and the four-form associated to a G_{2}-structure are special examples of stable forms. Hitchin considered the gradient flow of a certain functional defined on the space of closed stable four-forms in a fixed cohomology class. He obtained a theorem similar to the above one for the more general case that G H is replaced by a coclosed but not necessarily homogeneous G_{2}-manifold.

2. Although there always exists a short-time solution of (3.17) near a principal orbit, we will see that sometimes there are invariant metrics on the singular orbit which cannot be extended to a cohomogeneity-one metric of holonomy Spin 7 .

There is one special solution of the equation (3.17) which can be easily obtained:

Corollary 3.2.34. Let G H be a homogeneous space and let ω_{0} be a G-invariant G_{2}-structure on G H which is nearly parallel but not parallel. Then, there exists a cone M, g over G H such that

1. ω_{0} can be extended to a parallel G-invariant Spin 7 -structure Ω on M.

2. The metric associated to Ω is the cone metric g which has holonomy Spin 7 .

Furthermore, the extension of ω_{0} to a parallel G-invariant Spin 7 -structure is unique.

Proof : The uniqueness of Ω follows directly from the uniqueness of the solutions of the initial value problem (3.17), (3.18). We therefore only have to solve (3.17) for this special situation. Since ω_{0} is nearly parallel, we have

d_{G H}ω_{0} λ ω_{0}

for a constant λ 0. We equip the cone over G H with the Spin 7 -structure

Ω : \frac{λ^{4}t^{4}}{256} _{g_{0}} ω_{0} \frac{λ^{3}t^{3}}{64} ω_{0} dt

and the metric

g : \frac{λ^{2}t^{2}}{16} g_{0} dt^{2} .

It is easy to see that g is the associated metric to Ω. Ω satisfies equation (3.17). Furthermore, the restriction of the Spin 7 -structure to the principal orbit at t \frac{4}{λ} equals ω_{0}.

Remark 3.2.35. If G H is not a quotient of a sphere by a discrete group, the cone has a singularity at the tip which is not an orbifold singularity. In Chapter 5, we therefore will not consider the cone metrics in much detail.

Later on, we will construct concrete examples of parallel Spin 7 -manifolds M, Ω of cohomogeneity one. As usual, we denote the metric associated to Ω by g. We assume that we have shown the smoothness conditions for g from Theorem 3.2.18. Of course, we want the four-form Ω to have a smooth extension to the singular orbit, too. Furthermore, we want to prove this fact without manually checking any further smoothness conditions. Since g can be smoothly extended to all of M and its holonomy is a subgroup of Spin 7 , we know that there exists a parallel smooth four-form on M. We first consider the case where the holonomy equals Spin 7 . In this case, the four-form is up to a constant factor unique. Therefore, it has to equal a multiple of Ω and Ω therefore can be smoothly extended. If the holonomy is a proper subgroup of Spin 7 , we have a bigger family of smooth four-forms. One of those four-forms has to coincide with Ω on the union of all principal orbits. Since that four-form has a smooth extension to the singular orbit, Ω has to have a smooth extension to the singular orbit, too. All in all, we have shown that in the above situation Ω can be smoothly extended to the singular orbit if g has that property.

At the end of this section, we compare the solutions of equation (3.17) in the case where the metric is diagonal with the non-diagonal case. Before we start, we motivate why this comparison is interesting for us.

When we construct parallel Spin 7 -structures of cohomogeneity one, we will often make the following ansatz: First, we will choose a seven-dimensional homogeneous space G H admitting a cosymplectic G_{2}-structure as principal orbit. As usual, G and H are chosen as compact and the action of G shall leave the G_{2}-structure invariant. The union of all principal orbits we identify G-equivariantly with G H I. Next, we fix a basis e_{i} _{1 i 7} of the tangent space 𝔪. This basis will often be a special one, for example, a basis adapted to a splitting of 𝔪 into H-submodules or an orthogonal basis with respect to a normal metric q.

The calculations needed for the construction of metrics with holonomy Spin 7 will become easier if we assume that the restriction of the metric to any principal orbit is diagonal. This assumption is in many cases justified, for example if 𝔪 splits into pairwise inequivalent irreducible H-submodules. We consider the initial value problem (3.17), (3.18). The metric associated to the G_{2}-structure ω_{t} we denote by g_{t}. In Chapter 5, where we consider concrete principal orbits, we will rewrite our initial value problem in terms of a family of functions I ℝ. By considering the resulting systems of ordinary differential equations, we will often see that if g_{0} is diagonal, any g_{t} is diagonal, too. This is a further indication that our assumption is natural.

We now compare the non-diagonal with the diagonal case. We assume that e_{i} _{1 i 7} is qorthogonal. Let g be a not necessarily diagonal metric on G H. From linear algebra we know that there exists a q-orthogonal endomorphism ψ : 𝔪 𝔪 such that g is diagonal with respect to ψ^{1} e_{i} _{i 1,...,7} . It is possible to show that we can choose ψ even as an H-equivariant map. We remark that ψ only is defined on the fixed tangent space T_{p}G H we identify with 𝔪. Since ψ End 𝔪 is H-equivariant, it is possible to extend it to a G-invariant endomorphism field on all of G H, which we denote by ψ, too.

Let Ω be a Spin 7 -structure which was found as an solution of (3.17). We consider the principal orbit at t 0 and choose ψ : 𝔪 𝔪 as above. We can extend ψ to a t-independent endomorphism field on all of G H I. The best case would be if ψ^{}Ω was parallel, too. If this was true, we only would have to solve the equation (3.17) for the diagonal case. The non-diagonal solutions could be obtained by applying the pull-back ψ^{}. We will therefore compute d ψ^{}Ω under the assumption that dΩ 0.

First, we consider the special case that ψ is the differential of an isometry. This case will be important for our examples with holonomy SU 4 . In this situation, we obviously have d ψ^{}Ω 0 if dΩ 0.

We now consider the general case. Let X_{1}, …, X_{5} 𝔪 be arbitrary. According to equation (3.4), we have:

d ψ^{}Ω X_{1}, …, X_{5}
_{1 i j 5} 1 ^{i j} ψ^{}Ω X_{i}, X_{j} , X_{1}, …, { \atop X_{i}}, …, { \atop X_{j}}, …, X_{5}
_{1 i j 5} 1 ^{i j} Ω ψ X_{i}, X_{j} , ψ X_{1} , …, { \atop ψ X_{i}}, …, { \atop ψ X_{j}}, …, ψ X_{5} .

If ψ is the restriction of a Lie algebra homomorphism of 𝔤 to 𝔪, we therefore obtain:

d ψ^{}Ω X_{1}, …, X_{5} ψ^{}dΩ X_{1}, …, X_{5} .

Under this assumption, we further have analogously to above: d ψ^{}Ω X_{1}^{}, …, X_{4}^{}, \frac{}{t}

_{i 1}^{4} 1 ^{i 1} X_{i}^{}ψ^{}Ω X_{1}^{}, …, { \atop X_{i}^{}}, …, X_{4}^{}, \frac{}{t}

1 ^{5 1} \frac{}{t} ψ^{}Ω X_{1}^{}, …, X_{4}^{}

_{1 i j 4} 1 ^{i j} ψ^{}Ω X_{i}^{}, X_{j}^{}, X_{1}^{}, …, { \atop X_{i}^{}}, …, { \atop X_{j}^{}}, …, X_{4}^{}, \frac{}{t}

_{i 1}^{4} 1 ^{i 5} ψ^{}Ω X_{i}^{}, \frac{}{t} , X_{1}^{}, …, { \atop X_{i}^{}}, …, X_{4}^{}

_{i 1}^{4} 1 ^{i}X_{i}^{}ψ^{}ω X_{1}^{}, …, { \atop X_{i}^{}}, …, X_{4}^{}

\frac{}{t} ω ψ X_{1}^{}, …, ψ X_{4}^{}

_{1 i j 4} 1 ^{i j 1} ψ^{}ω X_{i}^{}, X_{j}^{}, X_{1}^{}, …, { \atop X_{i}^{}}, …, { \atop X_{j}^{}}, …, X_{4}^{}

d_{G H} ψ^{}ω X_{1}^{}, …, X_{4}^{}ψ^{}\frac{}{t} ω X_{1}^{}, …, X_{4}^{}
ψ^{}d_{G H}ω X_{1}^{}, …, X_{4}^{}ψ^{}\frac{}{t} ω X_{1}^{}, …, X_{4}^{}
ψ^{}dΩ X_{1}^{}, …, X_{4}^{}, \frac{}{t}

All in all, we have shown that in this situation d ψ^{}Ω ψ^{}dΩ. If dΩ 0, d ψ^{}Ω therefore has to vanish, too. We sum up our results:

Lemma 3.2.36. Let M, Ω be a parallel Spin 7 -manifold with a cohomogeneity-one G-action preserving Ω. The isotropy group of the G-action on the principal orbit we denote by H G. As usual, the tangent space of G H is identified with 𝔪. Furthermore, let ψ : 𝔤 𝔤 be a Lie algebra homomorphism which is H-equivariant and satisfies ψ 𝔪 𝔪. In this situation, the four-form ψ^{}Ω defined above satisfies d ψ^{}Ω 0.

Remark 3.2.37. 1. The Lie group G from the above lemma does not necessarily has to be compact or connected. Nevertheless, we will apply the lemma only in the case where G is compact and connected.

2. If G is semisimple, the automorphism group of 𝔤 is up to a discrete factor G itself acting on 𝔤 by the adjoint action. We assume that ψ Ad_{g} for a g G. Since ψ has to be H-equivariant, g has to commute with H. Therefore, the action of ψ^{}on Ω is in many cases induced by the action of Norm_{G}H on G H. Nevertheless, there are further possibilities for ψ. For example, ψ could be chosen as the differential of the geodesic symmetry of a symmetric space.

3. It is easy to see that the action of ψ^{}not only preserves the equation dΩ 0, but also the Einstein equation Ric λg. The reason for this is that the Ricci-tensor of a cohomogeneity-one metric can be expressed in terms of the Lie bracket. Therefore, we could state a lemma analogous to Lemma 3.2.36 on cohomogeneity-one Einstein metrics.

82 CHAPTER 3. SPACES WITH A LARGE ISOMETRY GROUP

Classification of the principal orbits

In this chapter, we classify the possible principal orbits of parallel cohomogeneity-one Spin 7 - structures. As we have seen in the previous chapter, those orbits are exactly the sevendimensional spaces carrying a homogeneous cosymplectic G_{2}-structure. We will therefore first classify all spaces admitting a homogeneous G_{2}-structure and then consider the question if any of those G_{2}-structures is cosymplectic. In Section 3.1, we have seen that a homogeneous space G H admits a G-invariant G_{2}-structure if and only if H G_{2}. In this formula, H acts on the tangent space by its isotropy representation and G_{2} acts as the stabilizer of the threeform. Before we classify the orbits G H, we therefore first have to classify the subgroups of G_{2}:

Lemma 4.1. As usual, let G_{2} be the simply connected Lie group whose Lie algebra is the compact real form of 𝔤_{2}. Furthermore, let H be a connected Lie subgroup of G_{2}. We denote the Lie algebra of H by 𝔥. The irreducible action of G_{2} on ℝ^{7} induces an action of H on ℝ^{7}. In this situation, 𝔥, H, and the action of H on ℝ^{7} are contained in the table below. Moreover, any two connected Lie subgroups of G_{2} whose action on ℝ^{7} is equivalent are conjugate not only by an element of GL 7 but by an element of G_{2}, too.

𝔥  H  Splitting of ℝ^{7} into irreducible summands
0  e
𝔲 1 U 1 𝕍_{a}^{ℂ} 𝕍_{b}^{ℂ} 𝕍_{c}^{ℂ} 𝕍_{0}^{ℝ} with a b c
2𝔲 1  U 1 ^{2} 𝕍_{1,0}^{ℂ} 𝕍_{0,1}^{ℂ} 𝕍_{1,1}^{ℂ} 𝕍_{0,0}^{ℝ}
𝔰𝔲 2  SU 2 𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ}
𝔰𝔲 2  SU 2 𝕍_{2}^{ℝ} 𝕍_{1}^{ℂ}
𝔰𝔲 2  SO 3 2𝕍_{2}^{ℝ} 𝕍_{0}^{ℝ}
𝔰𝔲 2  SO 3 𝕍_{6}^{ℝ}
𝔰𝔲 2 𝔲 1 U 2  𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ} w.r.t. 𝔰𝔲 2
𝔰𝔲 2 𝔲 1 U 2  𝕍_{2}^{ℝ} 𝕍_{1}^{ℂ} w.r.t 𝔰𝔲 2
2𝔰𝔲 2  SO 4 𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ} w.r.t. the first summand of 𝔥
𝕍_{2}^{ℝ} 𝕍_{1}^{ℂ} w.r.t. the second summand of 𝔥
𝔰𝔲 3  SU 3 𝕍_{1,0}^{ℂ} 𝕍_{0}^{ℝ}
𝔤_{2}  G_{2}  𝕍_{1,0}^{ℝ}

In the above table, the subscripts of the modules denote the weights of the H-action and the superscript indicates if the module is real or complex. Further details of the embeddings, in particular of those of U 2 and SO 4 into G_{2}, will be described in the proof below.

Proof : In the following, we will often consider G_{2} as the automorphism group of the octonions. Therefore, we will identify from now on ℝ^{7} with Im 𝕆 .

Since G_{2} is a compact Lie group, any closed subgroup of G_{2} has to be compact, too. 𝔥 therefore is the direct sum of an abelian and a semisimple Lie algebra. Furthermore, we have rank 𝔥 2 and dim 𝔥 14. The Lie algebras satisfying these criteria are:

• Rank 0: 0 ,

• Rank 1: 𝔲 1 , 𝔰𝔲 2 ,

• Rank 2: 2𝔲 1 , 𝔰𝔲 2 𝔲 1 , 2𝔰𝔲 2 , 𝔰𝔲 3 , 𝔰𝔬 5 , 𝔤_{2}.

The case 𝔥 𝔰𝔬 5 can be excluded as follows: We assume that 𝔰𝔬 5 𝔤_{2}. In this situation, 𝔰𝔬 5 would act on Im 𝕆 . Since the seven-dimensional representation of G_{2} is faithful, 𝔰𝔬 5 has to act non-trivially on Im 𝕆 . The spinor representation of 𝔰𝔬 5 is complex four-dimensional. Since there is no real spinor representation of 𝔰𝔬 5 , the only non-trivial irreducible real 𝔰𝔬 5 -module of dimension 7 is the 5-dimensional vector representation. Therefore, Im 𝕆 has to decompose into an irreducible five-dimensional module and a plane on which 𝔰𝔬 5 acts trivially. We assume without loss of generality that the two-dimensional trivial module is spanned by i and j. Since 𝔰𝔬 5 acts by automorphisms, k has to be invariant, too. This contradicts the fact that span i, j has to be the maximal trivial submodule of Im 𝕆 .

We will see that the remaining algebras from the above list can be embedded into 𝔤_{2}. Below, we describe the possible embeddings of those algebras into 𝔤_{2} in detail. Furthermore, we discuss the Lie subgroups of G_{2} associated to the Lie algebras. After that, we will consider the question if there are any further connected Lie subgroups of G_{2}. It will also be shown that the embeddings we have found are unique up to conjugation by an element of G_{2}. We begin with the program outlined above by considering each of the possible 𝔥 separately:

1. \underline{𝔥 0 :} The trivial algebra is clearly a subalgebra of 𝔤_{2}.

2. \underline{𝔥 𝔲 1 :} On page 22, we have described a Cartan subalgebra of 𝔤_{2} in detail. Any onedimensional Lie subalgebra of 𝔤_{2} is conjugate to a one-dimensional subalgebra of that Cartan subalgebra. Since G_{2} is compact, any one-dimensional Lie subgroup of G_{2} is isomorphic to U 1 .

3. \underline{𝔥 2𝔲 1 :} In this case, 𝔥 is the Cartan subalgebra described on page 22 or one of it conjugates. Any Lie subgroup of G_{2} whose Lie algebra is isomorphic to 2𝔲 1 is a maximal torus, which is isomorphic to U 1 ^{2}.

4. \underline{𝔥 𝔰𝔲 2 :} There are four subalgebras of 𝔤_{2} which are isomorphic to 𝔰𝔲 2 but pairwise non-conjugate. We will describe each of them in detail:

(a) We consider the automorphisms of 𝕆 which act trivially on the quaternions ℍ and map ϵ to hϵ, where h is a unit quaternion. These automorphisms can explicitly be described as:

x yϵ x hy ϵ for all x, y ℍ .

The proof that these maps are indeed automorphisms can be found in Cacciatori et al. [17]. The above formula defines a left action of Sp 1 on Im 𝕆 which is trivial on Im ℍ and irreducible on ℍϵ. Since the set of the union quaternions is isomorphic to SU 2 , we have described the first embedding SU 2 G_{2} stated in the lemma.

(b) We define for each unit quaternion h a map

Φ_{h} x yϵ : hxh^{1} yh^{1} ϵ for all x, y ℍ .

These maps define another left action of SU 2 on Im 𝕆 . In [17], it is proved that the Φ_{h} are automorphisms of 𝕆, too. SU 2 acts on Im ℍ as SO 3 on ℝ^{3}. On ℍϵ it acts irreducibly and faithfully. Therefore, the action of SU 2 on all of Im 𝕆 is faithful, too, and we have described the second embedding of SU 2 into G_{2} stated in the lemma. By identifying ℂ with ℝ^{2} we obtain a canonical embedding of 𝔰𝔲 2 into 𝔰𝔬 4 . We equip ϵℍ with the basis ϵ, iϵ, kϵ, jϵ . The matrices of the restriction of the above 𝔰𝔲 2 -action to ϵℍ with respect to that basis describe the orthogonal complement of 𝔰𝔲 2 𝔰𝔬 4 . Therefore, the first Lie subalgebra of type 𝔰𝔲 2 and this one commute.

(c) The automorphism group of ℍ is isomorphic to SO 3 . Let H be the set of those automorphisms of 𝕆 which map ℍ to itself and fix ϵ. H is obviously a group. Since any automorphism of ℍ can be extended to a unique automorphism φ of 𝕆 with φ ϵ ϵ, H is isomorphic SO 3 , too. It is easy to see that H acts irreducibly on Im ℍ and Im ℍ ϵ. This proves the decomposition of Im 𝕆 into 2𝕍_{2}^{ℝ} 𝕍_{0}^{ℝ} , which we have stated in the lemma.

(d) In a paper of Dynkin [31], it is proven that there are four subalgebras of 𝔤_{2} which are isomorphic to 𝔰𝔲 2 . In fact, the semisimple subalgebras of all semisimple Lie algebras are classified. Their embeddings are described in an abstract manner, which is not convenient for our considerations. Therefore, the work we do in this lemma is nevertheless necessary. The first three of the embeddings of 𝔰𝔲 2 into 𝔤_{2} which can be found in [31] are those we have described above. The fourth subalgebra acts irreducibly on Im 𝕆 and its associated Lie group is isomorphic to SO 3 . For our considerations, we only need the existence of this subgroup but not its explicit description. We therefore refer the reader to the literature for further details. Dynkin has classified the subalgebras up to inner automorphisms of the greater Lie algebra. Thus, we also know that there is up to conjugation by an element of G_{2} no other subalgebra of type 𝔰𝔲 2 which acts irreducibly on Im 𝕆 . For reasons of completeness, we will nevertheless explicitly prove that the other Lie algebras in the table of our lemma are uniquely determined up to conjugation by an automorphism of 𝕆.

In order to avoid confusion, we will index the Lie subalgebras of type 𝔰𝔲 2 by their non-zero weights. The above four algebras will therefore be denoted by 𝔰𝔲 2 _{1}, 𝔰𝔲 2 _{1,2}, 𝔰𝔲 2 _{2,2}, and 𝔰𝔲 2 _{6}.

5. \underline{𝔥 𝔰𝔲 2 𝔲 1 :} We describe the two subalgebras mentioned in the lemma in detail:

(a) We embed 𝔰𝔲 2 as 𝔰𝔲 2 _{1} into 𝔤_{2}. Furthermore, we choose a certain Φ 𝔤_{2}. Φ generates a Lie subalgebra of 𝔤_{2}, which is isomorphic to 𝔲 1 . The matrix representation of Φ with respect to the basis i, j, k, ϵ, iϵ, jϵ, kϵ shall be:

Φ : ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 & 2 \\ & 2 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \\ & & & & & 0 & 1 \\ & & & & & 1 & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; .

It is possible to show that Φ commutes with all of 𝔰𝔲 2 _{1} 𝔤_{2}. This proves that there is indeed a subalgebra of 𝔤_{2} which is isomorphic to 𝔰𝔲 2 𝔲 1 and acts in the same way as stated in the lemma on Im 𝕆 . The corresponding Lie group is finitely covered by SU 2 U 1 and the kernel of the covering map is ℤ_{2} : 1, 1 ; 1, 1 . Therefore, the Lie group which is associated to the Lie algebra we have constructed above is isomorphic to SU 2 U 1 ℤ_{2} U 2 .

(b) The 𝔰𝔲 2 -summand of the next subalgebra we consider is chosen as 𝔰𝔲 2 _{1,2}. We further choose the abelian summand as the Lie algebra generated by the following matrix:

Ψ : ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 & 0 \\ & 0 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \\ & & & & & 0 & 1 \\ & & & & & 1 & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; .

Since Ψ commutes with any element of 𝔰𝔲 2 _{1,2}, the direct sum 𝔰𝔲 2 _{1,2} 𝔲 1 is closed under the Lie bracket. The Lie subgroup of G_{2} whose Lie algebra is given by 𝔰𝔲 2 _{1,2} 𝔲 1 is isomorphic to U 2 for the same reasons as in the previous case.

6. \underline{𝔥 2𝔰𝔲 2 :} Again, we consider ℍ as a subalgebra of 𝕆. Let 𝔥 be the Lie algebra of all derivations of 𝕆 which leave ℍ invariant. It is easy to see that the first, second, and third subalgebra of type 𝔰𝔲 2 we have constructed above are contained in 𝔥. The first and the second of those algebras commute and their direct sum therefore is a Lie algebra isomorphic to 2𝔰𝔲 2 . The third Lie algebra 𝔰𝔲 2 _{2,2} is diagonally embedded into 2𝔰𝔲 2 . It can be shown that there is no subalgebra 𝔨 of 𝔤_{2} such that 2𝔰𝔲 2 𝔨 𝔤_{2}, i.e., 2𝔰𝔲 2 is a maximal subalgebra. From this fact, it follows that 𝔥 2𝔰𝔲 2 . We define a Lie group homomorphism φ by

\begin{matrix}φ : Sp 1 Sp 1 & G_{2} \\ φ h, k x yϵ & : hxh^{1} kyh^{1} ϵ . \end{matrix}

It is easy to see that the kernel of this homomorphism is 1, 1 ; 1, 1 . Therefore, the Lie group associated to 2𝔰𝔲 2 𝔤_{2} is isomorphic to SO 4 . Furthermore, we obtain the same splitting of Im 𝕆 with respect to the action of 2𝔰𝔲 2 as stated in the lemma. The idea for the construction of this subgroup we have taken from [17], too.

7. \underline{𝔥 𝔰𝔲 3 :} Let H be the group of all automorphisms of 𝕆 which fix i. For any h of unit length in the orthogonal complement of ℂ 𝕆 there is an automorphism Φ with Φ i i, Φ j h. We can further choose a unit octonion h^{}span 1, i, h, ih ^{}and require that Φ ϵ h^{}. The choice of Φ i , Φ j , and Φ ϵ makes Φ unique. Since any automorphism of 𝕆 which fixes i is of this type, H is a S^{3}-bundle over S^{5} and therefore simply connected. Furthermore, dim H dim S^{5} dim S^{3} 8. H acts irreducibly and faithfully on the six-dimensional space ℂ^{}and trivially on the one-dimensional space span i . The only possible action by a compact Lie group satisfying all these conditions is by SU 3 acting on Im 𝕆 as on ℂ^{3} ℝ.

8. \underline{𝔥 𝔤_{2}:} In this case, the statement of the lemma is trivially true.

It remains to show that there are up to conjugacy no other embeddings of the above groups into G_{2}. The cases 𝔥 0 and 𝔥 𝔤_{2} are trivial and the case where 𝔥 is abelian has already been handled above. Next, we consider the case 𝔥 𝔰𝔲 3 . The only real 𝔰𝔲 3 - representations of dimension 7 are the trivial one and the standard representation on ℂ^{3} ℝ^{6}. Since 𝔰𝔲 3 has to act faithfully on Im 𝕆 , the imaginary space has to decompose into a six- and a one-dimensional irreducible submodule. By conjugating 𝔥 by an element of G_{2}, we can assume that the trivial submodule is spanned by i. In this situation, H is a connected closed group of automorphisms which preserve i. Therefore, it has to be a subgroup of the group SU 3 G_{2} we have constructed above. Since dim 𝔥 8, H coincides with that group. The only remaining cases are 𝔥 𝔰𝔲 2 , 𝔰𝔲 2 𝔲 1 , or 2𝔰𝔲 2 . We start with the 𝔰𝔲 2 -case and first give a short introduction to the theory of real 𝔰𝔲 2 -representations:

Any irreducible complex representation of 𝔰𝔲 2 is equivalent to a symmetric power S_{ℂ}^{m} ℂ^{2} . These spaces either contain no non-trivial real 𝔰𝔲 2 -submodule or they split into two submodules of the same dimension. It is easy to see that S^{m} ℂ^{2} splits if and only if the conjugation map τ : S_{ℂ}^{m} ℂ^{2} S_{ℂ}^{m} ℂ^{2} commutes with the action of 𝔰𝔲 2 . In this situation, the two real 𝔰𝔲 2 -submodules are the eigenspaces of τ. By some calculations, we can prove that τ and 𝔰𝔲 2 commute if and only if m is even. In the even case, the Lie group associated to the action of 𝔰𝔲 2 is SO 3 and in the odd case it is SU 2 . We will denote the irreducible complex module on which 𝔰𝔲 2 acts with weight m shortly by 𝕍_{m}^{ℂ}. If 𝕍_{m}^{ℂ} splits into two real modules, we will denote each of them by 𝕍_{m}^{ℝ}. With help of the above considerations, we can show that the only irreducible real 𝔰𝔲 2 -modules of real dimension 7 can be described as follows:

Action of 𝔰𝔲 2  Name of the module Real dimension
Trivial action  𝕍_{0}^{ℝ}  1
Standard action of 𝔰𝔲 2 on ℂ^{2}  𝕍_{1}^{ℂ}  4
Standard action of 𝔰𝔬 3 on ℝ^{3}  𝕍_{2}^{ℝ}  3
Action of 𝔰𝔬 3 on the trace-free,  𝕍_{4}^{ℝ}  5

symmetric 3 3-matrices

Irreducible action of 𝔰𝔬 3 on Im 𝕆 𝕍_{6}^{ℝ}  7

as described above

We now prove that there are no splittings of Im 𝕆 with respect to a 𝔰𝔲 2 𝔤_{2} except those which we already have described. The case 𝕍_{4}^{ℝ} Im 𝕆 can be excluded by similar arguments as 𝔰𝔬 5 𝔤_{2}: We assume that 𝔰𝔲 2 is embedded into 𝔤_{2} such that Im 𝕆 contains 𝕍_{4}^{ℝ} as a 𝔰𝔲 2 -submodule. Since there exists no real two-dimensional irreducible 𝔰𝔲 2 -module, the orthogonal complement of 𝕍_{4}^{ℝ} is trivial. Without loss of generality, we can assume that the orthogonal complement is spanned by i and j. Since i and j are fixed by the action of the Lie group SO 3 G_{2}, which is associated to 𝔰𝔲 2 , and G_{2} acts by automorphisms, k is fixed, too. Therefore, Im 𝕆 contains a three-dimensional trivial submodule, which contradicts our assumption.

The only other case we have to exclude is Im 𝕆 𝕍_{2}^{ℝ} 4𝕍_{0}^{ℝ}. We assume that 𝔰𝔲 2 acts irreducibly on a three-dimensional subspace V of Im 𝕆 and trivially on its complement V^{}. Without loss of generality, we furthermore can assume that i and j are contained in V. Since i j, there exists a ψ 𝔰𝔲 2 with

\begin{matrix}ψ i & j \\ ψ j & i \\ ψ span i, j ^{} & 0 . \end{matrix}

ψ 𝔰𝔲 2 𝔤_{2} has to be a derivation. Therefore, we have:

ψ \begin{matrix}iϵ \\ \end{matrix}_{span i,j} ψ i ϵ iψ ϵ jϵ 0 .

Since this is a contradiction, the case Im 𝕆 𝕍_{2}^{ℝ} 4𝕍_{0}^{ℝ} can be excluded.

Before we can finish the 𝔰𝔲 2 -case, we have to prove that any two subalgebras of type 𝔰𝔲 2 which act by the same weights on Im 𝕆 are conjugate by an automorphism of 𝕆. We start with the subcase where the subalgebra, which we denote by 𝔥, splits Im 𝕆 into 𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ}. In this situation, there exists a four-dimensional subspace V Im 𝕆 and an orthonormal basis x, y, z, w of V such that the matrix representation of 𝔥 is the same as of 𝔰𝔲 2 𝔤𝔩 2, ℂ 𝔤𝔩 4, ℝ . If there exists an automorphism of 𝕆 which maps x, y, z, w to ϵ, iϵ, kϵ, jϵ , it follows that 𝔥 is conjugate to 𝔰𝔲 2 _{1} by an element of G_{2}. Since x, y is orthonormal, there is an automorphism which maps x, y to ϵ, iϵ . We therefore assume from now on that x ϵ and y iϵ. Let 𝔥 φ : V V be the linear map whose matrix representation with respect to x, y, z, w is

; ; ; \begin{aligned}0 & 1 \\ 1 & 0 \\ & & 0 & 1 \\ & & 1 & 0 \end{aligned} ; ; ; .

It is easy to see that

φ i φ ϵ iϵ φ ϵ iϵ ϵ φ iϵ iϵ iϵ ϵ ϵ 0 .

The space of all octonions on which φ acts trivially is a φ-invariant subspace of Im 𝕆 . Since φ 𝔰𝔬 7 , that subspace has to be orthogonal to V. The basis element z, which is orthogonal to span i, ϵ, iϵ , can be mapped by an automorphism to any other element of span i, ϵ, iϵ ^{}. We therefore can assume that z kϵ. Since ϵ, iϵ, kϵ is a basis triple, the action of any ψ 𝔥 on Im 𝕆 is determined by the matrix representation of ψ _{V} with respect to x, y, z, w . We finally have proven that there is up to conjugation by an element of G_{2} at most one subalgebra, namely 𝔰𝔲 2 _{1}, with the properties we have assumed. The uniqueness of 𝔰𝔲 2 _{1,2} and 𝔰𝔲 2 _{2,2} can be verified by similar arguments. For the 𝔰𝔲 2 _{6}-case, we already have referred to [31].

Next, we will show that there are no embeddings of U 2 SU 2 U 1 ℤ_{2} into G_{2} except those described above. Let x be a generator of the Lie algebra of U 1 . Since SU 2 and U 1 commute, the action φ_{x} of x on Im 𝕆 is a 𝔰𝔲 2 -equivariant map. We can apply Schur’s lemma and exclude some of the possible 𝔰𝔲 2 𝔤_{2} as the first summand of a Lie subalgebra isomorphic to 𝔰𝔲 2 𝔲 1 :

• \underline{𝔰𝔲 2 _{6}:} In this situation, φ_{x} has to be a multiplication by a real constant. The only multiple of the identity which is contained in 𝔤_{2} is 0. Since x has to act non-trivially, we can exclude this case.

• \underline{𝔰𝔲 2 _{2,2}:} Since x 𝔤_{2} 𝔰𝔬 7 , it has to act skew-symmetrically on Im 𝕆 . Therefore, the action of x on Im ℍ ϵIm ℍ has to be a multiple of

 ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 & & & 1 \\ & 0 & & & 1 \\ & & 0 & & & 1 \\ 1 & & & 0 \\ & 1 & & & 0 \\ & & 1 & & & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;

and the action on ϵ has to be trivial. If the φ_{x} defined above was a derivation, we had

φ_{x} i  iϵ
φ_{x} j  jϵ
φ_{x} k  kϵ

From the definition of the derivation property, it follows that:

φ_{x} ij φ_{x} i j iφ_{x} j
iϵ j i jϵ
2kϵ kϵ .

Therefore, we can exclude this case, too.

The above considerations prove that the only possibilities for the semisimple part of a Lie subalgebra of 𝔤_{2} which is isomorphic to 𝔰𝔲 2 𝔲 1 are those we have described in the lemma. Next, we will show that for any fixed choice of the 𝔰𝔲 2 -summand the abelian summand is unique up to conjugation by an element of G_{2}. Since the subalgebras 𝔰𝔲 2 _{1} and 𝔰𝔲 2 _{1,2} are unique up to conjugation, too, the uniqueness of the subalgebras 𝔰𝔲 2 _{1} 𝔲 1 and 𝔰𝔲 2 _{1,2} 𝔲 1 then will be proven. We first consider the 𝔰𝔲 2 _{1}-case. The abelian summand 𝔲 1 of 𝔰𝔲 2 _{1} 𝔲 1 has to act 𝔰𝔲 2 _{1}-equivariantly on Im 𝕆 . By the real version of Schur’s lemma, we see that the vector space of all 𝔰𝔲 2 _{1}-equivariant maps Im 𝕆 Im 𝕆 is given by M^{3 3} ℝ K. In the following, we will describe that space and in particular the summand K in more detail: The group SO 4 acts on ℍϵ. We consider its subgroup of type SU 2 whose Lie algebra commutes with 𝔰𝔲 2 _{1}. K acts on ℍϵ and is generated as a vector space by the above SU 2 and its real multiples. The first summand of M^{3 3} ℝ K acts canonically on Im ℍ . Since 𝔲 1 has to be a subalgebra of 𝔤_{2}, we have 𝔲 1 M^{3 3} ℝ K 𝔤_{2} 𝔰𝔲 2 _{1,2}. Conversely, we can take any one-dimensional subalgebra of 𝔰𝔲 2 _{1,2} as the abelian summand of 𝔰𝔲 2 _{1} 𝔲 1 . All in all, that summand is unique up to conjugation by an element of the Lie group SU 2 _{1,2} G_{2} which is associated to 𝔰𝔲 2 _{1,2} if the semisimple part of 𝔰𝔲 2 _{1} 𝔲 1 is fixed. The 𝔰𝔲 2 _{1,2} 𝔲 1 -case can be handled by similar arguments. The only thing we have to take care of is that 𝔰𝔲 2 _{1,2} acts on ℍϵ as the other summand of 𝔰𝔬 4 than 𝔰𝔲 2 _{1}.

Finally, we will show that 2𝔰𝔲 2 𝔤_{2} is unique up to conjugation by an automorphism. Let 𝔥 be an arbitrary subalgebra of 𝔤_{2} which is isomorphic to 2𝔰𝔲 2 . There is a subalgebra of 𝔥 which is isomorphic to 𝔰𝔲 2 𝔲 1 . After a conjugation, we can assume that this subalgebra is one of the two subalgebras of type 𝔰𝔲 2 𝔲 1 which we have constructed earlier in this proof. Therefore, either 𝔰𝔲 2 _{1} or 𝔰𝔲 2 _{1,2} is an ideal of 𝔥. We assume that 𝔰𝔲 2 _{1} ⊴ 𝔥. The second summand of 𝔥 has to commute with 𝔰𝔲 2 _{1}. We already have classified all derivations of 𝕆 which commute with 𝔰𝔲 2 _{1}. It therefore is easy to see that the second summand of 𝔥 has to be 𝔰𝔲 2 _{1,2}. If 𝔰𝔲 2 _{1,2} is an ideal of 𝔥, we can show by the same arguments that 𝔥 𝔰𝔲 2 _{1} 𝔰𝔲 2 _{1,2}. Therefore, the subalgebra 2𝔰𝔲 2 𝔤_{2} is unique up to conjugation, too.

We are now able to prove our theorem on the possible principal orbits. For each of the H G_{2} we have found, we search for the Lie groups G H such that dim G dim H 7 and H acts on the tangent space of G H in the same way as H G_{2} on Im 𝕆 . As we have stated in Convention 3.2.1, we require that G is compact and connected. G H is therefore compact and connected, too. Furthermore, it is determined up to a covering map by the Lie algebras 𝔤 and 𝔥 of G and H and the embedding of 𝔥 into 𝔤. Since G is compact, 𝔤 is the direct sum of a semisimple and an abelian Lie algebra. This will make the classification of the possible 𝔤 easier. We find the following list of homogeneous spaces admitting a G_{2}-structure:

Theorem 4.2. Let G H be a compact, connected seven-dimensional homogeneous space such that G is compact and connected, too. Furthermore, we assume that G H is the product of a simply connected space and a torus, G acts almost effectively on G H, and G H admits a G-invariant G_{2}-structure. Then, G H is either a product of a circle and a six-dimensional homogeneous space or G H cannot be described as a product of lower-dimensional homogeneous spaces. In the first case, G and H are up to a finite cover one of the groups in the table below, and G H can be found in that table, too:

G  H  G H
U 1 ^{7}  e  T^{7}
SU 2 U 1 ^{4}  e  S^{3} T^{4}
SU 2 ^{2} U 1  e  S^{3} S^{3} S^{1}
SU 2 ^{2} U 1 ^{2} U 1  S^{3} S^{3} S^{1}
SU 2 ^{2} U 1 ^{2} U 1  SU 2 ^{2} U 1 T^{2}
SU 2 ^{3} U 1 SU 2 S^{3} S^{3} S^{1}
SU 3 U 1 ^{2} SU 2  S^{5} T^{2}
SU 3 U 1 U 1 ^{2} SU 3 U 1 ^{2} S^{1}
Sp 2 U 1  Sp 1 U 1 ℂℙ^{3} S^{1}
G_{2} U 1  SU 3  S^{6} S^{1}

The information on the topology of G H given in the above table determines the embedding of H into G except in two cases. In the fourth (fifth) row, the embedding U 1 SU 2 ^{2} has to be special in order to make S^{3} S^{3} S^{1} (SU 2 ^{2} U 1 T^{2}) a space admitting a SU 2 ^{2} U 1 ^{2}-invariant G_{2}-structure. The detailed description of those embeddings and the space SU 2 ^{2} U 1 T^{2} can be found in the proof below.

If G H is not a product of lower-dimensional homogeneous spaces, we obtain the following table:

G  H  G H
SU 3  U 1  N^{k,l} with k, l ℤ
SO 5  SO 3  V^{5,2}
Sp 2  Sp 1  S^{7}
SO 5  SO 3  B^{7}
SU 2 ^{3}  U 1 ^{2}  Q^{1,1,1}
SU 3 U 1  U 1 ^{2}  N^{k,l} with k, l ℤ
SU 3 SU 2 SU 2 U 1 M^{1,1,0}
SU 3 SU 2 SU 2 U 1 N^{1,1}
Sp 2 U 1 Sp 1 U 1 S^{7}
Sp 2 Sp 1  Sp 1 Sp 1 S^{7}
SU 4  SU 3  S^{7}
Spin 7  G_{2}  S^{7}

As in the first case, G and H are only determined up to a finite cover. In the above table, the indices of M^{1,1,0} (Q^{1,1,1}, N^{k,l}) denote a special embedding of SU 2 U 1 (U 1 ^{2}, U 1 ) into SU 3 SU 2 (SU 2 ^{3}, SU 3 ). Those embeddings will be described explicitly in Section 5.3 (5.2, 5.4). V^{5,2} denotes the Stiefel-manifold of all orthonormal pairs in ℝ^{5}. B^{7} is the Berger space, which will be described in more detail in the proof below and in Section 5.1.

The converse of our statement is also true: Any of the above spaces G H admits a G-invariant G_{2}-structure.

Proof : In order to prove the theorem, we consider the different possibilities for the Lie algebra 𝔥 of H separately:

• \underline{𝔥 0 :} In this situation, G is simply a seven-dimensional Lie group with the properties we have required in the theorem. The only possibilities for G are:

1. U 1 ^{7}

2. SU 2 U 1 ^{4}

3. SU 2 ^{2} U 1

• \underline{𝔥 𝔲 1 :} In this case, G is an eight-dimensional compact Lie group. It therefore has to be up to a cover one the groups in the following list:

1. U 1 ^{8}

2. SU 2 U 1 ^{5}

3. SU 2 ^{2} U 1 ^{2}

4. SU 3

For our considerations, the following argument will often be helpful: Let 𝔥 be non-trivial. A Cartan subalgebra of 𝔥 has to act on the tangent space of G H as a subalgebra of the Cartan subalgebra (2.1) of 𝔤_{2}. Therefore, the subspace on which 𝔥 acts trivially is either one- or three-dimensional. Since 𝔥 acts trivially on the center of 𝔤, the dimension of the center is at most three. In the case 𝔥 𝔲 1 , which we now consider, we can exclude the first two possibilities for G by this argument.

Next, we investigate the case G SU 2 ^{2} U 1 ^{2}. Since 𝔥 has to act non-trivially on the tangent space, it cannot be a subgroup of the center. Therefore, the projection of 𝔥 onto 2𝔰𝔲 2 𝔤 has to be injective. We first consider the case where H SU 2 ^{2}. It follows from Lemma 3.1.12 that in this situation S^{3} S^{3} S^{1} is a H SU 2 ^{2} -fold cover of G H. Since we want G H to be the product of a simply connected space and a torus, H SU 2 ^{2} has to be trivial. In order to satisfy this condition, the embedding of H into SU 2 ^{2} U 1 ^{2} has to be of the following type:

e^{iφ} ψ_{1} e^{iφ} , ψ_{2} e^{iφ} ,

where ψ_{1} : U 1 SU 2 ^{2} and ψ_{2} : U 1 U 1 ^{2} are injective group homomorphisms. We assume without loss of generality that ψ_{2} e^{iφ} 1, e^{iφ} . There are infinitely many non-conjugate embeddings of U 1 into SU 2 ^{2} which can all be described up to conjugation by:

e^{iφ} ; ; ; \begin{aligned}e^{ikφ} & 0 \\ 0 & e^{ikφ} \\ & & e^{ilφ} & 0 \\ & & 0 & e^{ilφ} \end{aligned} ; ; ; with k, l ℤ .

We can prove by a short calculation that U 1 has in fact to be embedded by

ψ_{1} e^{iφ} : ; ; ; \begin{aligned}e^{iφ} & 0 \\ 0 & e^{iφ} \\ & & e^{iφ} & 0 \\ & & 0 & e^{iφ} \end{aligned} ; ; ;

Only in that case, H acts on the tangent space in the same way as a subgroup of G_{2} on Im 𝕆 . The quotient G H can be considered as SU 2 SU 2 U 1 , where SU 2 SU 2 U 1 acts by left-multiplication and ψ_{1} U 1 acts by right-multiplication. Next, we have to consider the case where H SU 2 ^{2}. In this case, the embedding of H into SU 2 has to be the same as above and G H is a coset space of type SU 2 ^{2} U 1 T^{2}.

G SU 3 is the only remaining group from the above list. In this case, we obtain the Aloff-Wallach spaces N^{k,l} which do all admit a homogeneous G_{2}-structure.

• \underline{𝔥 𝔰𝔲 2 :} If this is the case, G has to be 10-dimensional and the dimension of its center is 3. With help of the classification theorem for the compact Lie groups we see that the universal cover of G has to be from the following list

1. SU 2 ^{3} U 1

2. SU 3 U 1 ^{2}

3. Sp 2

We consider each case separately and start with the first one: No matter how we embed SU 2 into SU 2 ^{3}, the tangent space T_{p}G H consists only of three-dimensional irreducible and trivial submodules. The only 𝔰𝔲 2 𝔤_{2} which acts in that way on Im 𝕆 is 𝔰𝔲 2 _{2,2}. Therefore, T_{p}G H has to contain exactly two irreducible three-dimensional submodules. The only embedding of SU 2 into SU 2 ^{3} which induces this kind of action is described by

X { \atop }\begin{aligned}X \\ & X \\ & & X \end{aligned} { \atop }.

The resulting homogeneous space is diffeomorphic to S^{3} S^{3} S^{1}, where SU 2 ^{3} acts by a suitable action on S^{3} S^{3} and trivially on S^{1}.

In the second case, H has to be embedded into the SU 3 -factor. Any embedding of 𝔰𝔲 2 into 𝔰𝔲 3 induces a complex three-dimensional representation of 𝔰𝔲 2 . Since there is only one two- and one three-dimensional complex representation of 𝔰𝔲 2 , H is either SU 2 SU 3 or SO 3 SU 3 . We first consider the case SU 2 SU 3 . The quotient SU 3 SU 2 is a five-dimensional sphere. 𝔰𝔲 2 commutes with a certain onedimensional subalgebra of 𝔰𝔲 3 and acts irreducibly on the complement of 𝔲 2 𝔰𝔲 3 . The tangent space of SU 3 SU 2 U 1 ^{2} therefore splits into an irreducible fourdimensional and a trivial three-dimensional 𝔰𝔲 2 -module. Since this SU 2 -action is included in the list of Lemma 4.1, we have found another space admitting a homogeneous G_{2}-structure. Next, we consider the case SO 3 SU 3 . The only 3 3-matrices which commute with all of SO 3 are the multiples of the identity. Therefore, the orthogonal complement of 𝔰𝔬 3 𝔰𝔲 3 contains no trivial 𝔰𝔬 3 -module. The maximal trivial submodule of T_{p}SU 3 SO 3 U 1 ^{2} therefore is two-dimensional. By considering the possible embeddings of 𝔰𝔲 2 into 𝔤_{2}, which we have found in Lemma 4.1, we can exclude this case.

We finally consider the case where 𝔤 𝔰𝔭 2 𝔰𝔬 5 . An embedding of 𝔰𝔲 2 into 𝔰𝔬 5 is the same as a real five-dimensional orthogonal representation of 𝔰𝔬 3 . The possible splittings of ℝ^{5} with respect to 𝔰𝔬 3 are:

1. 𝕍_{1}^{ℂ} 𝕍_{0}^{ℝ}

2. 𝕍_{2}^{ℝ} 2𝕍_{0}^{ℝ}

3. 𝕍_{4}^{ℝ}

In the first case, 𝔰𝔬 3 is embedded into 𝔰𝔬 5 via the standard representation. The resulting simply connected homogeneous space SO 5 SO 3 is the Stiefel-manifold V^{5,2}. It can easily be seen that 𝔥 acts on the tangent space of V^{5,2} as 𝔰𝔲 2 _{2,2}. Therefore, we have to put this space on our list.

In the second case, 𝔰𝔲 2 is embedded into 𝔰𝔬 5 by 𝔰𝔲 2 𝔰𝔬 4 𝔰𝔬 5 . In order to make the resulting space simply connected, we choose G Sp 2 and H Sp 1 . The quotient G H is the sphere S^{7}. 𝔰𝔲 2 acts on its tangent space as 𝔰𝔲 2 _{1}. Therefore, S^{7} is a possible principal orbit, too.

We finally consider the case where 𝔰𝔬 3 acts on ℝ^{5} as on the trace-free symmetric 3 3-matrices. In this situation, 𝔥 acts irreducibly on the tangent space of the resulting simply connected space of type SO 5 SO 3 . Since 𝔰𝔲 2 _{6} is a subalgebra of 𝔤_{2}, we have found a further candidate for a principal orbit. The space we have constructed is the seven-dimensional Berger space B^{7}.

• \underline{𝔥 2𝔲 1 :} For any 𝔥 of rank 2 we have the following argument at hand: The Cartan subalgebra (2.1) of 𝔤_{2} splits Im 𝕆 into three two-dimensional irreducible modules and one one-dimensional module. Since 𝔥 has to act as a subalgebra of 𝔤_{2} on the tangent space, the subspace V on which 𝔥 acts trivially is at most one-dimensional. Moreover, dim V 1 is only possible if 𝔥 is contained in 𝔰𝔲 3 , which is the algebra of all derivations which vanish on i 𝕆. Since 𝔥 acts trivially on the center of 𝔤, the center is in that situation at most one-dimensional. If dim V 0, we even now that the center has to be trivial. Since we now assume that 𝔥 2𝔲 1 , we can conclude that 𝔤 has to be one of the following algebras:

1. 3𝔰𝔲 2

2. 𝔰𝔲 3 𝔲 1 There are several non-conjugate embeddings of U 1 ^{2} into SU 2 ^{3}. Quotients of SU 2 ^{3} by U 1 ^{2} will be considered in detail in Section 5.2. There, we will show that SU 2 ^{3} U 1 ^{2} admits only for a special embedding an invariant G_{2}-structure. We denote that homogeneous space by Q^{1,1,1}.

Next, we consider the case 𝔤 𝔰𝔲 3 𝔲 1 . There are two subcases. The first one is 𝔥 𝔰𝔲 3 . Since any Cartan subalgebra of 𝔰𝔲 3 is isomorphic to 2𝔲 1 , any two embeddings of 𝔥 into 𝔰𝔲 3 are conjugate. It can easily be proven that 2𝔲 1 acts by the same weights on the tangent space of SU 3 U 1 ^{2} U 1 as the Cartan subalgebra (2.1) of 𝔤_{2} on Im 𝕆 . Therefore, the space SU 3 U 1 ^{2} U 1 admits a homogeneous G_{2}-structure.

The other case we have to consider is 𝔥 𝔰𝔲 3 . We can assume by similar arguments as above that U 1 ^{2} is embedded into SU 3 U 1 by

e^{iφ_{1}}, e^{iφ_{2}} ψ e^{iφ_{1}} , e^{iφ_{2}} ,

where ψ : U 1 SU 3 is an injective group homomorphism. The resulting space G H is an Aloff-Wallach space with an action of SU 3 U 1 . It is easy to see that the action of U 1 ^{2} on the tangent space is the same as the action of the maximal torus of G_{2} on Im 𝕆 .

• \underline{𝔥 𝔰𝔲 2 𝔲 1 :} If 𝔥 is isomorphic to 𝔰𝔲 2 𝔲 1 , 𝔤 has to be 11-dimensional. Since the center of 𝔤 is at most one-dimensional, 𝔤 has to be one of the following:

1. 𝔰𝔲 3 𝔰𝔲 2

2. 𝔰𝔬 5 𝔲 1

The first case we consider is 𝔤 𝔰𝔲 3 𝔰𝔲 2 . In order to describe the possible embeddings of 𝔰𝔲 2 𝔲 1 into 𝔰𝔲 3 𝔰𝔲 2 , we introduce some notations: Let

\begin{aligned}i & : 𝔰𝔲 2 𝔰𝔲 3 \\ i A & : \begin{aligned}A \\ & 1 \end{aligned} \end{aligned}

Furthermore, let j : 𝔰𝔲 2 𝔰𝔲 3 be the differential of the embedding SO 3 SU 3 . Next, we describe the possible embeddings of 𝔰𝔲 2 𝔥 into 𝔤. We will deal with the embedding of the abelian summand of 𝔥 into 𝔤 later on. It is easy to see that there are only the following possibilities for 𝔰𝔲 2 𝔤:

1. 𝔰𝔲 2 is the second summand of 𝔰𝔲 3 𝔰𝔲 2 .

2. 𝔰𝔲 2 is embedded by the map i into the first summand of 𝔰𝔲 3 𝔰𝔲 2 .

3. 𝔰𝔲 2 is embedded by the map j into the first summand of 𝔰𝔲 3 𝔰𝔲 2 .

4. 𝔰𝔲 2 is embedded by a map i^{}into 𝔰𝔲 3 𝔰𝔲 2 , where the projection of i^{}onto the first summand is i and the projection onto the second summand is the identity.

5. 𝔰𝔲 2 is embedded by a map j^{}into 𝔰𝔲 3 𝔰𝔲 2 , where the projection of j^{}onto the first summand is j and the projection onto the second summand is the identity.

The first of these cases can easily be excluded, since in that situation 𝔰𝔲 2 would act trivially on the tangent space.

The homogeneous spaces we obtain in the second case will be considered in more detail in Section 5.3. In that section, we will see that the embedding of the center of 𝔰𝔲 2 𝔲 1 into 𝔰𝔲 3 𝔰𝔲 2 has to be special in order to make G H a space admitting a SU 3 SU 2 -invariant G_{2}-structure. The homogeneous space we obtain this way we denote by M^{1,1,0}. 𝔰𝔲 2 𝔲 1 acts in the same way as 𝔰𝔲 2 _{1} 𝔲 1 𝔤_{2} on the tangent space of M^{1,1,0}. In the proof of Lemma 4.1, we have shown that the kernel of the associated representation of SU 2 U 1 is isomorphic to ℤ_{2}. Nevertheless, we will describe M^{1,1,0} as a quotient of type SU 3 SU 2 SU 2 U 1 . This is possible, since we only require that G acts almost effectively on G H. The isotropy group SU 2 U 1 of the SU 3 SU 2 -action is not contained in G_{2}. Since the group SU 2 U 1 ℤ_{2}, which acts effectively on the tangent space, is a subgroup of G_{2}, this is not a contradiction to Lemma 3.1.47. The other homogeneous spaces which we obtain in the case 𝔥 𝔰𝔲 2 𝔲 1 we will also describe as quotients of type G SU 2 U 1 .

In the third case, we can apply similar arguments as in the case H SO 3 , G SU 3 U 1 ^{2}: It easily follows from Schur’s lemma that H SO 3 commutes with no elements of SU 3 except those in the center. Therefore, the orthogonal complement of 𝔰𝔲 2 𝔰𝔲 3 contains no trivial 𝔰𝔲 2 -submodule. The only five-dimensional 𝔰𝔲 2 - module which contains no trivial submodule is the irreducible one. 𝔤_{2} contains no subalgebra 𝔥 which is isomorphic to 𝔰𝔲 2 and acts by its five-dimensional irreducible representation on a subspace of Im 𝕆 . Therefore, we can exclude this case. The fifth embedding of 𝔰𝔲 2 into 𝔰𝔲 3 𝔰𝔲 2 can be excluded by the same arguments.

We now consider the fourth embedding of 𝔰𝔲 2 into 𝔰𝔲 3 𝔰𝔲 2 . Since both summands of 𝔰𝔲 2 𝔲 1 commute, the action of 𝔲 1 on the tangent space of G H has to be a 𝔰𝔲 2 -equivariant map. From this fact, we can easily conclude that the action of 𝔲 1 on the second summand of 𝔰𝔲 3 𝔰𝔲 2 is trivial. Therefore, the abelian factor of H is a subgroup of SU 3 . Since G SU 3 SU 2 , we have an SU 3 -action on the coset space G H. In the following, we will determine the orbit of this action. An element g SU 3 fixes the point eH G H if and only if g H. Since SU 3 H U 1 , the orbit of the SU 3 -action is an Aloff-Wallach space SU 3 U 1 . By a dimension argument, we see that the SU 3 -orbit has to be an open subset of G H. Since G H is connected, the orbit SU 3 U 1 has to be all of G H. Next, we will determine the embedding of U 1 into SU 3 . The U 1 -factor of H, which is a subgroup of SU 3 , is up to conjugation of the following type:

; ; { \atop }\begin{matrix}e^{ikt} & 0 & 0 \\ 0 & e^{ilt} & 0 \\ 0 & 0 & e^{i k l t} \end{matrix} { \atop } ; ; ; ; ; t ℝ ; ; with k, l ℤ .

Since H SU 2 U 1 is a direct product, the above abelian group has to commute with SU 2 SU 3 . This forces k to equal l. Therefore, G H has to be a so called exceptional Aloff-Wallach space, which we will denote as in Section 5.4 by N^{1,1}. In order to make sure that N^{1,1} carries a SU 3 SU 2 -invariant rather than only a SU 3 - invariant G_{2}-structure, we have to determine the action of SU 2 U 1 on the tangent space. By an explicit calculation, we see that the tangent space of N^{1,1} decomposes into two SU 2 -modules, which are isomorphic to 𝕍_{1}^{ℂ} and 𝕍_{2}^{ℝ}. Furthermore, it is possible to prove that the center of 𝔥 acts in the same way on the tangent space as the second summand of 𝔰𝔲 2 _{1,2} 𝔲 1 𝔤_{2} on Im 𝕆 . Therefore, we have found another description of N^{1,1} as a coset space, which we will include in our list.

We finally consider the case 𝔤 𝔰𝔬 5 𝔲 1 . The projection of the center of 𝔥 onto 𝔰𝔬 5 has to be injective. Otherwise, the action of G on G H would not be almost effective. First, we consider the case where the projection of the center onto both summands of 𝔰𝔬 5 𝔲 1 is injective. In that situation, it follows from Lemma 3.1.12 that G H is covered by Sp 2 Sp 1 . Since we want G H to be simply connected, G H has to be the seven-sphere equipped with an action of Sp 2 U 1 . Below, we will construct a transitive action of Sp 2 Sp 1 on S^{7}, which preserves a G_{2}-structure. Since the action we have described above is a restriction of that Sp 2 Sp 1 -action, it preserves a G_{2}-structure, too.

Next, we consider the case where 𝔰𝔲 2 𝔲 1 is embedded into the 𝔰𝔬 5 -summand of 𝔤. As we have seen earlier in this proof, there are three possible embeddings of 𝔰𝔲 2 into 𝔰𝔬 5 . We will consider each of those embeddings separately. If 𝔰𝔲 2 was embedded by its five-dimensional real representation into 𝔰𝔬 5 , it would act irreducibly on the center of 𝔥, which has to be a subalgebra of 𝔰𝔬 5 , too. The induced action of 𝔥 on ℝ^{5} therefore would be a 𝔰𝔲 2 -equivariant map. Since any of those maps has to be a multiple of the identity, we can exclude that case. Next, we assume that 𝔰𝔲 2 is embedded by its three-dimensional real representation into 𝔰𝔬 5 . In that situation, 𝔰𝔲 2 would act as 𝔰𝔲 2 _{2,2} on the tangent space of G H. Since the 𝔰𝔲 2 -summand of any subalgebra 𝔰𝔲 2 𝔲 1 𝔤_{2} does not act in this way on Im 𝕆 , we can exclude this case, too. The only remaining possibility is that 𝔰𝔲 2 is embedded by its twodimensional complex representation into 𝔰𝔬 5 . Since the rank of 𝔰𝔬 5 is two, there is only one choice for the embedding of the center of 𝔥 into 𝔰𝔬 5 . Since 𝔰𝔲 2 𝔰𝔬 5 splits its orthogonal complement into 𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ}, 𝔥 has to act as 𝔰𝔲 2 _{1} 𝔲 1 on the tangent space. By a short calculation, it is possible to show that this is indeed the case. We obtain G H Sp 2 Sp 1 U 1 U 1 . Since Sp 2 Sp 1 is S^{7}, Sp 2 Sp 1 U 1 U 1 is diffeomorphic to ℂℙ^{3} S^{1}.

• \underline{𝔥 2𝔰𝔲 2 :} In this case, G has to be a 13-dimensional compact Lie group containing a subgroup of type SO 4 . The action of 𝔥 on the tangent space of G H has to be the same as the action of 2𝔰𝔲 2 𝔤_{2} on Im 𝕆 , which we have described in Lemma 4.1. There is no non-trivial subspace of Im 𝕆 on which 2𝔰𝔲 2 𝔤_{2} acts trivially. Since 𝔥 acts trivially on the center of 𝔤, the center has to be 0 . By these arguments, we can easily exclude all candidates for 𝔤 except 𝔰𝔬 5 𝔰𝔲 2 . The tangent space of G H has to split into a four- and a three-dimensional irreducible submodule with respect to the action of 2𝔰𝔲 2 . On the four-dimensional submodule, 2𝔰𝔲 2 has to act as the standard representation of 𝔰𝔬 4 . On the three-dimensional submodule, one of the summands of 2𝔰𝔲 2 has to act as the standard representation of 𝔰𝔬 3 , and the other one has to act trivially. Let i : 2𝔰𝔲 2 𝔰𝔬 5 𝔰𝔲 2 be the embedding of 𝔥 into 𝔤. It is easy to see that the projection of i onto 𝔰𝔬 5 has to be the canonical embedding of 𝔰𝔬 4 into 𝔰𝔬 5 . The projection i^{}: 2𝔰𝔲 2 𝔰𝔲 2 of i onto the second summand has to be given by i^{}x, y x, 0 . We are able to describe the space G H explicitly:

Let S^{7} ℍ^{2} be the seven-sphere. Sp 2 acts on S^{7} by matrix multiplication. Any unit quaternion h Sp 1 acts on S^{7} by scalar multiplication by h^{1} . Since the scalar multiplication on a quaternionic vector space acts from the right, the Sp 1 -action we have defined is a left action. Since both actions commute, we have constructed an action of Sp 2 Sp 1 on S^{7}. By a straightforward calculation, we see that the isotropy group of this action at 1, 0 ^{T} S^{7} is isomorphic to Sp 1 Sp 1 . The tangent space of S^{7} at 1, 0 ^{T} is Im ℍ ℍ and the action of Sp 1 Sp 1 on the tangent space can explicitly be described by

h_{1}, h_{2}  \begin{matrix}x \\ y \end{matrix}  \begin{matrix}h_{1}xh_{1}^{1} \\ h_{2}yh_{1}^{1} \end{matrix}  ,

where h_{1} and h_{2} are unit quaternions. On the Lie algebra level, we obtain the same action as of 2𝔰𝔲 2 𝔤_{2} on Im 𝕆 . We therefore include the action of Sp 2 Sp 1 on S^{7} in the list of our theorem. The group Sp 2 Sp 1 does not act effectively but almost effectively on the sphere. Furthermore, Sp 1 Sp 1 ℤ_{2} is a subgroup of G_{2} but not its universal cover. For the same reasons as in the case 𝔥 𝔰𝔲 2 𝔲 1 , we nevertheless can put the action of Sp 2 Sp 1 instead of Sp 2 Sp 1 ℤ_{2} on our list.

• \underline{𝔥 𝔰𝔲 3 :} Since 𝔰𝔲 3 acts irreducibly on a six-dimensional subspace of Im 𝕆 , we can conclude by the usual arguments that the center of 𝔤 is at most one-dimensional. Moreover, 𝔤 has to be 15-dimensional. There are the following possibilities for 𝔤:

1. 𝔰𝔲 3 𝔰𝔲 2 𝔲 1

2. 𝔤_{2} 𝔲 1

3. 𝔰𝔲 4 ( 𝔰𝔬 6 )

The first case can be excluded, since in that situation 𝔰𝔲 3 would act trivially on the orthogonal complement of 𝔥 𝔤. In the second case, we have G H G_{2} SU 3 U 1 S^{6} S^{1} and in the third case G H SU 4 SU 3 S^{7}. In both cases, the tangent space of G H decomposes into a six-dimensional irreducible and a trivial onedimensional SU 3 -module. Therefore, both spaces admit a G-invariant G_{2}-structure.

• \underline{𝔥 𝔤_{2}:} If this is the case, 𝔤 is 21-dimensional and the center of 𝔤 vanishes. Furthermore, 𝔤 contains an ideal whose dimension is 14. Therefore, 𝔤 is one the following three algebras:

1. 𝔰𝔬 6 2𝔰𝔲 2

2. 𝔰𝔬 7

3. 𝔰𝔭 3

If 𝔤_{2} was a subalgebra of 𝔰𝔬 6 (𝔰𝔭 3 ), 𝔤_{2} would have a six-dimensional real (threedimensional quaternionic) representation. Since this is not the case, the only remaining possibility is that G H is the sphere S^{7} Spin 7 G_{2}. S^{7} admits a Spin 7 -invariant G_{2}-structure, which coincides at T_{1}S^{7} Im 𝕆 with the three-form ω ^{3} Im 𝕆 ^{} defined in Section 2.1. This invariant G_{2}-structure is unique up to multiplication by a constant.

Remark 4.3. 1. We call those principal orbits which are not a product of lower-dimensional homogeneous spaces irreducible principal orbits. The other principal orbits, which are contained in the first table of our theorem, we call reducible principal orbits.

2. Some spaces appear more than once in the lists of Theorem 4.2. Namely, this happens for the following spaces:

(a) S^{3} S^{3} S^{1} appears as SU 2 ^{2} U 1 , SU 2 ^{2} U 1 ^{2} U 1 , and SU 2 ^{3} U 1 SU 2 in our list.

(b) N^{k,l} can be described as SU 3 U 1 and as SU 3 U 1 U 1 ^{2}.

(c) The space N^{1,1} has a further description as SU 3 SU 2 SU 2 U 1 .

(d) On the sphere S^{7}, there are the following five transitive actions, which all leave a certain G_{2}-structure invariant: Sp 2 Sp 1 , Sp 2 U 1 Sp 1 U 1 , Sp 2 Sp 1 Sp 1 Sp 1 , SU 4 SU 3 , and Spin 7 G_{2}.

We can conclude from the existence of the different transitive actions that in some cases G is not the whole group of diffeomorphisms of G H which leave the G_{2}-structure invariant.

3. There are further examples of diffeomorphisms between the spaces which we have included in our theorem. For example, some of the Aloff-Wallach spaces are diffeomorphic, although the actions of SU 3 are inequivalent and should therefore be treated separately. Further details on this phenomenon can be found in Section 5.4 or in the paper of Kreck and Stolz [51].

4. It is not a priori clear if there are parallel cohomogeneity-one Spin 7 -manifolds with principal orbit G^{}H^{}S^{1} which are not a Riemannian product of a circle and a parallel cohomogeneity-one G_{2}-manifold. Therefore, we will mostly focus on the irreducible principal orbits.

Our next step is to prove which of the above spaces admit not only a homogeneous G_{2}- structure but also a coclosed homogeneous one. Fortunately, this work has been done in part by other authors:

Remark 4.4. 1. In a paper of Friedrich, Kath, Moroianu, and Semmelmann [37] all simply connected homogeneous spaces admitting a nearly parallel G_{2}-structure are classified. The list of those spaces coincides with the second list in the above theorem. Since any nearly parallel G_{2}-structure is coclosed, all of the spaces of those list are possible principal orbits.

2. Next, we consider those G H which are a product G^{}H^{}S^{1} of a circle and a sixdimensional homogeneous space. G H admits a G_{2}-structure if and only if G^{}H^{}admits a SU 3 -structure. This fact can be shown by the same methods as Lemma 3.2.27 and 3.2.28. The basic reason behind this is that the group of all automorphisms of 𝕆 fixing i Im 𝕆 is isomorphic to SU 3 . Six-dimensional spaces admitting a homogeneous SU 3 -structure are important in another context: Those spaces are possible principal orbits for cohomogeneity-one G_{2}-structures. For this reason, Cleyton and Swann [20] study that kind of spaces, too. They obtain a list of homogeneous spaces which includes all of the G^{}H^{}from our theorem except SU 2 ^{2} U 1 U 1 , which is included in our theorem, but seems to be missing in [20]. For many of those six-dimensional spaces it is known that they are also possible principal orbits for parallel cohomogeneity-one G_{2}-structures. (See, for example, the calculations carried out in [20].) Therefore, the Riemannian products of those G_{2}-structures with a circle are parallel Spin 7 -manifolds and G^{}H^{}S^{1} is a possible principal orbit for our considerations. Since the reducible principal orbits are not the main subject of this thesis, we will not prove the existence of a homogeneous coclosed G_{2}-structure on each of those spaces.

In the following, we will choose some of the G H from Theorem 4.2 and construct parallel cohomogeneity-one Spin 7 -structures with principal orbit G H.

Parallel Spin 7 -manifolds with irreducible principal orbits

5.1 The principal orbit B^{7}

In this chapter, we consider parallel Spin 7 -manifolds of cohomogeneity one whose principal orbit is irreducible. For many of the possible cases from the list of Theorem 4.2, we will deduce the equations for the holonomy reduction and discuss its possible solutions. Each of the following sections will deal with a certain principal orbit. The order of the sections will reflect the complexity of the techniques we have to apply.

We first take a look at cohomogeneity-one manifolds whose principal orbit is the Berger space B^{7} : SO 5 SO 3 . The embedding of SO 3 into SO 5 is given by the five-dimensional irreducible representation of SO 3 . First, we recall how this representation can be explicitly described. Let W be the space of all trace-free, symmetric 3 3-matrices. It is easy to see that W is five-dimensional and that the action of SO 3 on W by conjugation is irreducible and orthogonal with respect to the inner product X, Y tr XY on W. This action therefore induces an injective Lie group homomorphism SO 3 SO 5 .

The biinvariant metric on 𝔰𝔬 5 is as in Section 3.1 denoted by q and 𝔪 shall be the qorthogonal complement of 𝔰𝔬 3 𝔰𝔬 5 . The tangent space of B^{7} we identify with 𝔪. Next, we will describe how 𝔰𝔬 3 acts on the tangent space. The five-dimensional 𝔰𝔬 3 - module W has the weight 4 and 𝔰𝔬 5 is 𝔰𝔬 3 -equivariantly isomorphic to ^{2} W. With help of the Clebsch-Gordan formula, we decompose this module into the following irreducible submodules: ^{2} W 𝕍_{6}^{ℝ} 𝕍_{2}^{ℝ}, where 𝕍_{n}^{ℝ} denotes as usual the irreducible real 𝔰𝔬 3 -module with weight n. Since 𝕍_{2}^{ℝ} is the adjoint representation of 𝔰𝔬 3 , 𝔪 has to be isomorphic to 𝕍_{6}^{ℝ}. This proves that B^{7} is isotropy-irreducible. It easily follows from Schur’s lemma that there is up to a constant factor only one SO 5 -invariant metric on B^{7}. This metric is given by the restriction of q to 𝔪 and will be denoted by q, too.

The metric q plays an important role in another context: It is one of the few examples of a homogeneous metric with positive sectional curvature. This fact has been discovered by Berger [9], which explains the name ”Berger space”.

Our next step is to describe the set of all SO 5 -invariant G_{2}-structures on B^{7}. In order to do this, we first describe the space of all invariant three-forms. ^{3} 𝔪 decomposes into 𝕍_{12}^{ℝ} 𝕍_{8}^{ℝ} 𝕍_{6}^{ℝ} 𝕍_{4}^{ℝ} 𝕍_{0}^{ℝ}. Since ^{3} 𝔪 contains only one trivial submodule, there is up to a constant factor only one invariant three-form ω on B^{7}. In Chapter 4, we have already shown that B^{7} admits an invariant G_{2}-structure. Therefore, ω has to be a three-form defining a G_{2}-structure.

We now deduce the equation for the holonomy reduction. The 𝔰𝔬 3 -modules ^{3} 𝔪 and ^{4} 𝔪 are equivalent. An isomorphism of these modules is given by the Hodge star operator . The exterior differential d : ^{3} 𝔪 ^{4} 𝔪 is a 𝔰𝔬 3 -equivariant map. dω has to be an element of the trivial submodule of ^{4} 𝔪. We therefore have the equation:

dω λ ω

for a constant λ ℝ. Since ^{5} 𝔪 ^{2} 𝔪 𝕍_{6}^{ℝ} 𝕍_{2}^{ℝ} and d ω has to be 𝔰𝔬 3 -invariant, we have d ω 0. This proves that ω is a nearly parallel G_{2}-structure. As we have seen in Proposition 2.2.15, B^{7} is an Einstein manifold with Einstein constant \frac{3}{8}λ^{2}. λ has to be non-zero, since we have shown in Theorem 3.1.51 that a Ricci-flat homogeneous space is necessarily flat.

Because of Corollary 3.2.34, the only torsion-free Spin 7 -structures of cohomogeneity one with principal orbit B^{7} are cones over that space. Since B^{7} is not a sphere, the resulting space of cohomogeneity one has a singularity at the singular orbit.

5.2. THE PRINCIPAL ORBIT Q^{1,1,1} 103

5.2 The principal orbit Q^{1,1,1}

The topic of this section are parallel Spin 7 -manifolds with a cohomogeneity-one action of SU 2 ^{3} which preserves the associated four-form. From the proof of Theorem 4.2, it follows that the isotropy group of the action of SU 2 ^{3} on the principal orbits is U 1 ^{2}. It will turn out that a space of type SU 2 ^{3} U 1 ^{2} admits a homogeneous G_{2}-structure for a special embedding of U 1 ^{2} into SU 2 ^{3} only. This fact already has been stated in Theorem 4.2 but was left unproven. The homogeneous space of type SU 2 ^{3} U 1 ^{2} we obtain will be denoted by Q^{1,1,1}. We will search for parallel cohomogeneity-one Spin 7 -structures with this space as principal orbit. The metrics we will find have been considered in the literature before, for example, in Cvetič et al. [21], [25]. Nevertheless, we will introduce a new, more algebraic point of view. Furthermore, we will prove a classification result which guarantees that there are no further metrics of this kind.

Our first aim is to describe the special embedding of U 1 ^{2} into SU 2 ^{3} mentioned above. We will represent the elements of SU 2 ^{3} by complex 6 6-matrices:

SU 2 ^{3} : ; ; { \atop }\begin{aligned}X \\ & Y \\ & & Z \end{aligned} { \atop } ; ; ; ; ; X, Y, Z SU 2 ; ; .

Since U 1 ^{2} and SU 2 ^{3} are both connected, an embedding ı : U 1 ^{2} SU 2 ^{3} is determined by its differential dı _{e} : 2𝔲 1 3𝔰𝔲 3 . In order to describe dı _{e}, we first consider the possible Lie algebra morphisms 𝔲 1 3𝔰𝔲 3 . We assume that the image of 𝔲 1 is the Lie algebra of a closed subgroup of SU 2 ^{3}. In this situation, any of these morphisms is up to a conjugation and a multiplication by a constant given by:

\begin{aligned}i_{k,l,m} & : 𝔲 1 3𝔰𝔲 2 with k, l, m ℤ and \\ i_{k,l,m} ix & : ; ; ; ; ; ; ; \begin{aligned}ikx & 0 \\ 0 & ikx \\ & & ilx & 0 \\ & & 0 & ilx \\ & & & & imx & 0 \\ & & & & 0 & imx \end{aligned} ; ; ; ; ; ; ; . \end{aligned}

Without loss of generality, we can assume that k, l, m are coprime. Furthermore, we can even restrict ourselves to non-negative values of k, l, and m: Let ϕ_{P} : SU 2 ^{3} SU 2 ^{3} be defined by ϕ_{P} Q : P QP^{1} , where P SU 2 ^{3} is the following matrix:

P : ; ; ; ; ; ; ; \begin{aligned}0 & 1 \\ 1 & 0 \\ & & 1 & 0 \\ & & 0 & 1 \\ & & & & 1 & 0 \\ & & & & 0 & 1 \end{aligned} ; ; ; ; ; ; ; .

It is easy to see that ϕ_{P} maps i_{k,l,m} 𝔲 1 into i_{k,l,m} 𝔲 1 . By a similar choice of P, we can map i_{k,l,m} 𝔲 1 into i_{k, l,m} 𝔲 1 or i_{k,l, m} 𝔲 1 . Therefore, our restriction to the case where k, l, m 0 is justified. It is also possible to choose P as an arbitrary permutation of the three components of ℂ^{2} ^{3}. Since the group which is generated by the corresponding ϕ_{P} acts on k, l, m by permutations, too, we finally can assume that k l m 0. There is a deeper reason behind this argument: The symmetry group of the root system A_{1} A_{1} A_{1} is ℤ_{2} S_{3}. This group is also the group of outer isomorphisms of SU 2 ^{3}. It can be generated by the maps ϕ_{P} which we have constructed above. Since the automorphism group of SU 2 ^{3} is generated by the outer and the inner automorphisms, we have found the strongest possible restriction on k, l, m . We continue describing the embeddings dı _{e} : 2𝔲 1 3𝔰𝔲 2 . A Cartan subalgebra of 3𝔰𝔲 2 is given by:

; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}ix & 0 \\ 0 & ix \\ & & iy & 0 \\ & & 0 & iy \\ & & & & iz & 0 \\ & & & & 0 & iz \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; x, y, z ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

This algebra is denoted throughout this section by 3𝔲 1 . We can assume without loss of generality that dı _{e} 2𝔲 1 3𝔲 1 . The equation q X, Y : tr XY defines a biinvariant metric q on SU 2 ^{3}. The q-orthogonal complement of i_{k,l,m} 𝔲 1 3𝔲 1 we denote by 2𝔲 1 _{k,l,m} and the Lie subgroup of SU 2 ^{3} whose Lie algebra is 2𝔲 1 _{k,l,m} by U 1 _{k,l,m}^{2}. The quotient SU 2 ^{3} U 1 _{k,l,m}^{2} is called Q^{k,l,m}. We remark that since U 1 ^{2} SU 2 ^{3} is compact, i_{k,l,m} 𝔲 1 3𝔰𝔲 2 had to be chosen as a Lie subalgebra whose associated Lie subgroup is compact, too. Let e_{1}, …, e_{9} be the following basis of 3𝔰𝔲 2 :

e_{1} : ; ; ; ; ; ; ; ; \begin{aligned}0 & \frac{1}{2}i \\ \frac{1}{2}i & 0 \\ & & 0 \\ & & & 0 \\ & & & & 0 \\ & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; , e_{2} : ; ; ; ; ; ; ; ; \begin{aligned}0 & \frac{1}{2} \\ \frac{1}{2} & 0 \\ & & 0 \\ & & & 0 \\ & & & & 0 \\ & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ,

e_{3} : ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 \\ & & 0 & \frac{1}{2}i \\ & & \frac{1}{2}i & 0 \\ & & & & 0 \\ & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; , e_{4} : ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 \\ & & 0 & \frac{1}{2} \\ & & \frac{1}{2} & 0 \\ & & & & 0 \\ & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ,

e_{5} : ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 \\ & & 0 \\ & & & 0 \\ & & & & 0 & \frac{1}{2}i \\ & & & & \frac{1}{2}i & 0 \end{aligned} ; ; ; ; ; ; ; ; , e_{6} : ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 \\ & & 0 \\ & & & 0 \\ & & & & 0 & \frac{1}{2} \\ & & & & \frac{1}{2} & 0 \end{aligned} ; ; ; ; ; ; ; ; ,

e_{7} : ; ; ; ; ; ; ; \begin{aligned}\frac{1}{2}ki & 0 \\ 0 & \frac{1}{2}ki \\ & & \frac{1}{2}li & 0 \\ & & 0 & \frac{1}{2}li \\ & & & & \frac{1}{2}im & 0 \\ & & & & 0 & \frac{1}{2}im \end{aligned} ; ; ; ; ; ; ; , e_{8} : ; ; ; ; ; ; ; \begin{aligned}\frac{1}{2}li & 0 \\ 0 & \frac{1}{2}li \\ & & \frac{1}{2}ki & 0 \\ & & 0 & \frac{1}{2}ki \\ & & & & 0 & 0 \\ & & & & 0 & 0 \end{aligned} ; ; ; ; ; ; ; ,

e_{9} : ; ; ; ; ; ; ; \begin{aligned}\frac{1}{2}mki & 0 \\ 0 & \frac{1}{2}mki \\ & & \frac{1}{2}mli & 0 \\ & & 0 & \frac{1}{2}mli \\ & & & & \frac{1}{2} k^{2} l^{2} i & 0 \\ & & & & 0 & \frac{1}{2} k^{2} l^{2} i \end{aligned} ; ; ; ; ; ; ; .

The above basis is orthogonal with respect to q. As in Section 3.1, we denote the q-orthogonal complement of 2𝔲 1 _{k,l,m} 3𝔰𝔲 2 by 𝔪. The 2𝔲 1 _{k,l,m}-module 𝔪 can be equivariantly identified with the tangent space of Q^{k,l,m}. It is easy to see that e_{1}, …, e_{7} is a basis of 𝔪, and e_{8}, e_{9} is a basis of 2𝔲 1 _{k,l,m}. We determine the matrices of ad_{e_{8}} _{𝔪} and ad_{e_{9}} _{𝔪} with respect to the basis e_{1}, …, e_{7} . For ad_{e_{8}} _{𝔪} we obtain:

; ; ; ; ; ; ; ; ; ; \begin{aligned}0 & l \\ l & 0 \\ & & 0 & k \\ & & k & 0 \\ & & & & 0 & 0 \\ & & & & 0 & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; ,

and for ad_{e_{9}} _{𝔪} we obtain:

; ; ; ; ; ; ; ; ; ; \begin{aligned}0 & mk \\ mk & 0 \\ & & 0 & ml \\ & & ml & 0 \\ & & & & 0 & k^{2} l^{2} \\ & & & & k^{2} l^{2} & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; .

The action of the isotropy algebra 2𝔲 1 _{k,l,m} on 𝔪 yields the following subalgebra of 𝔤𝔩 𝔪 :

ad_{2𝔲 1 _{k,l,m}} _{𝔪} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 & x \\ x & 0 \\ & & 0 & y \\ & & y & 0 \\ & & & & 0 & z \\ & & & & z & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; kx ly mz 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

We equip ℝ^{7} with the action of the Cartan subalgebra

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & x \\ & x & 0 \\ & & & 0 & y \\ & & & y & 0 \\ & & & & & 0 & z \\ & & & & & z & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; x y z 0 ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; (5.1)

of 𝔤_{2}, which we have defined on page 22. We assume that there exists a SU 2 ^{3}-invariant G_{2}-structure on Q^{k,l,m}. Let p Q^{k,l,m} be arbitrary. An element of the fiber π^{1} p of the G_{2}-structure can be identified with a map ψ : 𝔪 ℝ^{7}. ψ has to be a 2𝔲 1 -equivariant isomorphism of the 2𝔲 1 -modules 𝔪 and ℝ^{7}. We therefore have:

ψ ad_{2𝔲 1 _{k,l,m}} _{𝔪} ψ^{1} 2𝔲 1 𝔤_{2} .  (5.2)

Since the weights of the action of 2𝔲 1 are invariant with respect to ψ, we also have

ψ ad_{2𝔲 1 _{k,l,m}} _{𝔪} ψ^{1} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 \\ & 0 & x \\ & x & 0 \\ & & & 0 & y \\ & & & y & 0 \\ & & & & & 0 & z \\ & & & & & z & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; k^{}x l^{}y m^{}z 0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ,

where k^{}, l^{}, m^{}equals k, l, m up to signs and a permutation. In order to satisfy (5.2), Q^{k,l,m} has to be one of the spaces Q^{1, 1, 1} . Since all of those spaces are SU 2 ^{3}-equivariantly diffeomorphic, it suffices to focus on the space Q^{1,1,1}. Consequently, whenever we consider a Lie subgroup U 1 ^{2} SU 2 ^{3} or a Lie subalgebra 2𝔲 1 3𝔰𝔲 2 in this section, the embedding shall be given by U 1 _{1,1,1}^{2} SU 2 ^{3} or 2𝔲 1 _{1,1,1} 3𝔰𝔲 2 . For our considerations, we have to classify the SU 2 ^{3}-invariant metrics on Q^{1,1,1}. 𝔪 splits into the following irreducible 2𝔲 1 -submodules:

V_{1} : span e_{1}, e_{2}
V_{2} : span e_{3}, e_{4}
V_{3} : span e_{5}, e_{6}
V_{4} : span e_{7}

In order to describe the possible metrics, we have to check if any pair of the above 2𝔲 1 - modules is equivalent. It is easy to see that V_{1}, V_{2}, and V_{3} are pairwise inequivalent, since on any pair of those spaces either the one-dimensional Lie algebra generated by e_{8} or the Lie algebra generated by e_{9} acts with different weights. V_{4} cannot be equivalent to one of the other modules, since its dimension is one.

We are now able to classify the SU 2 ^{3}-invariant metrics on Q^{1,1,1} by Schur’s lemma. Any such metric g can be identified with a 2𝔲 1 -equivariant endomorphism φ : 𝔪 𝔪 satisfying q φ X , Y g X, Y for all X, Y 𝔪. Since φ has to be symmetric with respect to q, the restriction of φ to any of the V_{i} has to be a multiple of the identity map. Therefore, the matrix representation of g with respect to the basis e_{1}, …, e_{7} has to be of type

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}a^{2} & 0 \\ 0 & a^{2} \\ & & b^{2} & 0 \\ & & 0 & b^{2} \\ & & & & c^{2} & 0 \\ & & & & 0 & c^{2} \\ & & & & & & f^{2} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; (5.3)

with a, b, c, f ℝ 0 . Conversely, any such matrix can be identified with a SU 2 ^{3}-invariant metric on Q^{1,1,1}. The G_{2}-structures on Q^{1,1,1} are most conveniently described by a three-form which contains odd powers of a, b, c, and f. Therefore, we allow those parameters to take negative values although this does not change the metric. Any cohomogeneity-one metric g with principal orbit Q^{1,1,1} is given by:

g g_{t} dt^{2} .

In this formula, g_{t} describes the invariant metric on the orbit and is identified with a matrix of the above type. If the orbit is singular, g_{t} degenerates. t denotes the coordinate in the direction of the geodesics which intersect all orbits perpendicularly.

Our next step is to construct a sufficiently generic Spin 7 -structure of cohomogeneity one whose principal orbit is Q^{1,1,1}. Let 𝔊_{G_{2}} be an invariant G_{2}-structure on Q^{1,1,1} with associated metric g. Any 𝔊_{G_{2}} π^{1} p ψ : 𝔪, g Im 𝕆 , , has to preserve the scalar product and satisfy (5.2). ψ defines a frame ψ^{1} i , …, ψ^{1} kϵ . This frame can canonically be extended to a frame f_{0}, …, f_{7} of a cohomogeneity-one Spin 7 -structure. A possible f_{0}, …, f_{7} which we can construct this way is:

\begin{matrix}f_{0} : \frac{}{t} & f_{1} : \frac{1}{f} e_{7} & f_{2} : \frac{1}{a}e_{1} & f_{3} : \frac{1}{a}e_{2} \\ f_{4} : \frac{1}{b}e_{3} & f_{5} : \frac{1}{b}e_{4} & f_{6} : \frac{1}{c}e_{6} & f_{7} : \frac{1}{c}e_{5} \end{matrix}  (5.4)

We obtain the following four-form Ω which is associated to this frame:

\begin{aligned}Ω & abcf e^{1357} abcf e^{1467} abcf e^{2367} abcf e^{2457} \\ & a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456} \\ & a^{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt \\ & abc e^{136} dt abc e^{145} dt abc e^{235} dt abc e^{246} dt \end{aligned}  (5.5)

It can be easily checked that this four-form is indeed SU 2 ^{3}-invariant. We want to express the equation dΩ 0 as a system of ordinary differential equations for the metric functions a, b, c, and f. In order to do this, we first compute the exterior derivatives of the one-forms e^{1}^{}, …, e^{7}^{}, which are dual to the Killing vector fields e_{1}^{}, …, e_{7}^{}. Then, we apply Remark 3.1.46 and obtain dΩ. We finally see that dΩ 0 is equivalent to:

\begin{matrix}a^{} & \frac{1}{6} \frac{f}{a} \\ b^{} & \frac{1}{6} \frac{f}{b} \\ c^{} & \frac{1}{6} \frac{f}{c} \\ f^{} & \frac{1}{6} \frac{f^{2}}{a^{2}} \frac{1}{6} \frac{f^{2}}{b^{2}} \frac{1}{6} \frac{f^{2}}{c^{2}} 3 \end{matrix}  (5.6)

The details of the calculations can be found in Appendix A.

By considering the above equations, we see that if f is non-negative and the function a, b, or c is positive, its first derivative is non-positive. By replacing \frac{}{t} by \frac{}{t} we could make the derivatives of those metric functions non-negative, but we will not do this change. Next, we discuss the solutions of the system (5.6). Since we have:

a^{}a b^{}b c^{}c \frac{1}{6}f ,

the functions a^{2}, b^{2}, and c^{2} differ only by a constant. Let the orbit on which we fix the initial conditions be at t 0 and let F be a function with F^{}f. By requiring F 0 0, we make F unique. We conclude from the above equation that

a^{2} a_{0}^{2} b^{2} b_{0}^{2} c^{2} c_{0}^{2} \frac{1}{3}F

where a_{0} : a 0 , b_{0} : b 0 and c_{0} : c 0 . Analogously, f 0 will from now on be denoted by f_{0}. We rewrite the fourth equation of (5.6):

f^{}\frac{1}{6} \frac{f^{2}}{\frac{1}{3}F a_{0}^{2}}  \frac{1}{6} \frac{f^{2}}{\frac{1}{3}F b_{0}^{2}}  \frac{1}{6} \frac{f^{2}}{\frac{1}{3}F c_{0}^{2}} 3 .

By replacing t by τ : F t , the four equations simplify to:

\begin{matrix}a & \frac{1}{6a} b \frac{1}{6b} c \frac{1}{6c} \\ ff & \frac{f^{2}}{2τ 6a_{0}^{2}}  \frac{f^{2}}{2τ 6b_{0}^{2}}  \frac{f^{2}}{2τ 6c_{0}^{2}} 3 \end{matrix}

where the dot denotes the derivation with respect to τ. The last equation is linear in f^{2} and can be solved explicitly. We first consider the homogeneous problem and obtain:

f τ ^{2} C \frac{1}{τ 3a_{0}^{2} τ 3b_{0}^{2} τ 3c_{0}^{2}} .

By variation of constants, we obtain the following solution of the inhomogeneous equation:

f τ ^{2} \frac{1}{τ 3a_{0}^{2} τ 3b_{0}^{2} τ 3c_{0}^{2}} C 3 _{0}^{τ} s 3a_{0}^{2} s 3b_{0}^{2} s 3c_{0}^{2} ds .

The constant C is fixed by the initial condition f 0 f_{0} at the value of 27a_{0}^{2}b_{0}^{2}c_{0}^{2}f_{0}^{2}. Therefore, we finally obtain:

\begin{aligned}f τ ^{2} & \frac{f_{0}^{2}}{1 \frac{τ}{3a_{0}^{2}} 1 \frac{τ}{3b_{0}^{2}} 1 \frac{τ}{3c_{0}^{2}}} \\ & \frac{3}{τ 3a_{0}^{2} τ 3b_{0}^{2} τ 3c_{0}^{2}} _{0}^{τ} s 3a_{0}^{2} s 3b_{0}^{2} s 3c_{0}^{2} ds . \end{aligned}  (5.7)

The resulting cohomogeneity-one metric is given by:

\begin{aligned}g & \frac{1}{3}τ a_{0}^{2} e^{1} e^{1} e^{2} e^{2}  \frac{1}{3}τ b_{0}^{2} e^{3} e^{3} e^{4} e^{4} \\ & \frac{1}{3}τ c_{0}^{2} e^{5} e^{5} e^{6} e^{6} f τ ^{2} e^{7} e^{7} dτ^{2} . \end{aligned}  (5.8)

We will check the compactness and completeness of the metrics (5.8) later on, when we have determined the possible initial values at the singular orbit.

The next question we will consider is if there are other parallel Spin 7 -structures with principal orbit Q^{1,1,1} which are not described by the frame (5.4) and the equations (5.6). According to Theorem 3.2.31, a parallel Spin 7 -structure of cohomogeneity one is determined by the G_{2}-structure on a fixed principal orbit. We will therefore classify all SU 2 ^{3}-invariant G_{2}- structures on Q^{1,1,1} which are compatible with an arbitrary but fixed metric and orientation. Because of Lemma 3.1.50, the space of those G_{2}-structures is described by:

Norm_{SO 7}U 1 ^{2} Norm_{G_{2}}U 1 ^{2} ,

where U 1 ^{2} can be chosen as the maximal torus of G_{2} with (5.1) as Lie algebra. Whenever we consider on the following pages a Lie subgroup U 1 ^{2} SO 7 without specifying the embedding, U 1 ^{2} shall be given by the maximal torus of G_{2} SO 7 . Analogously, 2𝔲 1 𝔰𝔬 7 will denote the Cartan subalgebra of 𝔤_{2} 𝔰𝔬 7 . In order to describe the space of invariant G_{2}-structures, we first consider the Lie algebra of Norm_{SO 7}U 1 ^{2}. Since U 1 ^{2} is connected, we have:

Norm_{SO 7}U 1 ^{2} h SO 7 h 2𝔲 1 h^{1} 2𝔲 1 .

Therefore, the Lie algebra of the normalizer is

Norm_{𝔰𝔬 7}2𝔲 1 : x 𝔰𝔬 7 ad_{x} 2𝔲 1 2𝔲 1 .

We denote the Cartan subalgebra

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & a \\ & a & 0 \\ & & & 0 & b \\ & & & b & 0 \\ & & & & & 0 & c \\ & & & & & c & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; a, b, c ℝ ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; (5.9)

of 𝔰𝔬 7 by 3𝔲 1 . Since we will not work with the Cartan subalgebra of 3𝔰𝔲 2 on the next pages, the notation ”3𝔲 1 ” is non-ambiguous. Obviously, 3𝔲 1 Norm_{𝔰𝔬 7}2𝔲 1 . We will show that ” ” is also true. Let x Norm_{𝔰𝔬 7}2𝔲 1 be arbitrary and let κ be the Killing form of 𝔰𝔬 7 . x has to satisfy

x, z 2𝔲 1 z 2𝔲 1 .

Since κ is associative, we have for any y 2𝔲 1 :

κ x, z , y κ x, z, y 0

and therefore x, z 0. This consideration proves that the normalizer equals the centralizer

C_{𝔰𝔬 7} 2𝔲 1 : x 𝔰𝔬 7 ad_{x} 2𝔲 1 0 .

For the following considerations, we complexify all Lie algebras and return to the real case later on. Our x can be considered as an element of Norm_{𝔰𝔬 7,ℂ} 2𝔲 1 ℂ . x has a Cartan decomposition

x x_{h} _{α Φ} μ_{α}g_{α} with x_{h} 3𝔲 1 ℂ , μ_{α} ℂ , g_{α} L_{α} ,  (5.10)

where Φ is the root system of 𝔰𝔬 7, ℂ and L_{α} is the eigenspace to the eigenvalue α : 3𝔲 1 ℂ ℂ of ad_{3𝔲 1 ℂ} . Let z 2𝔲 1 ℂ be arbitrary. Applying ad_{z} to (5.10) yields the following equation:

_{α Φ} μ_{α}α z g_{α} 0 .

We want to prove that the centralizer is 3𝔲 1 ℂ, or equivalently that there is no non-zero x in the orthogonal complement of 3𝔲 1 ℂ which commutes with 2𝔲 1 ℂ. If there exists an α Φ with α 2𝔲 1 ℂ 0, then z, g_{α} α z g_{α} 0 for all z 2𝔲 1 ℂ. Conversely, if there is no such α, we can easily prove for all 3𝔲 1 ℂ ^{}0 x _{α Φ} μ_{α}g_{α} that ad_{z} x 0 if z 2𝔲 1 ℂ satisfies α z 0 for an α with μ_{α} 0. In order to answer the question if there is an α with α 2𝔲 1 ℂ 0 we have to take a closer look at the root system of 𝔰𝔬 7, ℂ :

In Fulton, Harris [35], the complex Lie algebra 𝔰𝔬 7, ℂ is defined as the set of all endomorphisms which are skew-symmetric with respect to a bilinear form Q. The matrix representation of Q with respect to the standard basis of ℂ^{7} is given by:

; ; ; ; ; ; ; ; ; 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 ; ; ; ; ; ; ; ; ; .

It is easily possible to make a change of the basis such that the matrix representation of Q becomes the identity matrix. Fulton and Harris take the following basis of the Cartan subalgebra of 𝔰𝔬 7, ℂ :

L_{1} : diag 1, 0, 0, 1, 0, 0, 0
L_{2} : diag 0, 1, 0, 0, 1, 0, 0
L_{3} : diag 0, 0, 1, 0, 0, 1, 0

With respect to our new basis of ℂ^{7}, the matrix representation of L_{1}, L_{2}, and L_{3} changes: If we set in (5.9) a i and b c 0, we obtain the first element L_{1} of the Cartan subalgebra. Analogously, the second element L_{2} is determined by b i, a c 0, and the third element L_{3} by c i and a b 0. Unfortunately, these matrices are not contained in the real Lie algebra 𝔰𝔬 7 . Nevertheless, we obtain the same root system if we consider 𝔰𝔬 7, ℂ together with the complex basis L_{1}, L_{2}, L_{3} of 3𝔲 1 ℂ. Therefore, we can work with the root system which is described in [35]. This root system is given by:

θ_{j} 1 j 3 θ_{j} θ_{k} 1 j k 3 ,

where θ_{j} is the dual of L_{j} with respect to the Killing form. We redefine L_{j} and θ_{j} as i times the old basis elements L_{j} and θ_{j}, since this does not change our equations but guarantees that L_{j} is contained in the real Lie algebra 𝔰𝔬 7 . The Cartan subalgebra 2𝔲 1 ℂ of 𝔤_{2}^{ℂ} 𝔰𝔬 7, ℂ is the plane which is orthogonal to L_{1} L_{2} L_{3} (see page 106). The α Φ which vanish on 2𝔲 1 ℂ are precisely those which are multiples of θ_{1} θ_{2} θ_{3}. Since there is no root of 𝔰𝔬 7, ℂ with this property, we have proven that indeed Norm_{𝔰𝔬 7,ℂ} 2𝔲 1 ℂ 3𝔲 1 ℂ. By passing to the compact real form of 𝔰𝔬 7, ℂ , we can conclude that Norm_{𝔰𝔬 7}2𝔲 1 3𝔲 1 .

Let U 1 ^{3} be the maximal torus of SO 7 with Lie algebra 3𝔲 1 . Our next step is to describe the discrete group Γ : Norm_{SO 7}U 1 ^{2} U 1 ^{3}. In order to do this, we first consider the group Norm_{SO 7}U 1 ^{3} U 1 ^{3}, which is isomorphic to the Weyl group W_{𝔰𝔬 7} of 𝔰𝔬 7 . This isomorphism is given by xU 1 ^{3} Ad_{x}^{}_{3𝔲 1 ^{}}: 3𝔲 1 ^{}3𝔲 1 ^{}, where Ad^{}is the coadjoint action of SO 7 . This is a general statement, which is true for any semisimple Lie algebra and its Cartan subalgebra.

We will show that Norm_{SO 7}U 1 ^{2} Norm_{SO 7}U 1 ^{3}. Let us assume that there is an element h of SO 7 such that Ad_{h} leaves 2𝔲 1 invariant, but does not leave 3𝔲 1 invariant. Then 3𝔲 1 and Ad_{h} 3𝔲 1 are two distinct Cartan subalgebras whose intersection is 2𝔲 1 . In this situation, the centralizer of 2𝔲 1 is at least four-dimensional which is not the case. The quotient Γ we want to determine therefore is a subgroup of the Weyl group. More precisely, it is the subgroup of the Weyl group which leaves the plane 2𝔲 1 3𝔲 1 invariant. In order to describe this group, we have to introduce a few facts on the Weyl group of 𝔰𝔬 7 . W_{𝔰𝔬 7} is isomorphic to ℤ_{2}^{3} S_{3}, which is of order 48. The ℤ_{2}^{3}-factor of W_{𝔰𝔬 7} acts by changing the signs of the θ_{i}. The S_{3}-factor of the Weyl group consists of the permutations of θ_{1}, θ_{2}, θ_{3} . The Cartan subalgebra 2𝔲 1 of 𝔤_{2} is the plane of all x 3𝔲 1 satisfying:

θ_{1} x θ_{2} x θ_{3} x 0 .

By replacing L_{3} by L_{3} in the basis of 3𝔲 1 , we can change this equation to:

θ_{1} x θ_{2} x θ_{3} x 0 .

The subgroup of W_{𝔰𝔬 7} which leaves this equation invariant is generated by the permutations and the global change of all signs. Therefore, Γ is a semidirect product ℤ_{2} S_{3}. It is easy to see that Γ is in fact the Dieder group D_{6}. All in all, we have proven:

Norm_{SO 7}U 1 ^{2} U 1 ^{3} D_{6} .

Next, we have to determine Norm_{G_{2}}U 1 ^{2}. Since 2𝔲 1 is a Cartan subalgebra of 𝔤_{2}, Norm_{G_{2}}U 1 ^{2} U 1 ^{2} is the Weyl group W_{𝔤_{2}} of 𝔤_{2}. It is known that W_{𝔤_{2}} is isomorphic to D_{6}. The group Norm_{G_{2}}U 1 ^{2} therefore is a semidirect product U 1 ^{2} D_{6}. There is the following exact sequence:

π_{0} Norm_{G_{2}}U 1 ^{2} ^{π_{0} i} π_{0} Norm_{SO 7}U 1 ^{2} ^{π_{0} π} π_{0} Norm_{SO 7}U 1 ^{2} Norm_{G_{2}}U 1 ^{2} 0 ,

where π_{0} i is the map which is induced by the inclusion of Norm_{G_{2}}U 1 ^{2} into Norm_{SO 7}U 1 ^{2} and π_{0} π is induced by the projection map. Norm_{G_{2}}U 1 ^{2} has 12 connected components and Norm_{SO 7}U 1 ^{2} has 12 connected components, too. If we were able to prove that π_{0} i is surjective, we could conclude that Norm_{SO 7}U 1 ^{2} Norm_{G_{2}}U 1 ^{2} is connected. Since π_{0} Norm_{SO 7}U 1 ^{2} and π_{0} Norm_{G_{2}}U 1 ^{2} are both finite, we can equivalently prove the injectivity of π_{0} i . Let x Norm_{G_{2}}U 1 ^{2} with x U 1 ^{2} be arbitrary. Then x αx_{0} with α W_{𝔤_{2}} e and x_{0} U 1 ^{2}. Since α acts non-trivially on the dual of the Cartan subalgebra 2𝔲 1 , it cannot be an element of the maximal torus U 1 ^{3} of SO 7 . Therefore, x is not an element of the identity component of Norm_{SO 7}U 1 ^{2}. This consideration proves that π_{0} i is injective and Norm_{SO 7}U 1 ^{2} Norm_{G_{2}}U 1 ^{2} therefore is connected. Since the identity component of Norm_{SO 7}U 1 ^{2} is U 1 ^{3} and the identity component of Norm_{G_{2}}U 1 ^{2} is U 1 ^{2}, their quotient is isomorphic to U 1 .

The Weyl group of 𝔰𝔬 7, ℂ (or 𝔤_{2}^{ℂ}) leaves the Cartan subalgebra of the compact real form 𝔰𝔬 7 (or 𝔤_{2}) invariant. For this reason, there was no need to carry out the above calculations first on the complex and then on the real level. Instead, we were able to work directly with the real Lie algebras and groups.

Norm_{SO 7}U 1 ^{2} Norm_{G_{2}}U 1 ^{2} can be represented by the subgroup of U 1 ^{3} whose Lie algebra is the orthogonal complement of 2𝔲 1 3𝔲 1 . This group is given by:

; ; ; ; ; ; ; ; ; \begin{aligned}1 \\ & R_{θ} \\ & & R_{θ} \\ & & & R_{θ} \end{aligned} ; ; ; ; ; ; ; ; ; ; θ ℝ ; ; ; ; ; ; GL Im 𝕆 ,

where R_{θ} denotes the rotation in the plane around the angle θ. In order to construct the set of invariant G_{2}-structures which are associated to a given metric and orientation, we have to describe how the above group acts on 𝔪. We define:

T : ; ; ; ; ; ; ; ; ; \begin{aligned}R_{θ} \\ & R_{θ} \\ & & R_{θ} \\ & & & 1 \end{aligned} ; ; ; : T_{θ} ; ; ; ; ; ; ; θ ℝ ; ; ; ; ; ; GL 𝔪 .

In the above formula, the matrix representation of T_{θ} is with respect to the basis e_{1}, …, e_{7} . T commutes with the action of U 1 _{1,1,1}^{2} on 𝔪. Furthermore, T_{θ} is orthogonal with respect to g and orientation preserving. Therefore, T describes the action of Norm_{SO 7}U 1 ^{2} Norm_{G_{2}}U 1 ^{2} on 𝔪 and its action on a given invariant G_{2}-structure determines the set of all invariant G_{2}-structures. We consider the following subgroup of Norm_{SU 2 ^{3}}U 1 ^{2}:

S :  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}e^{i \frac{θ}{2}} & 0 \\ 0 & e^{i \frac{θ}{2}} \\ & & e^{i \frac{θ}{2}} & 0 \\ & & 0 & e^{i \frac{θ}{2}} \\ & & & & e^{i \frac{θ}{2}} & 0 \\ & & & & 0 & e^{i \frac{θ}{2}} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; : S_{θ}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; θ ℝ ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; .  (5.11)

Conjugation by S_{θ} is a well-defined diffeomorphism of Q^{1,1,1}. It is easy to see that it is even an orientation preserving isometry. By a short calculation we see that the differential of this map acts as T_{θ} on 𝔪. We sum up our results: Let ω be a SU 2 ^{3}-invariant G_{2}-structure on Q^{1,1,1}. Any other invariant G_{2}-structure on Q^{1,1,1} with the same associated metric and orientation as ω can be obtained by the action of S on ω. Furthermore, this action is isometric.

We are now able to determine the holonomy of the metrics which we obtain as solutions of (5.6): Let M be a cohomogeneity-one manifold with principal orbit Q^{1,1,1}, and let Ω be a parallel Spin 7 -structure which is described by (5.5) and (5.6). The union of all principal orbits we denote by M^{o}. We choose a diffeomorphism such that M^{o} is identified with Q^{1,1,1} I where I is an interval. Conjugation by S_{θ} on all Q^{1,1,1} t is an isometry of M^{o}. The action of S_{θ} maps Ω into another Spin 7 -structure, which is parallel with respect to the Levi-Civita connection, too. We therefore have found a one-parameter family of parallel Spin 7 - structures whose extension to a SO 8 -structure is the same. Furthermore, this family is diffeomorphic to a circle. Because of Lemma 2.3.17, the holonomy of the metric is contained in SU 4 . From the classification of the possible holonomy groups of a parallel Spin 7 - manifold, it follows that the holonomy is contained in SU 3 or is one of the groups SU 4 , Sp 2 , or SU 2 SU 2 . In the first case, there exists a parallel vector field on the manifold, since SU 3 acts trivially on a subspace of the tangent space. The holonomy bundle of a Riemannian manifold is invariant with respect to isometries. Since the Spin 7 -structure we have found is invariant, too, the extension of the holonomy bundle to a SU 3 -structure has to be preserved by the action of SU 2 ^{3}. Therefore, we can assume that the parallel vector field we search for is invariant. If there exists a parallel vector field X on the manifold, it has to be of type c_{1} \frac{1}{f} e_{7} c_{2} \frac{}{t}, where c_{1} and c_{2} depend on t only. We first consider the case where c_{1} 0. Since ∇_{\frac{}{t}} X 0 and the length of X is constant, the functions c_{1}, c_{2}, and f

have to be constant, too. We can conclude from the equations (5.6) that a^{2}, b^{2}, and c^{2} are strictly monotonous. More precisely, either all of them are strictly increasing or all of them are strictly decreasing. In any case, the right hand side of the fourth equation of (5.6) cannot be 0 for all values of t. This is a contradiction to f being constant. Next, we assume that c_{1} 0 and c_{2} 1. If \frac{}{t} is parallel, we have a^{}b^{}c^{}f^{}0, which is impossible, too. Therefore, the holonomy has to be SU 4 , Sp 2 , or SU 2 SU 2 and the metric thus is Kähler.

Since GL 4, ℂ Spin 7 SU 4 , we only have to describe the complex structure J or equivalently the Kähler form η in order to determine the SU 4 -structure. For the same reasons as above, the SU 4 -structure and η have to be invariant. The two-forms

e^{12} , e^{34} , e^{56} , e^{7} dt

span the space of all invariant two-forms on Q^{1,1,1}. Let f_{0}, …, f_{7} be a frame of the SU 4 - structure. Since that basis has to be orthonormal and we have η f_{2i}, f_{2i 1} 1 for i 0, …, 3, the Kähler form has to satisfy:

η ϵ_{1}a^{2}e^{12} ϵ_{2}b^{2}e^{34} ϵ_{3}c^{2}e^{56} fe^{7} dt with ϵ_{1}, ϵ_{2}, ϵ_{3} 1, 1 .  (5.12)

With help of the exterior derivatives of the one-forms which we have calculated in Appendix A, we obtain for the exterior derivative of η:

dη f \frac{1}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56} dt

ϵ_{1} 2a^{}a dt e^{12} ϵ_{1} a^{2} e^{27} e^{2} e^{1} e^{17}

ϵ_{2} 2b^{}b dt e^{34} ϵ_{2} b^{2} e^{47} e^{4} e^{3} e^{37}

ϵ_{3} 2c^{}c dt e^{56} ϵ_{3} c^{2} e^{67} e^{6} e^{5} e^{57}

\frac{1}{3}f e^{12} dt ϵ_{1}2a^{}a e^{12} dt

\frac{1}{3}f e^{34} dt ϵ_{2}2b^{}b e^{34} dt

\frac{1}{3}f e^{56} dt ϵ_{3}2c^{}c e^{56} dt

The metric g therefore has to satisfy the following equations:

f 6ϵ_{1}a^{}a 6ϵ_{2}b^{}b 6ϵ_{3}c^{}c .  (5.13)

By comparing (5.13) with the equations (5.6), we see that we have ϵ_{1} ϵ_{2} ϵ_{3} 1 and obtain:

η a^{2}e^{12} b^{2}e^{34} c^{2}e^{56} fe^{7} dt .  (5.14)

Since we already know that the SU 4 -structure is parallel, we do not have to prove that J is integrable, or that ∇η 0.

We now prove that the holonomy of the metrics which satisfy the equations (5.6) is all of SU 4 . Let us assume that the holonomy is a subgroup of Sp 2 . If this is the case, the metric is hyperkähler and there exists another symplectic form η^{}with the same properties as η such that η, η^{}is linearly independent. η^{}has to be of type (5.12), too. Therefore, the equations (5.13) have to be true for another choice of the signs ϵ_{i}. It easily follows that f 0, which cannot be the case.

We consider the behavior of the metric near a singular orbit. The following lemma yields the possible dimensions of the collapsing spheres:

Lemma 5.2.1. Let U 1 _{1,1,1}^{2} SU 2 ^{3} be chosen as in the beginning of this section. As usual, we denote this subgroup shortly by U 1 ^{2} and its Lie algebra by 2𝔲 1 . Furthermore, let K be a connected, closed group with U 1 ^{2} K SU 2 ^{3}. We denote the Lie algebra of K by 𝔨. In this situation, 𝔨 and K can be found in the table below. Furthermore, K U 1 ^{2} and SU 2 ^{3} K satisfy the following topological conditions:

𝔨  K  K U 1 ^{2}  SU 2 ^{3} K
3𝔲 1  U 1 ^{3}  S^{1}  S^{2} S^{2} S^{2}
2𝔲 1 𝔰𝔲 2 U 1 ^{2} SU 2  S^{3}  S^{2} S^{2}
𝔲 1 2𝔰𝔲 2 U 1 SU 2 ^{2} S^{5} Γ  S^{2}
3𝔰𝔲 2  SU 2 ^{3}  Q^{1,1,1} S^{7} Γ

where Γ is an arbitrary discrete subgroup of the orthogonal group.

Proof : 𝔨 is obviously a 2𝔲 1 -module. Since we have decomposed 𝔪 into its irreducible 2𝔲 1 -submodules, we are able to list all possibilities for 𝔨. We consider each case separately:

1. 𝔨 2𝔲 1 V_{4}: In this case, 𝔨 is a Lie algebra isomorphic to 3𝔲 1 . The corresponding Lie group K is U 1 ^{3}. K U 1 ^{2} therefore is a circle. For the quotient SU 2 ^{3} K we obtain SU 2 U 1 SU 2 U 1 SU 2 U 1 , which is due to the Hopf fibration diffeomorphic to S^{2} S^{2} S^{2}.

2. 𝔨 2𝔲 1 V_{i} with i 1, 2, 3 : We have e_{2i 1} , e_{2i} 3𝔲 1 , but e_{2i 1} , e_{2i} 2𝔲 1 . Therefore, 𝔨 is not closed under the Lie bracket and we can exclude this case.

3. 𝔨 2𝔲 1 V_{i} V_{4} with i 1, 2, 3 . Without loss of generality, we assume that i 1. 𝔨 is a Lie algebra isomorphic to 2𝔲 1 𝔰𝔲 2 . In order to describe K U 1 ^{2} explicitly, we first have to describe K and U 1 ^{2} in more detail: Let S_{θ} be the matrix

\begin{matrix}e^{iθ} & 0 \\ 0 & e^{iθ} \end{matrix} .

We have:

K ; ; { \atop }\begin{aligned}A \\ & S_{ϕ} \\ & & S_{ψ} \end{aligned} { \atop } ; ; ; ; ; A SU 2 , ϕ, ψ 0, 2π ; ; ,

and

U 1 ^{2} ; ; { \atop }\begin{aligned}S_{ψ ϕ} \\ & S_{ϕ} \\ & & S_{ψ} \end{aligned} { \atop } ; ; ; ; ; ϕ, ψ 0, 2π ; ; .

It is easy to see that an element of U 1 ^{2} can only be an element of the semisimple part of K if ϕ ψ 0. By applying Lemma 3.1.12 twice, it follows that K U 1 ^{2} SU 2 U 1 ^{2} U 1 ^{2} SU 2 U 1 U 1 SU 2 S^{3}. Finally, we can deduce from the above description of K that SU 2 ^{3} K is diffeomorphic to S^{2} S^{2}.

4. 𝔨 2𝔲 1 V_{i} V_{j} with i, j 1, 2, 3 and i j: Analogously to the second case, 𝔨 is not closed under the Lie bracket.

5. 𝔨 2𝔲 1 V_{i} V_{j} V_{4} with V_{i} and V_{j} as above: We can assume without loss of generality that i 1 and j 2. The group K is isomorphic to SU 2 SU 2 U 1 . As in the third case, we apply Lemma 3.1.12. In order to do this, we consider the following two subgroups of U 1 ^{2}:

\begin{aligned}U 1 _{1} & : ; ; { \atop }\begin{aligned}S_{ϕ} \\ & S_{ϕ} \\ & & S_{0} \end{aligned} { \atop } ; ; ; ; ; ϕ 0, 2π ; ; \\ U 1 _{2} & : ; ; { \atop }\begin{aligned}S_{ψ} \\ & S_{0} \\ & & S_{ψ} \end{aligned} { \atop } ; ; ; ; ; ψ 0, 2π ; ; \end{aligned}

Since U 1 _{1} U 1 _{2} I , the covering map from U 1 _{1} U 1 _{2} to U 1 ^{2} which maps ϕ, ψ to ϕ ψ is a diffeomorphism. An element of U 1 _{2} can be in SU 2 SU 2 K only for ψ 0. We therefore can rewrite K U 1 ^{2} as SU 2 SU 2 U 1 , where U 1 is given by:

\begin{aligned}S_{ϕ} \\ & S_{ϕ} \end{aligned} ; ; ; ϕ 0, 2π .

We assume that K U 1 ^{2} is a quotient of the sphere S^{5} by a discrete subgroup Γ of O 6 . S^{3} S^{3} is a circle bundle over K U 1 ^{2}. If our assumption was true, we would have the following exact sequence:

… π_{3} S^{1} π_{3} S^{3} S^{3} π_{3} S^{5} Γ ….

Since the higher homotopy groups of S^{5} and S^{5} Γ coincide, the above exact sequence becomes:

… 0 ℤ^{2} 0 ….

which is impossible. This proves that SU 2 ^{3} K is not a possible singular orbit for a smooth cohomogeneity-one manifold with principal orbit Q^{1,1,1}. We nevertheless note that this space is diffeomorphic to S^{2} for similar reasons as in the above cases.

6. 𝔨 2𝔲 1 V_{1} V_{2} V_{3}: For the same reasons as in the second and the fourth case, 𝔨 is not closed under the Lie bracket.

7. 𝔨 2𝔲 1 V_{1} V_{2} V_{3} V_{4}: In this case, K U 1 ^{2} Q^{1,1,1}. We assume that Q^{1,1,1} is homeomorphic to S^{7} Γ, where Γ O 8 is discrete. Q^{1,1,1} can be described as a circle bundle over SU 2 ^{3} U 1 ^{3} S^{2} ^{3}. Therefore, we obtain the following exact sequence:

… π_{2} S^{7} Γ π_{2} S^{2} ^{3} π_{1} S^{1} π_{1} S^{7} Γ …,

which can be explicitly written as

… 0 ℤ^{3} ℤ Γ ….

Since there is no injective group homomorphism from ℤ^{3} to ℤ, Q^{1,1,1} is not homeomorphic to S^{7} Γ.

Next, we investigate the geometric properties of the metrics, which satisfy the equations (5.6) and have a singular orbit. We start with a proof that none of those metrics is compact. Without loss of generality, we can assume that we have a singular orbit at t 0. The only possibilities for the singular orbit are S^{2} S^{2} and S^{2} S^{2} S^{2}. In both cases, we have f 0 0. We will see below that in the first case we have f^{}0 \frac{3}{2} and in the second one we have f^{}0 3. Therefore, f is negative near the singular orbit. If the manifold was compact, there would be two singular orbits. We denote the position of the second singular orbit by T. For the same reasons as above, we have f T 0, f^{}T 3, \frac{3}{2} , and f is positive near the second singular orbit. Therefore, there exists a s 0, T such that f s 0. Since there are only two singular orbits, we have obtained a contradiction.

In order to check the completeness of our metrics, we have to find out if the geodesics which intersect all orbits perpendicularly are of infinite length and if the metric can be smoothly extended to the singular orbit. Since f is negative near the singular orbit and we have f 0 outside of that orbit, τ : F t is negative, too. The initial value f_{0} equals zero and the metric therefore is determined by:

\begin{matrix}f τ ^{2} & \frac{3}{τ 3a_{0}^{2} τ 3b_{0}^{2} τ 3c_{0}^{2}} _{0}^{τ} s 3a_{0}^{2} s 3b_{0}^{2} s 3c_{0}^{2} ds \\ a τ ^{2} & \frac{1}{3}τ a_{0}^{2}  b τ ^{2}  \frac{1}{3}τ b_{0}^{2}  c τ ^{2}  \frac{1}{3}τ c_{0}^{2} . \end{matrix}

If the singular orbit is S^{2} S^{2} S^{2}, all of the values of a_{0}, b_{0}, and c_{0} are non-zero. If we have S^{2} S^{2} as singular orbit, one of those values has to vanish. Let I be the maximal interval on which we can define f. Since the formula for the metric contains the term f τ ^{2}dτ^{2}, we have to prove that _{I} f τ dτ . All zeros of s 3a_{0}^{2} s 3b_{0}^{2} s 3c_{0}^{2} are positive and we therefore have I , 0 . For negative values of τ, f^{2} τ behaves like O τ . We conclude that _{I} f τ dτ and thus have proven the first condition for the completeness.

Next, we will check if the solutions of (5.6) we have found satisfy the smoothness conditions from Theorem 3.2.18. We consider both possible singular orbits separately and start with S^{2} S^{2} S^{2}. In this case, we have 𝔨 3𝔲 1 and the tangent space of the singular orbit can be identified with 𝔭 V_{1} V_{2} V_{3}. For the following arguments, we need some information on S^{2} 𝔭 . We will see soon that we only need to consider those 3𝔲 1 -submodules of S^{2} 𝔭 which are trivial with respect to 2𝔲 1 _{1,1,1}. On page 107, we have classified the invariant metrics on Q^{1,1,1}. Since any of those metrics corresponds to a trivial 2𝔲 1 _{1,1,1}-submodule of S^{2} 𝔪 S^{2} 𝔭 V_{4} , the maximal trivial submodule of S^{2} 𝔭 decomposes into:

span e_{1} e_{1} e_{2} e_{2}  S^{2} V_{1} ,
span e_{3} e_{3} e_{4} e_{4}  S^{2} V_{2} ,
span e_{5} e_{5} e_{6} e_{6}  S^{2} V_{3} .

We remark that all of these modules are not only trivial with respect to 2𝔲 1 _{1,1,1} but also with respect to 3𝔲 1 . The normal space 𝔭^{}of the singular orbit has to be described as a 3𝔲 1 -module, too. Since 2𝔲 1 _{1,1,1} is the isotropy algebra of the cohomogeneity-one action, it has to leave \frac{}{t} invariant. The only two-dimensional 2𝔲 1 -module which contains invariant elements except 0 is the trivial one. Therefore, 𝔭^{}is a trivial 2𝔲 1 _{1,1,1}-module, too. At the beginning of this section, we have introduced the notation i_{1,1,1} 𝔲 1 for the orthogonal complement of 2𝔲 1 _{1,1,1} 3𝔲 1 . As on page 114, we denote the connected Lie subgroup of SU 2 ^{3} with Lie algebra i_{1,1,1} 𝔲 1 by S. We will need the weight of the action of S on 𝔭^{}, too. The orbit of a point in K U 1 _{1,1,1}^{2} with respect to the action of S is a loop in that space. Since π_{1} K U 1 _{1,1,1}^{2} π_{1} S^{1} ℤ, the homotopy class of that loop corresponds to an integer. For geometric reasons, it coincides with the weight of the action of S on 𝔭^{}. By a short calculation, we see that

S U 1 _{1,1,1}^{2} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}e^{\frac{2πik}{3}} & 0 \\ 0 & e^{\frac{2πik}{3}} \\ & & e^{\frac{2πik}{3}} & 0 \\ & & 0 & e^{\frac{2πik}{3}} \\ & & & & e^{\frac{2πik}{3}} & 0 \\ & & & & 0 & e^{\frac{2πik}{3}} \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; k ℤ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ℤ_{3} .

Therefore, the winding number of the S-orbit is 3 and the weight of the S-action on 𝔭^{}equals 3, too. We denote the irreducible 3𝔲 1 -module on which the group U 1 _{1} from the proof of Lemma 5.2.1 acts with weight r, U 1 _{2} with weight s and S with weight t by 𝕍_{r,s,t}. All of those spaces have complex dimension 1 except 𝕍_{0,0,0} which is real and one-dimensional. In contrast to Chapter 4, we will not index those spaces by a ℝ or ℂ. We decompose S^{m} 𝔭^{}, which can be written as S^{m} 𝕍_{0,0,3} , into irreducible submodules and obtain:

S^{m} 𝕍_{0,0,3}  𝕍_{0,0,3m} 𝕍_{0,0,3 m 2} … 𝕍_{0,0,0} if m is even 𝕍_{0,0,3m} 𝕍_{0,0,3 m 2} … 𝕍_{0,0,3}  if m is odd

The embedding of the summands into S^{m} 𝕍_{0,0,3} can be described as follows: We identify 𝕍_{0,0,3} with ℂ. The homogeneous polynomial Re z ^{2} Im z ^{2} of degree 2 is K-invariant. The module 𝕍_{0,0,3k} with k 2 can be identified with the orthogonal complement of z ^{2} S^{k 2} 𝕍_{0,0,3} in S^{k} 𝕍_{0,0,3} . Therefore, the embedding of 𝕍_{0,0,3k} into S^{m} 𝕍_{0,0,3} with m k can be described by p z ^{2 m k} p.

The dimensions of the spaces W_{m}^{v} Hom_{3𝔲 1} S^{m} 𝔭^{}, S^{2} 𝔭^{}and W_{m}^{h} Hom_{3𝔲 1} S^{m} 𝔭^{}, S^{2} 𝔭 easily follow from Schur’s lemma. We have dim Hom_{3𝔲 1}𝕍_{0,0,0} 1. If r, s, t 0, 0, 0 , we have dim Hom_{3𝔲 1}𝕍_{r,s,t} 2, since 𝕍_{r,s,t} is a complex space. S^{m} 𝔭^{}is a trivial 2𝔲 1 _{1,1,1}-module and thus we only have to take care of the trivial summands of S^{2} 𝔭 . We put our results together and obtain:

dim W_{m}^{h}  3 if m is even, 0 if m is odd.  dim W_{m}^{v}  ;  ; 1 if m 0, 0 if m is odd, 3 if m 2 is even.  (5.15)

These dimensions we have to interpret. The horizontal part of the metric is determined by the functions a^{2}, b^{2}, and c^{2}. Since dim W_{2k 1}^{h} 0 for all k ℕ_{0}, the values of the 2k 1 ^{st} derivatives of a^{2}, b^{2}, and c^{2} at t 0 have to be a fixed number. All of the conditions from Theorem 3.2.18 are linear and thus we have a^{2} ^{2k 1} 0 b^{2} ^{2k 1} 0 c^{2} ^{2k 1} 0 0. The values of the even derivatives a^{2} ^{2k} 0 , b^{2} ^{2k} 0 , and c^{2} ^{2k} 0 can be chosen freely, since W_{2k}^{h} is a three-dimensional space.

In Remark 3.2.21, we have seen that the dimension of W_{m}^{v} is not directly linked to the properties of f^{2} ^{m} 0 . One reason for this is that g \frac{}{t}, e_{7} and g \frac{}{t}, \frac{}{t} are fixed to 0 and 1 by our choice of the coordinates. Furthermore, dim W_{m}^{v} makes a statement on \frac{1}{t^{2}} f^{2} ^{m} 0 , since we implicitly have parameterized 𝔭^{}by polar coordinates. In order to find the smoothness conditions for f, we have to describe the metric on 𝔭^{}more explicitly. There is a unique inner product g_{e} on 𝔭^{}which is U 1 ^{3}-invariant and satisfies g_{e} \frac{}{t}, \frac{}{t} 1. Let g_{t}^{v} be the restriction of the cohomogeneity-one metric to the circle of radius t in 𝔭^{}. It follows from our formula for dim W_{m}^{v} that

g_{t}^{v} 1 0 t κ_{2}t^{2} 0 t^{3} κ_{4}t^{4} … g_{e} κ t g_{e}  (5.16)

for some coefficients κ_{2}, κ_{4}, … ℝ. Since f t ^{2} g_{t}^{v} e_{7}, e_{7} κ t g_{e} e_{7}, e_{7} , we have to consider e_{7} in more detail. Let γ be a geodesic which intersects all orbits perpendicularly. Instead of a tangent vector of Q^{1,1,1}, e_{7} should in this situation be considered as a vector field along γ. Since S intersects U 1 _{1,1,1}^{2} three times, its orbit can be parameterized by δ : 0, \frac{2π}{3} Q^{1,1,1} where

δ s : ; ; ; ; ; ; ; \begin{aligned}e^{is} & 0 \\ 0 & e^{is} \\ & & e^{is} & 0 \\ & & 0 & e^{is} \\ & & & & e^{is} & 0 \\ & & & & 0 & e^{is} \end{aligned} ; ; ; ; ; ; ; U 1 _{1,1,1}^{2}

We see that δ^{}0 2e_{7} and that the length of the circle in 𝔭^{}with radius t is with respect to g_{e} given by:

\frac{2π}{3} ; 0 \overline{g_{e} 2t e_{7}, 2t e_{7}}ds \frac{4π}{3} \overline{g_{e} e_{7}, e_{7}} t .

On the other hand, this length has to equal 2πt. Therefore, g_{e} e_{7}, e_{7} \frac{9}{4}t^{2} and we have:

f t ^{2} \frac{9}{4}t^{2} \frac{9}{4}κ_{2}t^{4} \frac{9}{4}κ_{4}t^{6} ….

Any choice of κ_{2}, κ_{4}, … satisfies the conditions from Theorem 3.2.18. Since a, b, c, and f are analytic, we have found the following necessary and sufficient conditions for the smoothness of the metric:

• a^{2}, b^{2}, and c^{2} are even functions. We conclude from a 0 , b 0 , c 0 0 that a, b, and c have to be even, too.

• f is an odd function.

• f^{}0 \frac{3}{2}.

We have to check if our metrics satisfy the above conditions. Let a t , b t , c t , f t be a solution of (5.6). It is easy to see that a t , b t , c t , f t is another solution of (5.6), whose value at t 0 is the same. Since (5.6) has a unique solution for any initial metric on the singular orbit S^{2} S^{2} S^{2}, the first two conditions are satisfied. It follows directly from our equations that f^{}0 3 if f 0 0 and a 0 , b 0 , c 0 0. Therefore, the third condition is not satisfied and our metrics have a singularity at the singular orbit.

As a by-product of the above considerations, we obtain new local examples of cohomogeneityone Einstein metrics with principal orbit Q^{1,1,1} and singular orbit S^{2} S^{2} S^{2}. Those metrics satisfy f^{}0 \frac{3}{2} and thus are not one of our metrics with holonomy SU 4 . Before we can apply Theorem 3.2.24, we have to check if Assumption 3.2.19 is satisfied. In our situation, that assumption states that 𝔭 and 𝔭^{}have no equivalent U 1 _{1,1,1}^{2}-submodules in common. Since 𝔭^{}is trivial with respect to the action of U 1 _{1,1,1}^{2} and 𝔭 decomposes into the three non-trivial modules V_{1}, V_{2}, and V_{3}, 3.2.19 is satisfied. The dimensions of the spaces W_{m}^{h} and W_{m}^{v} are known and we thus can calculate the number of cohomogeneity-one Einstein metrics with principal orbit Q^{1,1,1} and singular orbit S^{2} S^{2} S^{2}. We have stated above that dim W_{m}^{v} describes the degree of freedom of \frac{1}{t^{2}} f^{2} ^{m} 0 instead of f^{2} ^{m} 0 . If m 2, we can compute \frac{1}{t^{2}} f^{2} ^{m} 0 explicitly by l’Hôpital’s rule. We obtain a term which contains f 0 , f^{}0 , f^{}0 , and f^{}0 . Since the lower derivatives of f are fixed, we can take f^{}0 as the free parameter which is associated to W_{2}^{v}. All in all, there are the following parameters which are sufficient to determine a unique cohomogeneity-one Einstein metric on a tubular neighborhood of S^{2} S^{2} S^{2}:

1. a 0 , b 0 , and c 0 ,

2. f^{}0 , and

3. the Einstein constant.

Next, we have to prove the smoothness of the metrics with singular orbit S^{2} S^{2}. In order to keep our considerations short, we only prove a sufficient condition for the smoothness. We will see that the metrics which we have constructed satisfy that condition. The spaces W_{m}^{h} and W_{2}^{v} we will not have to describe in detail. Therefore, we will obtain only an existence result for cohomogeneity-one Einstein metrics which satisfy certain initial conditions, but no result on their uniqueness.

In our situation, we have f 0 0 and either a 0 0, b 0 0, or c 0 0. We assume without loss of generality that a 0 0. Motivated by our results for the S^{2} S^{2} S^{2}-case, we suppose that any metric with the following properties is smooth:

• b and c are even analytic functions.

• a and f are odd and analytic.

• a^{}0 and f^{}0 are chosen such that the sectional curvature of the collapsing sphere is \frac{1}{t^{2}} O \frac{1}{t} for small values of t.

We consider the third condition in more detail. The tangent space of the collapsing sphere can be identified with the first summand of the Lie algebra 3𝔰𝔲 2 . The collapsing sphere itself therefore should be considered as SU 2 . The metric on SU 2 with constant sectional curvature 1 is given by:

\begin{aligned}h & : 𝔰𝔲 2 𝔰𝔲 2 ℝ \\ h X, Y & : \frac{1}{2}tr XY . \end{aligned}

Since g e_{1}, e_{1} a^{2} t and h e_{1}, e_{1} \frac{1}{4}, a^{}0 has to be \frac{1}{2}. The orbit of a point on the collapsing sphere with respect to the group which is generated by e_{7} is a great circle on that sphere. Its length has to be 2πt O t^{2} for small t. By repeating the calculations which we have done in the previous case, we see that this is only possible if f^{}0 \frac{3}{2}.

We consider the system (5.6) under the assumption that a 0 f 0 0. With help of l’Hôpital’s rule, we obtain the following system for a^{}0 and f^{}0 :

a^{}0  \frac{1}{6} \frac{f^{}0}{a^{}0}
f^{}0  \frac{1}{6} \frac{f^{}0 ^{2}}{a^{}0 ^{2}} 3

Its solutions are given by a^{}0  \frac{1}{2} and f^{}0  \frac{3}{2}. Since we can change a t , b t , c t , f t

into a t , b t , c t , f t , we can assume that a^{}0 \frac{1}{2}. In any case, the third of our supposed conditions is satisfied.

In order to prove that our conditions are sufficient for the smoothness of the metric near the singular orbit, we again have to describe 𝔭 and 𝔭^{}as 𝔨-modules. The Lie algebra 𝔨 is in our situation isomorphic to 𝔰𝔲 2 2𝔲 1 . The corresponding Lie group is generated by SU 2 and U 1 _{1,1,1}^{2}. In most cases, we use the SU 2 -action for our arguments and turn only to the action of U 1 _{1,1,1}^{2} if necessary. The space 𝔭 is spanned by e_{3}, …, e_{6} . Since the first summand of 3𝔰𝔲 2 commutes with those e_{i}, 𝔭 is a trivial 𝔰𝔲 2 -module. The SU 2 -orbit of any v 𝔭^{}0 is a 3-sphere. 𝔭^{}therefore is a real four-dimensional irreducible 𝔰𝔲 2 -module.

As in Chapter 4, we will denote this module by 𝕍_{1}^{ℂ}. The action of U 1 _{1,1,1}^{2} leaves \frac{}{t} invariant. Thus, 𝔭^{}contains a trivial submodule. At the beginning of this section, we have seen that U 1 _{1,1,1}^{2} acts irreducibly on the space span e_{1}, e_{2} , which is tangent to the collapsing sphere. As a U 1 _{1,1,1}^{2}-module, 𝔭^{}therefore has to decompose into a trivial two-dimensional and an irreducible two-dimensional submodule, which is isomorphic to V_{1}. We see that 𝔭^{}and 𝔭, which decomposes into V_{2} V_{3}, have no equivalent U 1 _{1,1,1}^{2}-submodule in common and thus will be able to apply Theorem 3.2.24 later on.

We investigate the horizontal part of the metric, which is determined by b^{2} and c^{2}. With help of Theorem 3.2.18, we will prove that the metric is smooth if b^{2} and c^{2} are even. Since b 0 , c 0 0, we can conclude that in this situation b t b t and c t c t . The odd derivatives of any even function have to vanish at t 0. Since b^{2} and c^{2} are analytic, we thus only have to prove that we can choose b^{2} ^{2m} 0 and c^{2} ^{2m} 0 arbitrarily. As in Section 3.2, these initial values are associated to certain 𝔨-equivariant maps ϕ_{1}, ϕ_{2} : S^{2m} 𝔭^{}S^{2} 𝔭 . We will describe those maps in detail. Let { \atop e_{1}}, …, { \atop e_{4}} be an orthonormal basis of 𝔭^{}, such that the matrix representation of SU 2 with respect to this basis is the standard one. The radial coordinate t of 𝔭^{}satisfies

dt dt { \atop e^{1}} { \atop e^{1}} … { \atop e^{4}} { \atop e^{4}} .

b^{2} and c^{2} are the coefficients of e^{3} e^{3} e^{4} e^{4} and e^{5} e^{5} e^{6} e^{6}. Since b^{2} ^{2m} 0 and c^{2} ^{2m} 0 are derivatives in the t-direction, ϕ_{1} and ϕ_{2} are determined by

\begin{matrix}ϕ_{1}  ^{m} { \atop e^{1}} { \atop e^{1}} … { \atop e^{4}} { \atop e^{4}} & e^{3} e^{3} e^{4} e^{4} \\ ϕ_{2}  ^{m} { \atop e^{1}} { \atop e^{1}} … { \atop e^{4}} { \atop e^{4}} & e^{5} e^{5} e^{6} e^{6} \end{matrix}

On the orthogonal complement of ^{m} { \atop e^{1}} { \atop e^{1}} … { \atop e^{4}} { \atop e^{4}} in S^{2m} 𝔭^{}, both of these maps vanish. Since SU 2 U 1 ^{2} acts by isometries on 𝔭^{}, { \atop e^{1}} { \atop e^{1}} … { \atop e^{4}} { \atop e^{4}} is left invariant by this action. ϕ_{1} and ϕ_{2} thus map a 𝔰𝔲 2 2𝔲 1 -invariant object to another 𝔰𝔲 2 2𝔲 1 -invariant object. Any choice of b^{2} ^{2m} 0 and c^{2} ^{2m} 0 is associated to the map b^{2} ^{2m} 0 ϕ_{1} c^{2} ^{2m} 0 ϕ_{2}, which is 𝔰𝔲 2 2𝔲 1 -equivariant. From Theorem 3.2.18, it therefore follows that these initial values can be chosen arbitrarily.

The condition on a and f can be proven by similar means as above. For the same reasons as in the S^{2} S^{2} S^{2}-case, the 2m^{th} derivatives of \frac{1}{t^{2}} a^{2} and \frac{1}{t^{2}} f^{2} are associated to elements of W_{2m}^{v}. We therefore search for the maps S^{2m} 𝔭^{}S^{2} 𝔭^{}which represent \frac{1}{t^{2}} a^{2} ^{2m} 0 and \frac{1}{t^{2}} f^{2} ^{2m} 0 and prove that they are 𝔨-equivariant. Let γ be the geodesic with γ 0 0 𝔭^{} and γ^{}0 { \atop e_{1}}. In the coordinates which we have chosen, γ t t{ \atop e_{1}}. The tangent space of the collapsing sphere coincides with the orthogonal complement of γ^{}t . This space is spanned by { \atop e_{2}}, { \atop e_{3}}, { \atop e_{4}} . The tangent space of S^{3} is also spanned by e_{7}, e_{1}, e_{2} . From now on, we will identify both triples with each other. This is possible, since span { \atop e_{2}} and span e_{7} as well as span { \atop e_{3}}, { \atop e_{4}} and span e_{1}, e_{2} are equivalent with respect to the action of U 1 _{1,1,1}^{2}. We are interested in the value of

\frac{^{2m}}{{ \atop e_{1}^{2m}}} g { \atop e_{i}}, { \atop e_{i}} ,

where i 2, 3, 4 . For i 3, 4 , this term describes \frac{1}{t^{2}} a^{2} ^{2m} 0 and for i 2 it is \frac{1}{t^{2}} f^{2} ^{2m} 0 . We will extend the above derivatives to SU 2 -equivariant maps φ : 𝔭^{}ℝ. In order to do this, we first define the following function on SU 2 :

k \frac{^{2m}}{k { \atop e_{1}} ^{2m}}g k { \atop e_{i}}, k { \atop e_{i}} .

Since SU 2 S^{3} 𝔭^{}, we can extend this function by linearity to all of 𝔭^{}. We consider the case m 1. In this situation, we have to construct the two K-equivariant maps ϕ_{1}, ϕ_{2} : S^{2} 𝔭^{}S^{2} 𝔭^{}which correspond to \frac{1}{t^{2}} a^{2} ^{}0 and \frac{1}{t^{2}} f^{2} ^{}0 . With help of the function φ, we obtain the following relations for ϕ_{1} and ϕ_{2}:

\begin{matrix}ϕ_{1} L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} & L_{k}^{}{ \atop e^{3}} L_{k}^{}{ \atop e^{3}} L_{k}^{}{ \atop e^{4}} L_{k}^{}{ \atop e^{4}} \\ ϕ_{2} L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} & L_{k}^{}{ \atop e^{2}} L_{k}^{}{ \atop e^{2}} \end{matrix}

where L_{k}^{}denotes the pull-back of the left-multiplication with k SU 2 . By polarization, the ϕ_{i} become SU 2 -equivariant maps on all of S^{2} 𝔭^{}. Since { \atop e_{1}} { \atop e_{1}}, { \atop e_{2}} { \atop e_{2}} and { \atop e_{3}} { \atop e_{3}} { \atop e_{4}} { \atop e_{4}} are all U 1 _{1,1,1}^{2}-invariant, ϕ_{1} and ϕ_{2} are equivariant with respect to all of K. If m 2, we have to describe the K-equivariant maps ϕ_{1}^{m}, ϕ_{2}^{m} : S^{2m} 𝔭^{}S^{2} 𝔭^{}which are associated to \frac{1}{t^{2}} a^{2} ^{2m} 0 and \frac{1}{t^{2}} f^{2} ^{2m} 0 . These derivatives can be considered as the 2m^{th} derivatives of φ : 𝔭^{}ℝ in the radial direction. We therefore have:

\begin{matrix}ϕ_{1}^{m}  ^{m 1} dt dt L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} & L_{k}^{}{ \atop e^{3}} L_{k}^{}{ \atop e^{3}} L_{k}^{}{ \atop e^{4}} L_{k}^{}{ \atop e^{4}} \\ ϕ_{2}^{m}  ^{m 1} dt dt L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} & L_{k}^{}{ \atop e^{2}} L_{k}^{}{ \atop e^{2}} , \end{matrix}

where as usual

dt dt { \atop e^{1}} { \atop e^{1}} { \atop e^{2}} { \atop e^{2}} { \atop e^{3}} { \atop e^{3}} { \atop e^{4}} { \atop e^{4}}

and is the symmetric product. On the orthogonal complement of ^{m 1} dt dt S^{2} 𝔭^{} S^{2m} 𝔭^{}both of these maps vanish. Since dt dt is K-invariant, the K-equivariance of ϕ_{1}^{m} and ϕ_{2}^{m} directly follows from the equivariance of ϕ_{1} and ϕ_{2}. Again, we conclude with help of Theorem 3.2.18 that we can choose \frac{1}{t^{2}} a^{2} ^{2m} 0 and \frac{1}{t^{2}} f^{2} ^{2m} 0 for all m 1 arbitrarily. If \frac{1}{t^{2}} a^{2} and \frac{1}{t^{2}} f^{2} are even analytic functions, a^{}0 \frac{1}{2}, and f^{}0 \frac{3}{2}, all conditions from Theorem 3.2.18 on a and f are satisfied. Since a^{}0 , f^{}0 0, a and f have to be odd if \frac{1}{t^{2}} a^{2} and \frac{1}{t^{2}} f^{2} are even. By similar arguments as in the S^{2} S^{2} S^{2}-case, we see that our solutions of (5.6) with singular orbit S^{2} S^{2} satisfy all the conditions from page 122. All in all, we have proven that all of our solutions are smooth, complete, and non-compact.

In the S^{2} S^{2}-case, we have not calculated the dimension of the spaces W_{m}^{h} and W_{2}^{v}. Nevertheless, we have constructed some elements of those spaces and thus are able to make some statements on the cohomogeneity-one Einstein metrics with singular orbit S^{2} S^{2}. According to Theorem 3.2.24, the invariant metric on the singular orbit can be chosen arbitrarily. Since the maps S^{2} 𝔭^{}S^{2} 𝔭^{}which we have described above correspond to a^{}0 and f^{}0 , these initial values can be chosen freely, too. We finally sum up the results of this section:

Theorem 5.2.2. Let M, Ω be a parallel Spin 7 -manifold with a cohomogeneity-one action of SU 2 ^{3} which preserves Ω. In this situation, the following statements are true:

1. The principal orbits are SU 2 ^{3}-equivariantly diffeomorphic to Q^{1,1,1}.

2. The metric g which is associated to Ω is special Kähler and its holonomy is all of SU 4 .

3. g has the matrix representation (5.3) with respect to the basis e_{1}, …, e_{7} from page 104. The metric functions satisfy the equations (5.6), whose solutions are described by (5.7) and (5.8). The Kähler form is given by (5.14).

If M has a singular orbit, which has to be the case if M, g is complete, it is S^{2} S^{2} or S^{2} S^{2} S^{2}. Any SU 2 ^{3}-invariant metric on the singular orbit can be extended to a unique cohomogeneity-one metric with holonomy SU 4 , which is non-compact. If the singular orbit is S^{2} S^{2}, the metric is smooth and complete. If the singular orbit is S^{2} S^{2} S^{2}, the metric is not differentiable at the singular orbit. Outside of the singular orbit, the metric is smooth and any geodesic which does not intersect the singular orbit can be infinitely extended. In any of these cases, M is a vector bundle over the singular orbit.

With help of Theorem 3.2.24, we have also obtained results on cohomogeneity-one Einstein metrics with principal orbit Q^{1,1,1}:

Theorem 5.2.3. Let M be a cohomogeneity-one manifold, whose principal orbit is Q^{1,1,1}. We assume that M has a singular orbit, which has to be either S^{2} S^{2} S^{2} or S^{2} S^{2}. Any SU 2 ^{3}-invariant metric on M is described by a matrix of type (5.3) or equivalently by the four metric functions a, b, c, f : I ℝ, where I is an interval. Without loss of generality, we assume that the singular orbit is at 0 I. In this situation, the following statements are true:

1. Let the singular orbit be S^{2} S^{2} S^{2}. For any choice of a_{0}, b_{0}, c_{0}, f_{3}, λ ℝ, there exists a unique SU 2 ^{3}-invariant Einstein metric on a tubular neighborhood of S^{2} S^{2} S^{2} such that:

(a) a 0 a_{0}, b 0 b_{0}, c 0 c_{0},

(b) f^{}0 f_{3}, and

(c) the Einstein constant is λ.

2. Let the singular orbit be S^{2} S^{2}. For any choice of b_{0}, c_{0}, a_{3}, f_{3}, λ ℝ, there exists a SU 2 ^{3}-invariant Einstein metric on a tubular neighborhood of S^{2} S^{2} such that:

(a) a 0 0, b 0 b_{0}, c 0 c_{0},

(b) a^{}0 a_{3}, f^{}0 f_{3}, and

(c) the Einstein constant is λ.

Remark 5.2.4. 1. It is likely that the Kähler metrics with singular orbit S^{2} S^{2} S^{2} can be modified in such a way that they become smooth. If we replace the principal orbit by a space of type Q^{1,1,1} ℤ_{2}, where ℤ_{2} is contained in the isotropy group of the singular orbit, we obtain a new space of cohomogeneity one with singular orbit S^{2} S^{2} S^{2}.

The length of the collapsing circle gets multiplied by \frac{1}{2} and the smoothness condition on f^{}0 is satisfied. Since the isotropy group of the principal orbit has changed, it has to be checked if the other smoothness conditions have changed, too. If ℤ_{2} preserves the SU 4 -structure, it turns into a SU 4 -structure on the new space. Otherwise, we obtain a SU 4 ℤ_{2}-structure. In Section 5.4, we will describe a similar construction in detail.

2. The metrics with holonomy SU 4 which we have constructed have also been considered by Cvetič, Gibbons, Lü, and Pope in [25]. In that paper, the authors construct Ricci-flat Kähler-metrics on holomorphic vector bundles over a product of several Einstein-Kähler manifolds. As a special case, they obtain Ricci-flat Kähler metrics of cohomogeneity one with principal orbit Q^{1,1,1} and a single singular orbit of type S^{2} S^{2} or S^{2} S^{2} S^{2}. The authors also prove that those metrics are non-compact and complete away from the singular orbit. In [21], the same authors consider those metrics in another context. Both papers are based on earlier works by Berard-Bergery [8], Page and Pope [58], and Stenzel [64]. The examples with singular orbit S^{2} S^{2} S^{2} have also been considered by Herzog and Klebanov [41]. Although the metrics with holonomy SU 4 which we have constructed are known, the proof that any parallel Spin 7 -manifold with a cohomogeneity-one action of SU 2 ^{3} is one of our examples is new. Our results prove that it is impossible to deform the Kähler examples into metrics with holonomy Spin 7 without loosing the cohomogeneity-one property. Moreover, we have proven that all (not necessarily parallel) Spin 7 -structures of cohomogeneity one with principal orbit Q^{1,1,1} have to reduce to a SU 4 -structure. This fact has not been mentioned in the literature before, either. Finally, we remark that the Einstein metrics from the above theorem which are not included in Theorem 5.2.2 have, as far as the author knows, not been considered in the literature before. All in all, we hope to have introduced an interesting, more algebraic approach to the issue of special cohomogeneity-one metrics with principal orbit Q^{1,1,1}.

5.3. THE PRINCIPAL ORBIT M^{1,1,0} 127

5.3 The principal orbit M^{1,1,0}

In this section, we will investigate parallel cohomogeneity-one Spin 7 -structures whose principal orbit is of type SU 3 SU 2 SU 2 U 1 , where the semisimple part of SU 2 U 1 is embedded into SU 3 SU 3 SU 2 . These homogeneous spaces will be denoted by M^{k,l,m}. The indices k, l, and m describe the embedding of the smaller group into the larger. As in the previous section, we will see that M^{k,l,m} carries only for a special choice of k, l, and m a homogeneous G_{2}-structure. This fact already has been stated in Theorem 4.2. We will construct three classes of metrics with holonomy Spin 7 which are distinguished by their singular orbit. The first one was found by Cvetič et al. in [21], and the second and third class have been considered by the same authors in [25]. Most of the results we find in this section are analogous to those of Section 5.2.

Our first step is to describe the possible embeddings of SU 2 U 1 into SU 3 SU 2 . In the literature, there are two conventions how to denote these embeddings. The first one is used by Friedrich, Kath, Moroianu, and Semmelmann in [37]. Castellani et al. [19] and Fabbri [33] use the second convention. For our considerations, we will take the notation of Castellani.

In order to explain the meaning of the indices k, l, and m, we first consider spaces of type SU 3 SU 2 U 1 SU 2 U 1 U 1 instead of SU 3 SU 2 SU 2 U 1 . These spaces have been important in research on Kaluza-Klein supergravity in 11 dimensions (see [19], [33]). Since we want to distinguish the different abelian factors, we will denote the newly introduced spaces by SU 3 SU 2 U 1 SU 2 U 1 ^{}U 1 ^{}. The inclusion ı : SU 2 U 1 ^{}U 1 ^{}SU 3 SU 2 U 1 is determined by its differential dı _{e} : 𝔰𝔲 2 𝔲 1 ^{}𝔲 1 ^{}𝔰𝔲 3 𝔰𝔲 2 𝔲 1 , since both groups are connected. We describe the Lie algebra 𝔰𝔲 3 𝔰𝔲 2 𝔲 1 by the following matrices:

; ; { \atop }\begin{aligned}X \\ & Y \\ & & Z \end{aligned} { \atop } ; ; ; ; ; X 𝔰𝔲 3 , Y 𝔰𝔲 2 , Z 𝔲 1 ; ; .

𝔰𝔲 2 𝔰𝔲 2 𝔲 1 is embedded into this Lie algebra by:

dı _{e} ; ; ; \begin{aligned}ti & z \\ z̄ & ti \\ & & 0 \\ & & & 0 \end{aligned} ; ; ; : ; ; ; ; ; ; ; \begin{aligned}ti & z & 0 \\ z̄ & ti & 0 \\ 0 & 0 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \\ & & & & & 0 \end{aligned} ; ; ; ; ; ; ; .

We have to map 𝔲 1 ^{}and 𝔲 1 ^{}in such a way into 𝔰𝔲 3 𝔰𝔲 2 𝔲 1 that the sum of dı _{e} 𝔲 1 ^{}, dı _{e} 𝔲 1 ^{}, and dı _{e} 𝔰𝔲 2 is direct, and the three summands commute with each other. We choose the following Cartan subalgebra of 𝔰𝔲 3 𝔰𝔲 2 𝔲 1 :

𝔱 : ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}xi & 0 & 0 \\ 0 & yi & 0 \\ 0 & 0 & x y i \\ & & & zi & 0 \\ & & & 0 & zi \\ & & & & & wi \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; x, y, z, w ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

We may assume without loss of generality that dı _{e} 𝔲 1 ^{}, dı _{e} 𝔲 1 ^{}, and the image of the standard Cartan subalgebra of 𝔰𝔲 2 are subalgebras of 𝔱. The orthogonal complement of dı _{e} 𝔰𝔲 2 𝔲 1 ^{}𝔲 1 ^{}𝔱 in 𝔱 with respect to the Killing form is a one-dimensional Lie algebra. This algebra we will denote by 𝔲 1 ^{}. It is easy to see that ı is determined by 𝔲 1 ^{} up to a conjugation. Moreover, there is a one-to-one correspondence between the conjugacy classes of the possible ı and the one-dimensional subalgebras of 𝔱 which are orthogonal to dı _{e} 𝔰𝔲 2 . These subalgebras can be described by three numbers k, l, and m. We define:

𝔲 1 k,l,m^{}: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}\frac{k}{2}ix & 0 & 0 \\ 0 & \frac{k}{2}ix & 0 \\ 0 & 0 & kix \\ & & & \frac{l}{2}ix & 0 \\ & & & 0 & \frac{l}{2}ix \\ & & & & & mix \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; x ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

Any choice of k, l, and m determines an embedding of SU 2 U 1 ^{}U 1 ^{}into SU 3 SU 2 U 1 which we denote by ı_{k,l,m}. The coset space SU 3 SU 2 U 1 ı_{k,l,m} SU 2 U 1 ^{}U 1 ^{}is denoted by M^{k,l,m}. In order to make the exponential of dı _{e} 𝔲 1 ^{}𝔲 1 ^{} a closed subgroup, we have to choose k, l, and m as rational numbers. Since 𝔲 1 nk,nl,nm^{} equals 𝔲 1 k,l,m^{}, we assume from now on that k, l, m consists of coprime integers. Castellani et al. have proven in [19] that

M^{k,l,m} M^{k,l,0} ℤ_{r} with r \frac{3k^{2} l^{2}}{gcd 2mk, ml, 3k^{2} l^{2}} .

It therefore suffices to consider only the spaces of type M^{k,l,0}. Since in this case the abelian factor of SU 3 SU 2 U 1 is a subgroup of U 1 ^{}U 1 ^{}, M^{k,l,0} is a space of type SU 3 SU 2 SU 2 U 1 . In the notation of [37], the space M^{k,l,0} is denoted by M^{3k,2l}. For the rest of this section, we will shortly denote ı_{k,l,0} SU 2 U 1 by SU 2 U 1 and dı_{k,l,0} _{e} 𝔰𝔲 2 𝔲 1 by 𝔰𝔲 2 𝔲 1 , as long as the meaning of this abbreviated notation is clear from the context. We can restrict the values of k and l even more. Let

P : ; ; ; ; ; \begin{aligned}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ & & & 0 & i \\ & & & i & 0 \end{aligned} ; ; ; ; ; SU 3 SU 2 ,

and let ϕ_{P} : SU 3 SU 2 SU 3 SU 2 be defined by ϕ_{P} Q : P QP^{1} . ϕ_{P} is an automorphism of SU 3 SU 2 and maps ı_{k,l,0} SU 2 U 1 into ı_{k, l,0} SU 2 U 1 . Therefore, the spaces M^{k,l,0} and M^{k, l,0} are SU 3 SU 2 -equivariantly diffeomorphic. Since M^{k,l,0} and M^{k, l,0} are the same, too, we can assume without loss of generality that k, l ℕ_{0} ℕ_{0} 0, 0 .

Next, we decide which M^{k,l,0} admit an invariant G_{2}-structure. In order to answer this question, we first choose a basis e_{1}, …, e_{11} of 𝔰𝔲 3 𝔰𝔲 2 :

e_{1} : ; ; ; ; ; \begin{aligned}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; , e_{2} : ; ; ; ; ; \begin{aligned}0 & 0 & i \\ 0 & 0 & 0 \\ i & 0 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; , e_{3} : ; ; ; ; ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; ,

e_{4} : ; ; ; ; ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & i \\ 0 & i & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; , e_{5} : ; ; ; ; ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \end{aligned} ; ; ; ; ; , e_{6} : ; ; ; ; ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ & & & 0 & i \\ & & & i & 0 \end{aligned} ; ; ; ; ; ,

e_{7} : ; ; ; ; ; \begin{aligned}\frac{k}{2}i & 0 & 0 \\ 0 & \frac{k}{2}i & 0 \\ 0 & 0 & ki \\ & & & \frac{l}{2}i & 0 \\ & & & 0 & \frac{l}{2}i \end{aligned} ; ; ; ; ; , e_{8} : ; ; ; ; ; \begin{aligned}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; ,

e_{9} : ; ; ; ; ; \begin{aligned}0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; , e_{10} : ; ; ; ; ; \begin{aligned}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 0 \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned} ; ; ; ; ; ,

e_{11} : ; ; ; ; ; \begin{aligned}\frac{l}{3}i & 0 & 0 \\ 0 & \frac{l}{3}i & 0 \\ 0 & 0 & \frac{2l}{3}i \\ & & & ki & 0 \\ & & & 0 & ki \end{aligned} ; ; ; ; ; .

The bilinear form q on 𝔰𝔲 3 𝔰𝔲 2 which is defined by q X, Y : tr XY determines a biinvariant metric on SU 3 SU 2 . It is easy to see that e_{1}, …, e_{11} is a q-orthogonal basis of 𝔰𝔲 3 𝔰𝔲 2 , and that e_{8}, …, e_{11} spans 𝔰𝔲 2 𝔲 1 . The other elements of our basis span the q-orthogonal complement 𝔪 of 𝔰𝔲 2 𝔲 1 in 𝔰𝔲 3 𝔰𝔲 2 . As usual, we identify the tangent space of M^{k,l,0} at a fixed point with 𝔪. Our next step is to describe the isotropy action of 𝔰𝔲 2 𝔲 1 on 𝔪. We compute the matrix representations of ad_{e_{8}} _{𝔪}, …, ad_{e_{11}} _{𝔪} with respect to the basis e_{1}, …, e_{7} and obtain:

ad_{e_{8}} _{𝔪} ; ; ; ; ; ; ; ; ; \begin{aligned}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ & & & & 0 & 0 \\ & & & & 0 & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; , ad_{e_{9}} _{𝔪} ; ; ; ; ; ; ; ; ; \begin{aligned}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ & & & & 0 & 0 \\ & & & & 0 & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ,

ad_{e_{10}} _{𝔪} ; ; ; ; ; ; ; ; ; \begin{aligned}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ & & & & 0 & 0 \\ & & & & 0 & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; , ad_{e_{11}} _{𝔪} ; ; ; ; ; ; ; ; ; \begin{aligned}0 & l & 0 & 0 \\ l & 0 & 0 & 0 \\ 0 & 0 & 0 & l \\ 0 & 0 & l & 0 \\ & & & & 0 & 2k \\ & & & & 2k & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; .

We consider the Cartan subalgebra of 𝔰𝔲 2 𝔲 1 which is generated by e_{10} and e_{11}. Its adjoint action on 𝔪 is given by:

ad_{span e_{10},e_{11}} _{𝔪} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 & x \\ x & 0 \\ & & 0 & y \\ & & y & 0 \\ & & & & 0 & z \\ & & & & z & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; { \atop }\begin{matrix}k \\ k \\ l \end{matrix} { \atop }, { \atop }\begin{matrix}x \\ y \\ z \end{matrix} { \atop }0 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

We assume that there exists a SU 3 SU 2 -invariant G_{2}-structure on M^{k,l,0}. As in Section 5.2, we compare the action of span e_{10}, e_{11} on 𝔪 with the action of the standard Cartan subalgebra 2𝔲 1 𝔤_{2} on Im 𝕆 . 𝔪 and ℝ^{7} have to be equivalent 2𝔲 1 -modules. By considering the matrix representation of 2𝔲 1 𝔤_{2} from page 106, we can easily deduce that k 1 and l 1. Therefore, we will restrict ourselves from now on to the principal orbit M^{1,1,0}.

By a short calculation, we see that the product of SU 2 and the one-dimensional Lie group generated by e_{11} is a direct one. Thus, M^{1,1,0} indeed can be written as a coset space of type SU 3 SU 2 SU 2 U 1 . We could prove by another calculation that SU 2 U 1 acts as SU 2 U 1 ℤ_{2} on 𝔪. The action of SU 3 SU 2 on M^{1,1,0} therefore is only almost effective. Since G_{2} contains a subgroup of type SU 2 U 1 ℤ_{2}, M^{1,1,0} admits a homogeneous G_{2}-structure although SU 2 U 1 G_{2}.

We classify the SU 3 SU 2 -invariant metrics on M^{1,1,0}. Before we can do this, we have to decompose the tangent space into irreducible 𝔰𝔲 2 𝔲 1 -modules. The semisimple part of 𝔰𝔲 2 𝔲 1 obviously acts by its two-dimensional complex representation on span e_{1}, …, e_{4} . This space is a 𝔲 1 -module, too, although not an irreducible one. 𝔰𝔲 2 acts trivially on span e_{5}, e_{6} , while 𝔲 1 acts irreducibly on it. Both Lie algebras act trivially on span e_{7} . All in all, 𝔪 splits into three pairwise inequivalent 𝔰𝔲 2 𝔲 1 -modules V_{1}, V_{2}, and V_{3}:

V_{1} : span e_{1}, e_{2}, e_{3}, e_{4}
V_{2} : span e_{5}, e_{6}
V_{3} : span e_{7}

Any invariant metric g on M^{1,1,0} can be identified by the formula

q φ X , Y : g X, Y X, Y 𝔪

with a 𝔰𝔲 2 𝔲 1 -equivariant map φ : 𝔪 𝔪, which has to be q-symmetric and positive definite. We classify the possible φ by Schur’s lemma. V_{1} can be identified with ℂ^{2} on which SU 2 U 1 acts as U 2 by its standard representation. The only equivariant endomorphisms of ℂ^{2} are the complex multiples of the identity map. Since φ has to be symmetric, φ _{V_{1}} has in fact to be a real multiple of the identity. The other components of φ can be determined in a similar manner. We see that the matrix representation of g with respect to the basis e_{1}, …, e_{7} has to be given by:

 ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}a^{2} & 0 & 0 & 0 \\ 0 & a^{2} & 0 & 0 \\ 0 & 0 & a^{2} & 0 \\ 0 & 0 & 0 & a^{2} \\ & & & & b^{2} & 0 \\ & & & & 0 & b^{2} \\ & & & & & & c^{2} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ; with a, b, c ℝ 0 .  (5.17)

Conversely, any such matrix corresponds to an invariant metric on M^{1,1,0}. As in Section 5.2, we also allow negative values of a, b, and c, since a change of their signs changes the frame of the Spin 7 -structure, although it leaves the metric invariant. Let M, g be a cohomogeneityone manifold with an isometric SU 3 SU 2 -action whose principal orbit is M^{1,1,0}. We can identify the union of all principal orbits by a suitable diffeomorphism with M^{1,1,0} I, where I is an open interval. With this identification, g can be described by:

g g_{t} dt^{2} ,

where t is the coordinate directed along I and g_{t} is an invariant metric on the principal orbit at t. Near a singular orbit, g_{t} will degenerate.

Next, we construct a frame f_{0}, …, f_{7} of a SU 3 SU 2 -invariant Spin 7 -structure on M. We recall that the SU 2 -factor of SU 2 U 1 acts by its two-dimensional complex representation on V_{1} and trivially on V_{2} V_{3}. SU 2 also acts on ℍϵ Im 𝕆 by leftmultiplication by unit-quaternions. Since this is the only SU 2 -action by a subgroup of G_{2} which splits Im 𝕆 into 𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ}, we assume that the last four elements of the frame are in V_{1}. f_{0} is as usual chosen as \frac{}{t}. f_{1} will be a multiple of e_{7}, since i 𝕆 is invariant with respect to the subalgebra 𝔰𝔲 2 _{1} 𝔲 1 of 𝔤_{2} from Chapter 4. span f_{2}, f_{3} therefore will coincide with V_{2}. We compare the action of span e_{10}, e_{11} on 𝔪 with the action of 2𝔲 1 𝔤_{2} on Im 𝕆 . span e_{1}, e_{2} and span e_{3}, e_{4} are both irreducible with respect to the action of span e_{10}, e_{11} and thus will be identified with span f_{4}, f_{5} and span f_{6}, f_{7} . Since the above both actions have to be equivariant, we have to change the order of some of the pairs e_{2i 1} , e_{2i} . A frame of a Spin 7 -structure also has to be orthonormal with respect to g. Motivated by these considerations, we choose the following frame:

\begin{matrix}f_{0} : \frac{}{t} & f_{1} : \frac{1}{c}e_{7} & f_{2} : \frac{1}{b}e_{6} & f_{3} : \frac{1}{b}e_{5} \\ f_{4} : \frac{1}{a}e_{1} & f_{5} : \frac{1}{a}e_{2} & f_{6} : \frac{1}{a}e_{4} & f_{7} : \frac{1}{a}e_{3} \end{matrix}  (5.18)

By a short calculation, we see that the matrix representation of the isotropy action with respect to the above basis is the same as of 𝔰𝔲 2 _{1} 𝔲 1 𝔤_{2}. The four-form Ω which is associated to the above frame is given by:

\begin{aligned}Ω & a^{4} e^{1234} a^{2}b^{2} e^{1256} a^{2}b^{2} e^{3456} \\ & a^{2}bc e^{1367} a^{2}bc e^{1457} a^{2}bc e^{2357} a^{2}bc e^{2467} \\ & a^{2}b e^{135} dt a^{2}b e^{146} dt a^{2}b e^{236} dt a^{2}b e^{245} dt \\ & a^{2}c e^{127} dt a^{2}c e^{347} dt b^{2}c e^{567} dt \end{aligned}  (5.19)

In order to calculate dΩ, we apply the same methods as in Section 5.2. After that, we can express the condition dΩ 0 as a system of ordinary differential equations for the metric functions a, b, and c:

\begin{matrix}\frac{a^{}}{a} & \frac{3}{8} \frac{c}{a^{2}} \\ \frac{b^{}}{b} & \frac{1}{4} \frac{c}{b^{2}} \\ \frac{c^{}}{c} & 8\frac{1}{c} \frac{1}{4} \frac{c}{b^{2}} \frac{3}{4} \frac{c}{a^{2}} \end{matrix}  (5.20)

The details of the calculations can be found in Appendix B. Next, we discuss the solutions of the above system.

Without loss of generality, we assume that the orbit on which we choose the initial conditions is at t 0. We denote the initial values a 0 , b 0 , and c 0 by a_{0}, b_{0}, and c_{0}. Furthermore, we define C t : _{0}^{t} c s ds. The first two equations of the system (5.20) can be simplified to:

\begin{matrix}a^{2} & \frac{3}{4}C a_{0}^{2} \\ b^{2} & \frac{1}{2}C b_{0}^{2} \end{matrix}

By inserting this into the last equation, we obtain:

c^{}8 \frac{c^{2}}{2C 4b_{0}^{2}} \frac{3c^{2}}{3C 4a_{0}^{2}} .

As in Section 5.2, we define τ : C t and denote the derivation with respect to τ by a dot. With this notation, the above equation becomes:

cc 8 \frac{c^{2}}{2τ 4b_{0}^{2}} \frac{3c^{2}}{3τ 4a_{0}^{2}} .

Since this is a linear differential equation for c^{2}, it can be explicitly solved. For the homogeneous problem, we obtain the following solution:

c τ ^{2} K \frac{1}{τ 2b_{0}^{2} τ \frac{4}{3}a_{0}^{2} ^{2}} ,

where K is a constant. The most general solution of the inhomogeneous equation for c is:

c τ ^{2} \frac{1}{τ 2b_{0}^{2} τ \frac{4}{3}a_{0}^{2} ^{2}} K 8 _{0}^{τ} s 2b_{0}^{2} s \frac{4}{3}a_{0}^{2} ^{2}ds .

Next, we have to determine the value of K. By inserting τ 0 into the above equation, we obtain K \frac{32}{9}a_{0}^{4}b_{0}^{2}c_{0}^{2}:

\begin{aligned}c τ ^{2} & \frac{32a_{0}^{4}b_{0}^{2}c_{0}^{2}}{9 τ 2b_{0}^{2} τ \frac{4}{3}a_{0}^{2} ^{2}} \\ & \frac{8}{τ 2b_{0}^{2} τ \frac{4}{3}a_{0}^{2} ^{2}} _{0}^{τ} s 2b_{0}^{2} s \frac{4}{3}a_{0}^{2} ^{2}ds . \end{aligned}  (5.21)

We are finally able to give an explicit description of the cohomogeneity-one metric:

\begin{aligned}g & : \frac{3}{4}τ a_{0}^{2} e^{1} e^{1} e^{2} e^{2} e^{3} e^{3} e^{4} e^{4} \\ & \frac{1}{2}τ b_{0}^{2} e^{5} e^{5} e^{6} e^{6} c τ ^{2} e^{7} e^{7} dτ^{2} . \end{aligned}  (5.22)

As in Section 5.2, we put back the question of the compactness and completeness until we have determined the possible initial values at the singular orbit.

Our next question is if there are any other cohomogeneity-one metrics with principal orbit M^{1,1,0} and a holonomy contained in Spin 7 . We fix an invariant metric g and an orientation on M^{1,1,0} and consider the set of all invariant G_{2}-structures with the same associated metric and orientation. According to Lemma 3.1.50, this set can be described by:

Norm_{SO 7} 𝔰𝔲 2 _{1} 𝔲 1 Norm_{G_{2}} 𝔰𝔲 2 _{1} 𝔲 1 ,

where Norm_{G} 𝔰𝔲 2 _{1} 𝔲 1 is defined as:

g G g 𝔰𝔲 2 _{1} 𝔲 1 g^{1} 𝔰𝔲 2 _{1} 𝔲 1 .

We do not need the explicit form of the normalizers, since we can apply the knowledge which we have acquired in Section 5.2: Any homogeneous G_{2}-structure on M^{1,1,0} corresponds to a 𝔰𝔲 2 _{1} 𝔲 1 -equivariant map ψ : 𝔪 ℝ^{7}. As usual, let 2𝔲 1 𝔰𝔲 2 _{1} 𝔲 1 be the standard Cartan subalgebra of 𝔤_{2}, which acts on 𝔪 and on ℝ^{7}. ψ is obviously also 2𝔲 1 -equivariant. The set of all 2𝔲 1 -invariant G_{2}-structures on the tangent space 𝔪 which can be extended to a fixed SO 7 -structure is generated by a U 1 -action on one of those G_{2}-structures. This fact was proven in Section 5.2. The U 1 -action on the G_{2}-structures is generated by another U 1 -action on 𝔪. This action is not unique, since we can change it by elements of G_{2}. We have seen in Section 5.2 that one possibility for the action on 𝔪 is described with respect to e_{1}, …, e_{7} by the following matrices:

S :  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}R_{θ} \\ & R_{θ} \\ & & R_{θ} \\ & & & 1 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; θ ℝ ;  ;  ;  ;  ;  ; ,  (5.23)

where R_{θ} denotes the rotation in the plane around an angle of θ. Conjugation by any element of S leaves not only 2𝔲 1 but also 𝔰𝔲 2 _{1} 𝔲 1 invariant. Therefore, the action of S on a fixed SU 3 SU 2 -invariant G_{2}-structure ω indeed generates the set of all invariant G_{2}- structures which have the same extension to a SO 7 -structure as ω. These G_{2}-structures also can be obtained as the pull-back of ω by certain isometries of M^{1,1,0}. We consider the group:

T : ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}e^{\frac{iφ}{2}} & 0 & 0 \\ 0 & e^{\frac{iφ}{2}} & 0 \\ 0 & 0 & e^{iφ} \\ & & & e^{\frac{iφ}{2}} & 0 \\ & & & 0 & e^{\frac{iφ}{2}} \end{aligned} ; ; ; ; ; ; ; ; ; : T_{φ} ; ; ; ; ; ; ; ; ; ; ; ; ; φ ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

T is a subgroup of Norm_{SU 3 SU 2} SU 2 U 1 . Therefore, conjugation by T_{φ} induces a well-defined diffeomorphism of M^{1,1,0}. We prove that this diffeomorphism is even an isometry. The neutral element e of SU 3 SU 2 is fixed by conjugation by T_{φ}. The differential of this map at e is determined by the adjoint action of T_{φ} SU 3 SU 2 restricted to 𝔪. Since the tangent space of T is spanned by e_{7} and T is connected, we only have to prove that ad_{e_{7}} _{𝔪} is skew-symmetric with respect to g. We obtain by a straightforward calculation:

ad_{e_{7}} _{𝔪} ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 & \frac{3}{2} \\ \frac{3}{2} & 0 \\ & & 0 & \frac{3}{2} \\ & & \frac{3}{2} & 0 \\ & & & & 0 & 1 \\ & & & & 1 & 0 \\ & & & & & & 0 \end{aligned} ; ; ; ; ; ; ; ; ; ;

with respect to the basis e_{1}, …, e_{7} . This endomorphism is clearly skew-symmetric. It is easy to see that it is orientation-preserving, too. Unfortunately, the adjoint action of T on 𝔪 does not coincide with the action (5.23). Nevertheless, it generates the same family of G_{2}-structures as the action of S on 𝔪. The reason for this simply is that S and the action of T both are contained in Norm_{SO 7} 𝔰𝔲 2 _{1} 𝔲 1 , but not in Norm_{G_{2}} 𝔰𝔲 2 _{1} 𝔲 1 . In fact, we could choose T as any connected one-dimensional subgroup of the standard maximal torus of SU 3 SU 2 which is not contained in SU 2 U 1 .

To sum up, we have proven that the one-parameter family of all invariant G_{2}-structures with the same associated metric and orientation as ω can be obtained by the action of certain isometries. Furthermore, this family is diffeomorphic to a circle. We conclude with help of similar arguments as in Section 5.2 that the holonomy of our metrics is SU 4 . The existence of a parallel vector field can be excluded as in 5.2, too. Therefore, the holonomy is either SU 4 , Sp 2 , or SU 2 SU 2 and the metric thus is Kähler.

The Kähler form η has to be SU 3 SU 2 -invariant. The space of all two-forms on 𝔪 which are invariant with respect to the U 1 -factor of SU 2 U 1 is spanned by:

e^{12} , e^{34} , e^{56} , e^{7} dt .

η therefore has to be of the following type:

η ϵ_{1}a^{2} e^{12} ϵ_{2}a^{2} e^{34} ϵ_{3}b^{2} e^{56} c e^{7} dt with ϵ_{1}, ϵ_{2}, ϵ_{3} 1, 1 .  (5.24)

The exterior derivative of η is given by:

dη ϵ_{1} 2a^{}a dt e^{12} ϵ_{1} a^{2} 3e^{272} 3e^{117}
ϵ_{2} 2a^{}a dt e^{34} ϵ_{2} a^{2} 3e^{474} 3e^{337}
ϵ_{3} 2b^{}b dt e^{56} ϵ_{3} b^{2} 2e^{676} 2e^{557}
c \frac{3}{4}e^{12} \frac{3}{4}e^{34} \frac{1}{2}e^{56} dt

The condition dη 0 is thus equivalent to:

\frac{a^{}}{a} ϵ_{1} \frac{3}{8} \frac{c}{a^{2}} , \frac{a^{}}{a} ϵ_{2} \frac{3}{8} \frac{c}{a^{2}} , \frac{b^{}}{b} ϵ_{3} \frac{1}{4} \frac{c}{b^{2}} .

In order to make these equations consistent with (5.20), we have to choose ϵ_{1} ϵ_{2} 1 and ϵ_{3} 1. We obtain:

η a^{2} e^{12} a^{2} e^{34} b^{2} e^{56} c e^{7} dt .  (5.25)

If η is closed, no other two-form of type (5.24) can be closed, too. Otherwise, we had c 0, which is impossible, since the metric has to be positive definite. By the same arguments as in Section 5.2, it therefore follows that the holonomy of the metrics which satisfy (5.20) has to be all of SU 4 .

We finally consider the behavior of our metrics near a singular orbit. Any singular orbit is a homogeneous space of type SU 3 SU 2 K. The possibilities for K are described by the following lemma:

Lemma 5.3.1. Let SU 2 U 1 be embedded into SU 3 SU 2 by the map ı_{1,1,0}, which we have defined at the beginning of this section. Furthermore, let K be a connected, closed group with SU 2 U 1 K SU 3 SU 2 . The Lie algebra of K we denote by 𝔨. In this situation, 𝔨 and K can be found in the table below. Furthermore, K SU 2 U 1 and SU 3 SU 2 K satisfy the following topological conditions:

𝔨  K  K SU 2 U 1  SU 3 SU 2 K
𝔰𝔲 2 2𝔲 1 U 2 U 1  S^{1}  ℂℙ^{2} S^{2}
2𝔰𝔲 2 𝔲 1 U 2 SU 2  S^{3}  ℂℙ^{2}
𝔰𝔲 3 𝔲 1  SU 3 U 1  S^{5} ℤ_{3}  S^{2}
𝔰𝔲 3 𝔰𝔲 2 SU 3 SU 2  M^{1,1,0} S^{7} Γ

where Γ O 8 is an arbitrary discrete subgroup and the group ℤ_{3} which we divide out is λ Id_{ℂ^{3}} λ^{3} 1 .

Proof : 𝔰𝔲 2 𝔲 1 acts on 𝔰𝔲 3 𝔰𝔲 2 by the restriction of the adjoint action. 𝔨 obviously is a 𝔰𝔲 2 𝔲 1 -submodule of 𝔰𝔲 3 𝔰𝔲 2 . Since we have decomposed 𝔰𝔲 3 𝔰𝔲 2 into its submodules, we see that there are only few possibilities for 𝔨:

1. 𝔨 𝔰𝔲 2 𝔲 1 V_{3}: Since e_{7}, 𝔰𝔲 2 𝔲 1 0, K is covered by SU 2 U 1 U 1 , where the last factor is generated by e_{7}. 𝔨 contains the standard Cartan subalgebra of 𝔰𝔲 3 𝔰𝔲 2 . We therefore can rewrite 𝔨 as 𝔰𝔲 2 W_{1} W_{2}, where W_{1} is a onedimensional Lie algebra which is generated by an element of 𝔰𝔲 3 commuting with 𝔰𝔲 2 and W_{2} is generated by an element of the 𝔰𝔲 2 -summand of 𝔰𝔲 3 𝔰𝔲 2 . As we will see in the third case, the intersection of SU 2 and the Lie group which is generated by W_{1} is isomorphic to ℤ_{2}. The Lie group which corresponds to 𝔰𝔲 2 W_{1} therefore is isomorphic to SU 2 U 1 ℤ_{2} U 2 . Since the intersection of the Lie groups which are associated to 𝔰𝔲 2 W_{1} and W_{2} is trivial, K is isomorphic to U 2 U 1 . The quotient K SU 2 U 1 is a one-dimensional Lie group, which is diffeomorphic to a circle. Furthermore, we can conclude that SU 3 SU 2 K SU 3 S U 2 U 1 SU 2 U 1 ℂℙ^{2} S^{2}.

2. 𝔨 𝔰𝔲 2 𝔲 1 V_{2}: We can easily find elements x, y V_{2} such that q x, y , e_{7} 0. Since e_{7} 𝔰𝔲 2 𝔲 1 V_{2}, 𝔨 is not closed under the Lie bracket.

3. 𝔨 𝔰𝔲 2 𝔲 1 V_{2} V_{3}: It is easy to see that V_{2}, V_{2} span e_{7}, e_{11} , V_{2}, V_{3} V_{2}, V_{3}, V_{3} 0, and of course 𝔰𝔲 2 𝔲 1 , V_{2} V_{3} V_{2} V_{3}. These relations prove that 𝔨 is closed under the Lie bracket. We obtain 𝔨 𝔰𝔲 2 𝔰𝔲 2 𝔲 1 . The first summand is spanned by e_{8}, e_{9}, e_{10} . The second summand coincides with the second summand of 𝔰𝔲 3 𝔰𝔲 2 and the last summand is spanned by:

 ;  ;  ;  ;  ; \begin{aligned}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 2i \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned}  ;  ;  ;  ;  ; .  (5.26)

Let i_{1} : SU 2 SU 3 SU 2 be the map which is induced by the embedding of the first 𝔰𝔲 2 -summand. Analogously, we denote the map which is induced by the embedding of the second 𝔰𝔲 2 -summand by i_{2} and define a map i_{3} which is associated to the embedding of the 𝔲 1 -summand into 𝔰𝔲 3 𝔰𝔲 2 . The map

\begin{aligned}π & : SU 2 SU 2 U 1 K \\ π X, Y, Z & : i_{1} X i_{2} Y i_{3} Z \end{aligned}

is a covering map of K. Since its kernel is

1 0 0 1 , 1 0 0 1 , 1 ,  \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} , 1 0 0 1 , 1  ,

SU 2 SU 2 U 1 is a two-fold cover of K. More precisely, we have K SU 2 U 1 ℤ_{2} SU 2 U 2 SU 2 . For our considerations, we need some information on the topology of K H. In order to keep our notation short, we denote the isotropy group SU 2 U 1 of the SU 3 SU 2 -action shortly by H. Let π^{1} H _{e} be the identity component of π^{1} H . The abelian factor of H is

; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}e^{\frac{it}{3}} & 0 & 0 \\ 0 & e^{\frac{it}{3}} & 0 \\ 0 & 0 & e^{\frac{2it}{3}} \\ & & & e^{it} & 0 \\ & & & 0 & e^{it} \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; t ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .

Therefore,

π^{1} H _{e} X, \begin{matrix}e^{3it} & 0 \\ 0 & e^{3it} \end{matrix} , e^{it} ; ; ; X SU 2 , t ℝ .

Since ker π π^{1} H _{e}, π^{1} H is connected. We take a look at the quotient π^{1} K π^{1} H SU 2 SU 2 U 1 SU 2 U 1 . Since we can cancel the first SU 2 -factor, this quotient is a space of type SU 2 U 1 U 1 . The group

1 0 0 1 , \begin{matrix}e^{3it} & 0 \\ 0 & e^{3it} \end{matrix} , e^{it} ; ; ; t ℝ ,

by which we divide, intersects the second factor of SU 2 SU 2 U 1 only at t 2πℤ. By Lemma 3.1.12, we see that π^{1} K π^{1} H S^{3}. The map

π^{}: π^{1} K π^{1} H K H ; \begin{matrix}π^{}k π^{1} H \\ : π k H \end{matrix}

is a diffeomorphism, since π^{1} e H e π^{1} H . Thus, K H S^{3}. From the explicit description of K, which we have found above, we obtain SU 3 SU 2 K SU 3 S U 2 U 1 SU 2 SU 2 ℂℙ^{2}.

4. 𝔨 𝔰𝔲 2 𝔲 1 V_{1}: As in the second case, we can easily find elements x, y V_{1} such that q x, y , e_{7} 0. Therefore, 𝔨 is not closed under the Lie bracket.

5. 𝔨 𝔰𝔲 2 𝔲 1 V_{1} V_{3}: In this situation, we have V_{1}, V_{1} 𝔰𝔲 2 𝔲 1 V_{1} V_{3}, V_{1}, V_{3} V_{1} 𝔰𝔲 2 , V_{3}, V_{3} 0, and 𝔰𝔲 2 𝔲 1 , V_{1} V_{3} V_{1} V_{3}. These facts prove that 𝔨 is closed under the Lie bracket. We have K SU 3 U 1 , where the second factor is:

; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ & & & e^{it} & 0 \\ & & & 0 & e^{it} \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; t ℝ ; ; ; ; ; ; ; ; ; ; .

Let U 1 ^{}be the abelian factor of the isotropy group SU 2 U 1 SU 3 SU 2 . The intersection U 1 ^{}SU 3 is a group which is isomorphic to ℤ_{3} and is generated by:

 ;  ; \begin{matrix}e^{\frac{2πi}{3}} & 0 & 0 \\ 0 & e^{\frac{2πi}{3}} & 0 \\ 0 & 0 & e^{\frac{4πi}{3}} \end{matrix}  ;  ; e^{\frac{2πi}{3}} Id_{ℂ^{3}} .  (5.27)

We conclude with help of Lemma 3.1.12 that K SU 2 U 1 SU 3 SU 2 ℤ_{3} S^{5} ℤ_{3}. Furthermore, we have SU 3 SU 2 K SU 2 U 1 S^{2}.

6. 𝔨 𝔰𝔲 2 𝔲 1 V_{1} V_{2}: Analogously to the fourth case, we see that 𝔨 is not closed under the Lie bracket.

7. 𝔨 𝔰𝔲 2 𝔲 1 V_{1} V_{2} V_{3}: In this case, 𝔨 𝔰𝔲 3 𝔰𝔲 2 , K SU 3 SU 2 , and K SU 2 U 1 M^{1,1,0}. We assume that M^{1,1,0} is a quotient of S^{7} by a discrete group Γ. For our argument, we need the second homotopy group of ℂℙ^{2}. We have:

… \begin{matrix}π_{2} S^{5} \\  \end{matrix}_{0} π_{2} ℂℙ^{2}  \begin{matrix}π_{1} S^{1} \\  \end{matrix}_{ℤ} \begin{matrix}π_{1} S^{5} \\  \end{matrix}_{0} …

and therefore π_{2} ℂℙ^{2} ℤ. M^{1,1,0} is a circle bundle over the space ℂℙ^{2} S^{2}. Hence, we have the following exact sequence:

… \begin{matrix}π_{2} S^{7} Γ \\ \end{matrix}_{0} \begin{matrix}π_{2} ℂℙ^{2} S^{2} \\ \end{matrix}_{ℤ^{2}} \begin{matrix}π_{1} S^{1} \\ \end{matrix}_{ℤ} ….

Since there exists no injective group homomorphism from ℤ^{2} to ℤ, the above exact sequence is impossible.

Remark 5.3.2. In the case K SU 3 U 1 , the space K SU 2 U 1 is a quotient of a sphere by an effective action of a non-trivial discrete group. The resulting space of cohomogeneity one therefore is an orbifold instead of a manifold.

We investigate the geometric properties of the cohomogeneity-one metrics satisfying (5.20) which have a singular orbit. Without loss of generality, we assume that the singular orbit we consider is at t 0. For each of the three possible cases from the above lemma we have c 0 0. On the following pages, we will see that c^{}0 is either 8, 4, or \frac{8}{3}. If the underlying space of cohomogeneity one was compact, there would be exactly two singular orbits. Near the orbit at t 0, c would have to be positive and near the second one it would be negative. Therefore, there would exist a third orbit on which c vanishes. Since this is impossible, the space is non-compact.

We will check if the metrics which we have found are complete. As in Section 5.2, we only have to investigate the smoothness conditions from Theorem 3.2.18 and if _{0}^{}c τ dτ in order to do this. The reason why we consider the integral is that _{0}^{}c τ dτ measures the length of the geodesics which intersect all orbits perpendicularly. On the previous pages, we have seen that our metrics are described by the equations (5.21) and (5.22). Since c 0 vanishes, we have

c τ ^{2} \frac{8}{τ 2b_{0}^{2} τ \frac{4}{3}a_{0}^{2} ^{2}} _{0}^{τ} s 2b_{0}^{2} s \frac{4}{3}a_{0}^{2} ^{2}ds .

The above term behaves as O τ . Therefore, we have proven one of the conditions for the completeness. We are going to check the smoothness conditions for each of the possible singular orbits separately and start with ℂℙ^{2} S^{2}. The tangent space 𝔭 of ℂℙ^{2} S^{2} decomposes into V_{1} V_{2} with respect to the action of SU 2 U 1 . This splitting is also preserved by the action of K. For our considerations, we need some information on the K-module S^{2} 𝔭 . As in Section 5.2, it suffices to know the trivial submodules of this space. The maximal trivial SU 2 U 1 -submodule of S^{2} 𝔭 is:

span e_{1} e_{1} e_{2} e_{2} e_{3} e_{3} e_{4} e_{4}, e_{5} e_{5} e_{6} e_{6} .

It is easy to see that this space is trivial with respect to K, too. The normal space 𝔭^{}is of dimension two. Moreover, our chosen geodesic which intersects all orbits perpendicularly is fixed by the action of SU 2 U 1 . Since there is a straight line in 𝔭^{}which is left invariant, 𝔭^{}is a trivial module with respect to SU 2 U 1 . The subgroup of K which is generated by e_{7} acts on 𝔭^{}with a non-zero weight r, which we need not to specify for now. We decompose S^{m} 𝔭^{}into irreducible K-modules and obtain similar formulas as for the S^{1}-collapse in Section 5.2:

S^{m} 𝔭^{}𝕍_{rm} 𝕍_{r m 2} … 𝕍_{0} if m is even 𝕍_{rm} 𝕍_{r m 2} … 𝕍_{r} if m is odd

In this formula, 𝕍_{s} denotes the irreducible K-module on which e_{7} acts with weight s and SU 2 U 1 acts trivially on. We take a look at the space W_{m}^{h} which consists of all Kequivariant maps from S^{m} 𝔭^{}to S^{2} 𝔭 . Since any element of S^{m} 𝔭^{}is SU 2 U 1 - invariant, its image has to be SU 2 U 1 -invariant, too. Furthermore, since the SU 2 U 1 -invariant elements of S^{2} 𝔭 are also K-invariant, their preimages have to be K-invariant, too. The above considerations prove that W_{m}^{h} is trivial if and only if m is odd. If m is even, W_{m}^{h} is two-dimensional and the ϕ W_{m}^{h} are given by:

ϕ  ^{\frac{m}{2}} { \atop e_{1}} { \atop e_{1}} { \atop e_{2}} { \atop e_{2}}  λ_{1} e_{1} e_{1} e_{2} e_{2} e_{3} e_{3} e_{4} e_{4} λ_{2} e_{5} e_{5} e_{6} e_{6} .

{ \atop e_{1}}, { \atop e_{2}} denotes an orthonormal basis of 𝔭^{}, λ_{1} and λ_{2} are in ℝ, and ϕ shall vanish on the orthogonal complement of ^{m} { \atop e_{1}} { \atop e_{1}} { \atop e_{2}} { \atop e_{2}} . Therefore, we have:

dim W_{m}^{h}  2 if m is even 0 if m is odd

The interpretation of these numbers is that the odd derivatives of a^{2} and b^{2} have to vanish and that we can choose a^{2} ^{2m} 0 and b^{2} ^{2m} 0 arbitrarily. Therefore, a and b are either odd or even functions. Since a 0 , b 0 0, a and b have to be even functions in order to make the metric smooth. We turn to the vertical part of the metric, which is described by W_{m}^{v} Hom_{K} S^{m} 𝔭^{}, S^{2} 𝔭^{}. S^{2} 𝔭^{}decomposes into an irreducible two-dimensional and a trivial one-dimensional module. As in Section 5.2, we can deduce that

dim W_{m}^{v} ; ; 1 if m 0 0 if m is odd 3 if m 2 is even

We have to discuss the dimensions of the spaces W_{m}^{v}, too. Let g_{t}^{v} be the metric on the collapsing circle. The elements of W_{m}^{v} correspond to the possible values of lim_{t 0} \frac{1}{t^{2}} g_{t}^{v} ^{m}. In the case m 0, we thus obtain the degree of freedom for c^{}0 . By the choice of our coordinates, we have fixed \frac{}{t} to be 1. Therefore, g_{t}^{v} has to coincide up to first order with the unique K-invariant metric q on 𝔭^{}which satisfies q \frac{}{t}, \frac{}{t} 1. It follows that c^{}0 is in fact fixed. We recall the arguments we have made for the S^{1}-collapse in Section 5.2.

Analogously to that case, 2 c^{}0 has to coincide with the cardinality of U 1 ^{}SU 2 U 1 , where U 1 ^{}is generated by e_{7}. We have:

U 1 ^{} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}e^{\frac{it}{2}} & 0 & 0 \\ 0 & e^{\frac{it}{2}} & 0 \\ 0 & 0 & e^{it} \\ & & & e^{\frac{it}{2}} & 0 \\ & & & 0 & e^{\frac{it}{2}} \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; t ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

and

SU 2 U 1 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & e^{\frac{2it}{3}} \\ & & & e^{it} & 0 \\ & & & 0 & e^{it} \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{matrix}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & e^{\frac{2it}{3}} \end{matrix} ; ; SU 3 , t ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ;

By comparing the third and the fourth line of the above two matrices, we obtain the equations

e^{is} e^{\frac{2it}{3}} and e^{\frac{is}{2}} e^{it}, which the elements of U 1 ^{}SU 2 U 1 have to satisfy for some s, t ℝ. These conditions are equivalent to the following system:

s \frac{2}{3}t  2πk_{1}
\frac{1}{2}s t  2πk_{2}

where k_{1}, k_{2} ℤ. Its solution is

s  \frac{3}{2}πk_{1} πk_{2}
t  \frac{3}{4}πk_{1} \frac{3}{2}πk_{2}

The smallest possible non-zero value of s is \frac{π}{2} . Therefore

U 1 ^{}SU 2 U 1 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{aligned}e^{\frac{ik}{4} π} & 0 & 0 \\ 0 & e^{\frac{ik}{4} π} & 0 \\ 0 & 0 & e^{\frac{ik}{2} π} \\ & & & e^{\frac{ik}{4} π} & 0 \\ & & & 0 & e^{\frac{ik}{4} π} \end{aligned} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; k ℤ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

The cardinality of this group is 8. Thus, we have c^{}0 4. Next, we consider the spaces W_{2m}^{v} with m 1. The three dimensions of those spaces correspond to the higher derivatives of g_{t} \frac{}{t}, \frac{}{t} , g_{t} \frac{}{t}, e_{7} , and g_{t} e_{7}, e_{7} . By the choice of our coordinates, we have g_{t} \frac{}{t}, \frac{}{t} 1 and g_{t} \frac{}{t}, e_{7} 0. Therefore, our only freedom is in the choice of g_{t} e_{7}, e_{7} . By the same arguments as on page 120, we see that c^{2} is an even function. Since c^{}0 0, c has to be odd. All in all, we have found the following necessary and sufficient conditions for the smoothness of a metric of type (5.17) where a, b, and c are analytic functions:

• a and b are even.

• c is odd.

• c^{}0 4.

If a t , b t , c t is a solution of the equations (5.20), a t , b t , c t solves those equations, too. Moreover, the initial values of both solutions are the same. The metrics with singular orbit ℂℙ^{2} S^{2} which we have found thus satisfy the first two of our conditions. It directly follows from (5.20) that for any metric with a 0 , b 0 0 and c 0 0 we have c^{}0 8. Therefore, our metrics do not satisfy the conditions from Theorem 3.2.18 and are not smooth at the singular orbit.

We will classify the cohomogeneity-one Einstein metrics with principal orbit M^{1,1,0} and singular orbit ℂℙ^{2} S^{2}. Those metrics are not one of our examples with holonomy SU 4 , since they satisfy c^{}0 4. 𝔭^{}is a trivial SU 2 U 1 -module and 𝔭 decomposes into the non-trivial modules V_{1} and V_{2}. Assumption 3.2.19 thus is satisfied. Furthermore, we have calculated the dimension of W_{m}^{h} for all m ℕ_{0} and of W_{2}^{v}. Therefore, we are able to apply Theorem 3.2.24. With help of that theorem, we see that in the horizontal direction there are two initial conditions of 0^{th} order which we can prescribe. They correspond to a 0 and b 0 . The free parameter in the vertical direction, which corresponds to W_{2}^{v}, is for similar reasons as in the previous section given by c^{}0 .

Next, we take a look at the case 𝔨 𝔰𝔲 2 𝔰𝔲 2 𝔲 1 . Instead of describing a necessary and sufficient condition for the smoothness of a metric with singular orbit ℂℙ^{2}, we only prove a sufficient condition. We claim that any metric of type (5.17) for which

• a, b, and c are analytic,

• a is even,

• b and c are odd,

• b^{}0 1, and c^{}0 4,

is smooth at the singular orbit. We will be able to prove that there exist cohomogeneity-one Einstein metrics with certain initial conditions. The uniqueness of those metrics could only be proven by calculating dim W_{2}^{v} and dim W_{m}^{h}, which we will not do. As in the proof of Lemma 5.3.1, the first summand of 𝔨 denotes the subalgebra of 𝔰𝔲 3 𝔰𝔲 3 𝔰𝔲 2 which is contained in the isotropy algebra. The second summand coincides with the second summand of 𝔰𝔲 3 𝔰𝔲 2 and the abelian summand is generated by

 ;  ;  ;  ;  ; \begin{aligned}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 2i \\ & & & 0 & 0 \\ & & & 0 & 0 \end{aligned}  ;  ;  ;  ;  ;

The tangent space 𝔭 of the singular orbit can be identified with V_{1}. The first summand of 𝔨 acts by the complex two-dimensional representation of 𝔰𝔲 2 on 𝔭 and the second one acts trivially. When we have classified the invariant metrics on M^{1,1,0}, we have seen that the only trivial 𝔰𝔲 2 𝔲 1 -submodule of S^{2} 𝔭 is

span e_{1} e_{1} e_{2} e_{2} e_{3} e_{3} e_{4} e_{4} .

This space is of course invariant with respect to the second summand of 𝔰𝔲 3 𝔰𝔲 2 , too, and therefore is a K-module. The normal space 𝔭^{}of the singular orbit is four-dimensional. We discuss the action of the three summands of 𝔨 on 𝔭^{}separately. The first 𝔰𝔲 2 -summand fixes one point of the collapsing sphere, since it is contained in the isotropy algebra. The tangent space of the sphere at that point can be identified with the complement of 𝔰𝔲 2 𝔲 1 in 𝔨. This complement is given by span e_{5}, e_{6}, e_{7} . Since the first 𝔰𝔲 2 -summand commutes with e_{5}, e_{6}, and e_{7}, its action on the tangent space is trivial. Therefore, the action of the first summand on 𝔭^{}is trivial, too. The second 𝔰𝔲 2 -summand of 𝔨 cannot act trivially on 𝔭^{}. Otherwise, the K-orbits of the non-zero elements of 𝔭^{}would not be spheres. Since the collapsing sphere is a coset space of type SU 2 U 1 U 1 , we moreover can deduce that the second summand acts by the two-dimensional complex representation of 𝔰𝔲 2 on 𝔭^{}. 𝔨 can be generated by 2𝔰𝔲 2 𝔨 and the abelian summand of the isotropy algebra 𝔰𝔲 2 𝔲 1 . In order to describe the action of all of 𝔨 on 𝔭^{}, it suffices to investigate how the above summand acts on 𝔭^{}instead of the abelian part of 𝔨. The action of the abelian summand of 𝔰𝔲 2 𝔲 1 is determined by the action of e_{7} on the tangent space V_{2} V_{3} of the collapsing sphere. The weights with which e_{7} act can be easily calculated. Since we will not need their value, we will not do that calculation. Instead of decomposing S^{m} 𝔭^{}into its submodules, we will explicitly describe the 𝔨-equivariant maps which correspond to the derivatives of the metric we are interested in. Let { \atop e_{1}}, …, { \atop e_{4}} be an orthonormal basis of 𝔭^{} such that the matrix representation of the second 𝔰𝔲 2 -summand with respect to { \atop e_{1}}, …, { \atop e_{4}} is the standard one. As in the previous section, we have

dt dt { \atop e^{1}} { \atop e^{1}} { \atop e^{2}} { \atop e^{2}} { \atop e^{3}} { \atop e^{3}} { \atop e^{4}} { \atop e^{4}} .

Since we want a to be even, we have to consider the value of a^{2} ^{2m} 0 . The derivative of a^{2} is in the t-direction and we thus obtain the following map ϕ : S^{2m} 𝔭^{}S^{2} 𝔭 which is associated to a^{2} ^{2m} 0 :

ϕ  ^{m} { \atop e^{1}} { \atop e^{1}} { \atop e^{2}} { \atop e^{2}} { \atop e^{3}} { \atop e^{3}} { \atop e^{4}} { \atop e^{4}}  e^{1} e^{1} e^{2} e^{2} e^{3} e^{3} e^{4} e^{4} .

As usual, we assume that ϕ vanishes on the orthogonal complement of ^{m} { \atop e^{1}} { \atop e^{1}} { \atop e^{2}} { \atop e^{2}} { \atop e^{3}} { \atop e^{3}} { \atop e^{4}} { \atop e^{4}} . Since ϕ maps a K-invariant object to another K-invariant object, it is a K-equivariant map. Therefore, we can choose a^{2} ^{2m} 0 arbitrarily. In particular, a can be chosen as any even analytic function.

Next, we have to consider the vertical part of the metric. Since we have a group of type SU 2 acting irreducibly on the four-dimensional space 𝔭^{}, we are in a similar situation as in Section 5.2, when we have considered the S^{3}-collapse. By the same method as in that case, we are able to construct the maps which are associated to the odd derivatives of b and c. More precisely: Let γ be our chosen geodesic which intersects all orbits perpendicularly. We assume without loss of generality that γ^{}0 { \atop e_{1}}. The tangent space of the collapsing sphere is spanned by { \atop e_{2}}, { \atop e_{3}}, { \atop e_{4}} . Since that tangent space can also be identified with the complement of 𝔰𝔲 2 𝔲 1 in 𝔨, we will work with the basis e_{5}, e_{6}, e_{7} instead of { \atop e_{2}}, { \atop e_{3}}, { \atop e_{4}} . span e_{5}, e_{6}, e_{7} is a SU 2 U 1 -module, which decomposes into V_{2} V_{3}. We can extend any term of type

b^{2} e^{5} e^{5} e^{6} e^{6} c^{2} e^{7} e^{7}

by the action of K to a well-defined K-invariant metric on the sphere. Let ϕ_{1}^{m}, ϕ_{2}^{m} : S^{2m} 𝔭^{} S^{2} 𝔭^{}be the maps which correspond to \frac{1}{t^{2}} b^{2} ^{2m} 0 and \frac{1}{t^{2}} c^{2} ^{2m} 0 . Motivated by the arguments from Section 5.2 and the above considerations, we see that these maps satisfy:

\begin{matrix}ϕ_{1}^{m}  ^{m 1} dt dt L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} & L_{k}^{}e^{5} L_{k}^{}e^{5} L_{k}^{}e^{6} L_{k}^{}e^{6} \\ ϕ_{2}^{m}  ^{m 1} dt dt L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} & L_{k}^{}e^{7} L_{k}^{}e^{7} , \end{matrix}  (5.28)

where L_{k}^{}is the pull-back of the left-multiplication by k SU 2 and the ϕ_{i}^{m} with i 1, 2 are extended by polarization to all of ^{m 1} dt dt S^{2} 𝔭^{}. On the orthogonal complement of that space, both maps vanish. By their definition, ϕ_{1}^{m} and ϕ_{2}^{m} are equivariant with respect to the second 𝔰𝔲 2 -summand. With respect to the first 𝔰𝔲 2 -summand, they are equivariant, too, since that algebra acts trivially on 𝔭^{}. The abelian part of the isotropy algebra 𝔰𝔲 2 𝔲 1 leaves the terms on the left and the right hand side of (5.28) invariant. The maps ϕ_{1}^{m} and ϕ_{2}^{m} thus are 𝔨-equivariant. We have proven that the value of \frac{1}{t^{2}} b^{2} ^{2m} 0 and \frac{1}{t^{2}} c^{2} ^{2m} 0 can be chosen arbitrarily for all m 1. Therefore, the 2m 1 ^{th} derivatives of b and c can be chosen arbitrarily, too, if m 1.

It remains to prove that b^{}0 and c^{}0 have the value which we have claimed. As we have remarked in the previous section, the metric h on SU 2 with constant sectional curvature 1 is determined by h X, Y \frac{1}{2}tr XY for all X, Y 𝔰𝔲 2 . The following matrices are an orthonormal basis of 𝔰𝔲 2 with respect to h:

\begin{matrix}i & 0 \\ 0 & i \end{matrix}  ,  \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ,  \begin{matrix}0 & i \\ i & 0 \end{matrix}

The second and the third matrix correspond to e_{5} and e_{6}, since it is the second factor of SU 3 SU 2 which acts irreducibly on 𝔭^{}. We want the metric on the collapsing sphere to have sectional curvature \frac{1}{t^{2}} O \frac{1}{t} for small values of t. This is only possible if b^{}0 1. The matrix e_{7} whose length is measured by c generates the one-dimensional Lie group U 1 ^{}. The action of the group U 1 ^{}generates a great circle on the collapsing sphere which coincides with the collapsing circle in the previous case. We thus can repeat the arguments which we have made in that case. We again obtain c^{}0 4. All in all, we have proven that the criteria which we have proposed are indeed sufficient for the smoothness of the metric.

In order to complete our considerations, we have to prove that the solutions of (5.20) satisfy these sufficient conditions. By a short calculation, we see that any solution of (5.20) with b 0 c 0 0 satisfies b^{}0 1, 1 and c^{}0 4. We remark that it suffices to restrict ourselves to the case b^{}0 1, since we can replace b by b without changing our system of differential equations. Let a t , b t , c t be a solution of (5.20). Then a t , b t , c t is a solution of those equations, too. Furthermore, its values at t 0 coincide with the initial values of a t , b t , c t . Since the metric we obtain from the initial value problem is in both cases the same, we have proven that a is even and b as well as c is odd.

As a by-product of our considerations, we can prove the existence of certain cohomogeneityone Einstein metrics with singular orbit ℂℙ^{2}. The semisimple part of the isotropy group SU 2 U 1 acts irreducibly on 𝔭, which coincides with V_{1}, and trivially on 𝔭^{}. Therefore, Assumption 3.2.19 is satisfied and we can work with Theorem 3.2.24. Since we always can prescribe the metric on the singular orbit in the setting of Theorem 3.2.24, the initial value a 0 can be chosen freely. The maps (5.28) are two linearly independent elements of W_{2}^{v}. Since those maps correspond to the functions b and c, we also can prescribe b^{}0 and c^{}0 arbitrarily.

We finally consider the case where 𝔨 𝔰𝔲 3 𝔲 1 and we have S^{2} as singular orbit. As in the previous case, we only prove a sufficient condition for the smoothness of the metric. Our claim is that any metric of type (5.17), where

• a, b, and c are analytic,

• b is even,

• a and c are odd,

• a^{}0 1, and c^{}0 4,

is a smooth metric on the underlying orbifold. Again, we will prove the existence of cohomogeneity-one Einstein metrics with singular orbit S^{2} and certain initial values. In our situation, the tangent space 𝔭 of the singular orbit can be identified with V_{2}. The Lie algebra 𝔰𝔲 3 clearly acts trivially on that space. It is easy to see that the submodule

span e_{5} e_{5} e_{6} e_{6}

of S^{2} 𝔭 is trivial with respect to all of 𝔰𝔲 3 𝔲 1 and that it is the only trivial submodule of S^{2} 𝔭 . The normal space 𝔭^{}of the singular orbit is a quotient of a six-dimensional vector space by a discrete group which is isomorphic to ℤ_{3}. Since that group acts by complex multiples of the identity map, our considerations are essentially the same as if 𝔭^{}was a vector space and the underlying space was a manifold. In particular, we can apply the results of Eschenburg and Wang [32]. Since 𝔭^{}is six-dimensional and the orbit of any non-zero element of 𝔭^{}has to be of type S^{5} ℤ_{3}, 𝔰𝔲 3 acts by its six-dimensional irreducible representation on 𝔭^{}.

In order to prove our claim, we have to describe the K-equivariant maps which correspond to the derivatives of the horizontal and vertical part of the metric. Let { \atop e_{1}}, …, { \atop e_{6}} be an orthonormal basis of 𝔭^{}such that the matrix representation of 𝔰𝔲 3 with respect to this basis is the same as of 𝔰𝔲 3 𝔤𝔩 6, ℂ . The map ϕ : S^{2m} 𝔭^{}S^{2} 𝔭 which corresponds to b^{2} ^{2m} 0 is analogously to the previous case determined by:

ϕ  ^{m} { \atop e^{1}} { \atop e^{1}} … { \atop e^{6}} { \atop e^{6}}  e^{5} e^{5} e^{6} e^{6} .

On the rest of S^{2m} 𝔭^{}, ϕ vanishes. Since this map is 𝔨-equivariant, we can choose the even derivatives of b^{2} arbitrarily. Thus, b can be chosen as any analytic even function. In order to prove that we can choose the odd derivatives of a and c freely, we make a similar argument as in the case 𝔨 𝔰𝔲 2 𝔰𝔲 2 𝔲 1 . Again, we assume that the geodesic γ which we have fixed satisfies γ^{}0 { \atop e_{1}}. The normal space of { \atop e_{1}} can be identified with the tangent space V_{1} V_{3} of S^{5} ℤ_{3}. Any invariant metric on S^{5} ℤ_{3} is of the following type:

a^{2} e^{1} e^{1} e^{2} e^{2} e^{3} e^{3} e^{4} e^{4} c^{2} e^{7} e^{7} .

We define the following map ϕ : S^{2m} 𝔭^{}S^{2} 𝔭^{}:

\begin{matrix}ϕ ^{m 1} dt dt L_{k}^{}{ \atop e^{1}} L_{k}^{}{ \atop e^{1}} : a^{2} & L_{k}^{}e^{1} L_{k}^{}e^{1} L_{k}^{}e^{2} L_{k}^{}e^{2} \\ & L_{k}^{}e^{3} L_{k}^{}e^{3} L_{k}^{}e^{4} L_{k}^{}e^{4} \\ & c^{2} L_{k}^{}e^{7} L_{k}^{}e^{7} \end{matrix}

L_{k}^{}denotes the pull-back of the left-multiplication by k SU 3 . The above map is as usually extended by polarization and vanishes on the complement of ^{m 1} dt dt S^{2} 𝔭^{}. If k is the neutral element, the terms on both sides of the above equation are invariant with respect to the isotropy group SU 2 U 1 of the K-action. Therefore, the above map is well-defined. ϕ is SU 3 -equivariant by its definition. We have to prove if it is also equivariant with respect to the abelian summand of 𝔨. Since 𝔨 can be generated by 𝔰𝔲 3 and the abelian part of the isotropy algebra 𝔰𝔲 2 𝔲 1 , we can prove instead the equivariance with respect to the abelian summand. We have already remarked that ϕ is equivariant with respect to SU 2 U 1 and thus have proven the K-equivariance. ϕ is the map which is associated to \frac{1}{t^{2}} g_{t}^{v} ^{2m} 0 , where g_{t}^{v} is the metric on the collapsing sphere. We therefore can conclude by the usual arguments that a^{2m 1} 0 and c^{2m 1} 0 can be chosen freely for m 1.

The values of a^{}0 and c^{}0 which would make our metric smooth up to first order can be obtained as in the previous case. The metric h on SU 3 SU 2 with constant sectional curvature 1 is induced by h X, Y \frac{1}{2}tr XY for all X and Y in the orthogonal complement of 𝔰𝔲 2 𝔰𝔲 3 . Since V_{1} is contained in 𝔰𝔲 3 , we can conclude that a^{}0 h e_{1}, e_{1} 1. For the same reasons as in the other cases, we again obtain c^{}0 4.

We check if our solutions of (5.20) satisfy the conditions for the smoothness which we have found. It is easy to see that indeed b is even and a as well as c are odd. By a short calculation, we obtain a^{}0 1 and c^{}0 \frac{8}{3} as initial values. Since c^{}0 4, the metrics with singular orbit S^{2} have are not smooth at the singular orbit.

As in the case 𝔨 𝔰𝔲 2 𝔰𝔲 2 𝔲 1 , we can prove with help of Theorem 3.2.24 the existence of certain cohomogeneity-one Einstein metrics. 𝔭 can be identified with V_{2} and 𝔭^{}decomposes with respect to SU 2 U 1 into V_{1} and a two-dimensional trivial module. Therefore, 𝔭 and 𝔭^{}have no equivalent SU 2 U 1 -submodules in common and Assumption 3.2.19 is satisfied. It follows by similar arguments as in the previous case that for any choice of b 0 , a^{}0 , and c^{}0 there exists a cohomogeneity-one Einstein metric with principal orbit M^{1,1,0} and singular orbit S^{2}. Since those metrics satisfy c^{}0 4, our solutions of (5.20) are not one of them. Finally, we sum up the results of this section:

Theorem 5.3.3. Let M, Ω be a parallel Spin 7 -orbifold with a cohomogeneity-one action of SU 3 SU 2 which preserves Ω. We assume that the principal orbit is a coset space of type M^{k,l.0}. In this situation, the following statements are true:

1. The principal orbit is SU 3 SU 2 -equivariantly diffeomorphic to M^{1,1,0}.

2. The metric g which is associated to Ω is special Kähler and its holonomy is all of SU 4 .

3. g has the matrix representation (5.17) with respect to the basis e_{1}, …, e_{7} from page 129. The metric functions satisfy the equations (5.20), whose solutions are described by (5.21) and (5.22). The Kähler form is given by (5.25).

If M has a singular orbit, which has to be the case if M, g is complete, it is ℂℙ^{2} S^{2}, ℂℙ^{2}, or S^{2}. Any SU 3 SU 2 -invariant metric on the singular orbit can be extended to a unique cohomogeneity-one metric with holonomy SU 4 , which is non-compact.

1. If the singular orbit is ℂℙ^{2} S^{2}, M is a manifold, but the metric g is not differentiable at the singular orbit. Outside of the singular orbit, g is smooth and any geodesic which does not intersect the singular orbit can be infinitely extended.

2. If the singular orbit is ℂℙ^{2}, M is a manifold and the metric is smooth and complete.

3. If the singular orbit is S^{2}, M is an orbifold but not a manifold. At the singular orbit, the metric is not differentiable, i.e. it is not a smooth orbifold metric. Outside of the singular orbit, g is smooth and any geodesic which does not intersect the singular orbit can be infinitely extended.

In the first two cases M is a vector bundle over the singular orbit and in the third case it is a ℂ^{3} ℤ_{3}-bundle over S^{2}.

With help of Theorem 3.2.24, we have also obtained results on cohomogeneity-one Einstein metrics with principal orbit M^{1,1,0}:

Theorem 5.3.4. Let M be a cohomogeneity-one orbifold whose principal orbit is M^{1,1,0}. We assume that M has a singular orbit, which has to be ℂℙ^{2} S^{2}, ℂℙ^{2}, or S^{2}. Any SU 3 SU 2 - invariant metric on M is described by a matrix of type (5.17) or equivalently by the three metric functions a, b, c : I ℝ, where I is an interval. Without loss of generality, we assume that the singular orbit is at 0 I. In this situation, the following statements are true:

1. Let the singular orbit be ℂℙ^{2} S^{2}. For any choice of a_{0}, b_{0}, c_{3}, λ ℝ, there exists a unique SU 3 SU 2 -invariant Einstein metric on a tubular neighborhood of ℂℙ^{2} S^{2} such that:

(a) a 0 a_{0}, b 0 b_{0},

(b) c^{}0 c_{3}, and

(c) the Einstein constant is λ.

2. Let the singular orbit be ℂℙ^{2}. For any choice of a_{0}, b_{3}, c_{3}, λ ℝ, there exists a SU 3 SU 2 -invariant Einstein metric on a tubular neighborhood of ℂℙ^{2} such that:

(a) a 0 a_{0},

(b) b^{}0 b_{3}, c^{}0 c_{3}, and

(c) the Einstein constant is λ.

3. Let the singular orbit be S^{2}. For any choice of a_{3}, b_{0}, c_{3}, λ ℝ, there exists a SU 3 SU 2 -invariant Einstein metric on a tubular neighborhood of S^{2} such that:

(a) b 0 b_{0},

(b) a^{}0 a_{3}, c^{}0 c_{3}, and

(c) the Einstein constant is λ.

Remark 5.3.5. 1. It is likely that the Ricci-flat Kähler metrics with singular orbit ℂℙ^{2} S^{2} can be modified by a similar construction as we have described at the end of Section 5.2 in such a way that they become smooth.

2. The Kähler metrics with singular orbit ℂℙ^{2} S^{2} have been considered by Herzog and Klebanov in [41] and by Cvetič et al. in [21]. The examples with singular orbit ℂℙ^{2} or S^{2} are mentioned by the same authors in [21] and [25]. Our considerations prove that any complete parallel Spin 7 -orbifold of cohomogeneity-one with principal orbit M^{1,1,0} is one of our examples. This result and those of the above Einstein metrics, which are not contained in Theorem 5.3.3, are, as far as the author knows, new.

5.4 The Aloff-Wallach spaces as principal orbits

The subject of this section are parallel cohomogeneity-one Spin 7 -structures whose principal orbit is of type SU 3 U 1 . A coset space of that kind is called an Aloff-Wallach space. There are infinitely many distinct Aloff-Wallach spaces depending on the embedding of U 1 into SU 3 . For two special kinds of embeddings, we obtain coset spaces which have different features than the other ones. They are called exceptional Aloff-Wallach spaces. Any Aloff-Wallach space admits a cosymplectic G_{2}-structure. Therefore, there exist (not necessarily complete) parallel cohomogeneity-one Spin 7 -structures Ω whose principal orbit can be chosen as any Aloff-Wallach space. By considering the equation dΩ 0 near a singular orbit, we obtain local examples of parallel Spin 7 -structures. They include those which have been found by Cvetič, Gibbons, Lü, and Pope [24], by Gukov and Sparks [39], and by Kanno and Yasui [48], [49]. We also will construct new examples with an exceptional Aloff-Wallach space as principal orbit. In the setting of this section, it is possible to prove classification results for the cohomogeneity-one metrics with reduced holonomy and the Einstein metrics, which are similar to the Theorems 5.2.2, 5.2.3, 5.3.3, and 5.3.4. In the course of our considerations, it will turn out that the calculations needed for the proof of such theorems would be more difficult than those in Section 5.2 and 5.3. The main reason for this is the existence of non-diagonal metrics on the exceptional Aloff-Wallach spaces. This makes the space of homogeneous metrics and G_{2}-structures larger than in the previous sections. We therefore will omit some cases, but still cover the most important aspects. Another difference to the previous sections is that the equations for the holonomy reduction have no explicit solutions except in some special cases. Therefore, it is not a priori clear if our examples can be extended to complete metrics. This question has been considered by Cvetič et al. in [24] and in the recent paper of Bazaikin [5], but here we mostly focus on the classification of the local metrics.

Before we construct any examples of parallel Spin 7 -structures or Einstein metrics, we have to describe the Aloff-Wallach spaces themselves in more detail. The possible embeddings of U 1 into SU 3 are up to conjugacy given by:

\begin{matrix}i_{k,l} : U 1 & SU 3 \\ i_{k,l} e^{iφ} & : { \atop }\begin{matrix}e^{ikφ} & 0 & 0 \\ 0 & e^{ilφ} & 0 \\ 0 & 0 & e^{i k l φ} \end{matrix} { \atop }with k, l ℤ . \end{matrix}

We denote the image of U 1 with respect to i_{k,l} by U 1 _{k,l} and the quotient SU 3 U 1 _{k,l} by N^{k,l}. For different pairs k, l we sometimes obtain the same space N^{k,l}.

Let, for example, λ gcd k, l ^{1} ℤ be arbitrary. The spaces N^{k,l} and N^{λk,λl} coincide. Therefore, we will assume from now on that k and l are coprime. Let σ be a permutation of the triple k, l, k l . It is easy to see that the spaces N^{k,l} and N^{σ k ,σ l}are SU 3 -equivariantly diffeomorphic. This diffeomorphism could be explicitly described by conjugation by the permutation matrix of σ. The S_{3}-action on the indices k, l, k l also has a deeper meaning: It is well-known that the root system of the Lie algebra 𝔰𝔲 3 is of type A_{2}. Its Weyl group is the permutation group S_{3}, which acts on the dual of any Cartan subalgebra of 𝔰𝔲 3 . In this section, we fix the following Cartan subalgebra:

𝔱 : ; ; { \atop }\begin{matrix}ix & 0 & 0 \\ 0 & iy & 0 \\ 0 & 0 & i x y \end{matrix} { \atop } ; ; ; ; ; x, y ℝ ; ;

and identify 𝔱 with its dual space by the Killing form. The action on 𝔱 by a σ in the Weyl group maps the Lie algebra 𝔲 1 _{k,l} of U 1 _{k,l} into 𝔲 1 _{σ k ,σ l}. Therefore, the action of S_{3} on k, l, k l is induced by the action of the Weyl group. Our considerations motivate the following convention:

Convention 5.4.1. Let k, l, k^{}, l^{}ℤ and k, l, k l λ σ k^{}, σ l^{}, σ k^{}l^{}for a λ gcd σ k^{}, σ l^{}^{1} ℤ and a σ S_{3}. Since the Aloff-Wallach spaces N^{k,l} and N^{k^{},l^{}} are SU 3 -equivariantly diffeomorphic, we will write in this situation N^{k,l} N^{k^{},l^{}}. ” ” is obviously an equivalence relation. The Aloff-Wallach spaces in the equivalence class of N^{1,0} or N^{1,1} are called exceptional and the other ones are called generic Aloff-Wallach spaces. Since we have to consider only one representant in the equivalence class of N^{k,l}, we may assume that k and l are coprime and k l k l. If l 0, we replace N_{k,l} by N_{k l, l} . Therefore, it is permitted to assume without loss of generality that k and l are both non-negative. On the next pages, we will always use the convention k l 0. Later on, we will turn to another convention which will be introduced at that point.

We briefly review the history of mathematical research on the Aloff-Wallach spaces: The spaces N^{k,l} have been first considered by Simon Aloff and Nolan Wallach. They have proven in their paper [2] that on any N^{k,l} except on N^{1,0} there exists a homogeneous metric with strictly positive sectional curvature.

There are infinitely many homotopy types of Aloff-Wallach spaces. This follows from the fact that

H^{4} N^{k,l}, ℤ ℤ_{k^{2} lk l^{2}} .

Another interesting feature of the Aloff-Wallach spaces is that some of them are homeomorphic but not diffeomorphic. Examples of this fact are given in a paper of Kreck and Stolz [51].

On any Aloff-Wallach space there are up to homothety exactly two SU 3 -invariant Einstein metrics. First examples of those metrics have been constructed in the papers of Kobayashi [52], Jensen [46], and Wang [65]. It turned out that on the spaces N^{k,l} there exist two Einstein metrics, which depend on k and l. Those metrics all have weak holonomy G_{2} (see Cvetič et al. [24]). Therefore, the cones over those Riemannian manifolds have a holonomy, which is contained in Spin 7 . Since we will see that the Aloff-Wallach spaces are not diffeomorphic to any quotient of a sphere by a discrete group, the cones have a singularity at the tip, which is not an orbifold singularity. Page and Pope [57] have proven that there are no further homogeneous Einstein metrics on the generic Aloff-Wallach spaces. In a paper of Nikonorov [56], the exceptional case is treated. The author proves that on N^{1,1} the above two Einstein metrics are the only homogeneous Einstein metrics, too. On the space N^{1,0} the two Einstein metrics are homothetic. Surprisingly, there exists another non-diagonal homogeneous Einstein metric on N^{1,0}, which is not homothetic to the other ones.

Our next step is to describe all SU 3 -invariant metrics on the Aloff-Wallach spaces. We first choose a basis e_{1}, …, e_{8} of 𝔰𝔲 3 :

e_{1} : { \atop }\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} { \atop }, e_{2} : { \atop }\begin{matrix}0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{matrix} { \atop },

e_{3} : { \atop }\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} { \atop }, e_{4} : { \atop }\begin{matrix}0 & 0 & i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{matrix} { \atop },

e_{5} : { \atop }\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} { \atop }, e_{6} : { \atop }\begin{matrix}0 & 0 & 0 \\ 0 & 0 & i \\ 0 & i & 0 \end{matrix} { \atop },

e_{7} : { \atop }\begin{matrix}2l k i & 0 & 0 \\ 0 & 2k l i & 0 \\ 0 & 0 & k l i \end{matrix} { \atop }, e_{8} : { \atop }\begin{matrix}ki & 0 & 0 \\ 0 & li & 0 \\ 0 & 0 & k l i \end{matrix} { \atop }

We fix the biinvariant background metric q on 𝔰𝔲 3 which is defined by q X, Y : tr XY for all X, Y 𝔰𝔲 3 . It is easy to see that e_{1}, …, e_{8} is orthogonal with respect to q. Let 𝔪 be the q-orthogonal complement of 𝔲 1 _{k,l} in 𝔰𝔲 3 . 𝔪 is spanned by e_{1}, …, e_{7} , and 𝔲 1 _{k,l} is spanned by e_{8}. As usual, we fix a p N^{k,l} and identify T_{p}N^{k,l} by a U 1 _{k,l}-equivariant map with 𝔪. The matrix representation of ad_{e_{8}} _{𝔪} with respect to the basis e_{1}, …, e_{7} of 𝔪 is:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 & k l \\ k l & 0 \\ & & 0 & 2k l \\ & & 2k l & 0 \\ & & & & 0 & k 2l \\ & & & & k 2l & 0 \\ & & & & & & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; (5.29)

It is easy to see that 𝔪 decomposes into the following irreducible 𝔲 1 _{k,l}-submodules:

V_{1} : span e_{1}, e_{2}
V_{2} : span e_{3}, e_{4}
V_{3} : span e_{5}, e_{6}
V_{4} : span e_{7}

(5.30)

Later on, we will need the weight of the isotropy action on the above modules. Let z U 1 ℂ and w ℂ be arbitrary. The U 1 -action on ℂ with weight n is defined by z.w : z^{n}w. By identifying the complex number a bi : z^{n} with the matrix

\begin{matrix}a & b \\ b & a \end{matrix}

we obtain an action of weight n on ℝ^{2}. The weight of a U 1 -action is determined only up to the sign, since we can replace n by n by a change of the basis of ℝ^{2}. In this section, we will make the following convention: If an irreducible U 1 -module comes with a basis such that the action of U 1 has the above matrix representation, we denote the weight by n. With respect to the bases which we have chosen in (5.30), the action of U 1 _{k,l} on V_{1}, V_{2}, and V_{3} has the following weights:

k l, 2k l, k 2l .

If we choose k l 1, the module V_{1} is trivial. Therefore, it decomposes into two onedimensional submodules. Furthermore, the modules V_{2} and V_{3} are equivalent in this case. In the case k 1, l 0, the modules V_{1} and V_{3} have the same weight. In the other cases, all of the three modules are irreducible and pairwise inequivalent. This is the reason why N^{1,1} and N^{1,0} are called the exceptional Aloff-Wallach spaces. In the following, we consider the generic Aloff-Wallach spaces, N^{1,0}, and N^{1,1} separately.

We now start describing the possible SU 3 -invariant metrics on the Aloff-Wallach spaces. As in the previous two sections, we can identify any such metric g on N^{k,l} with a 𝔲 1 _{k,l}- equivariant, q-symmetric, positive definite endomorphism φ : 𝔪 𝔪, which is defined by q φ X , Y : g X, Y . Conversely, any such φ defines an invariant metric on N^{k,l}. The possible φ can be classified by Schur’s lemma. We first consider the generic Aloff-Wallach spaces. Since the submodules in which 𝔪 decomposes are pairwise inequivalent, the possible metrics are precisely those whose matrix representation with respect to the basis e_{1}, …, e_{7} is of type:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}a^{2} & 0 \\ 0 & a^{2} \\ & & b^{2} & 0 \\ & & 0 & b^{2} \\ & & & & c^{2} & 0 \\ & & & & 0 & c^{2} \\ & & & & & & f^{2} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; with a, b, c, f ℝ 0 .  (5.31)

As in Section 5.2 and 5.3, we allow both signs for a, b, c, and f, since the four-forms, which we will consider, contain odd powers of these parameters. Next, we consider N^{1,0}. The invariant metrics on this space can be described with respect to the basis e_{1}, e_{2}, e_{5}, e_{6}, e_{3}, e_{4}, e_{7} by the following matrices:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}a^{2} & 0 & β_{1,5} & β_{1,6} \\ 0 & a^{2} & β_{1,6} & β_{1,5} \\ β_{1,5} & β_{1,6} & c^{2} & 0 \\ β_{1,6} & β_{1,5} & 0 & c^{2} \\ & & & & b^{2} & 0 \\ & & & & 0 & b^{2} \\ & & & & & & f^{2} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}with & a, b, c, f, β_{1,5}, β_{1,6} ℝ , \\ & a^{2}c^{2} β_{1,5}^{2} β_{1,6}^{2} , b 0 , f 0 . \end{aligned}

(5.32)

The last three conditions are necessary, since the determinant of the above matrix is a^{2}c^{2} β_{1,5}^{2} β_{1,6}^{2} ^{2}b^{4}f^{2}. It should be noted that the upper left part of this matrix can be considered as the Hermitian matrix

\begin{matrix}a^{2} & β_{1,5} β_{1,6}i \\ β_{1,5} β_{1,6}i & c^{2} \end{matrix} .

Finally, we describe the SU 3 -invariant metrics on N^{1,1}. These metrics are precisely those whose matrix representation with respect to e_{1}, e_{2}, e_{7}, e_{3}, e_{4}, e_{5}, e_{6} is a positive definite matrix of the following type:

 ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}a_{1}^{2} & β_{1,2} & β_{1,7} \\ β_{1,2} & a_{2}^{2} & β_{2,7} \\ β_{1,7} & β_{2,7} & f^{2} \\ & & & b^{2} & 0 & β_{3,5} & β_{3,6} \\ & & & 0 & b^{2} & β_{3,6} & β_{3,5} \\ & & & β_{3,5} & β_{3,6} & c^{2} & 0 \\ & & & β_{3,6} & β_{3,5} & 0 & c^{2} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ; (5.33)

In Lemma 3.2.36 and Remark 3.2.37 we have proven that the action of the normalizer on each principal orbit preserves the equation dΩ 0 and the cohomogeneity-one Einstein equation. It therefore may be possible to reduce the number of metrics which we have to consider by some parameters. In order to check if this is possible we have to determine the normalizer Norm_{SU 3}U 1 _{k,l} for all k and l.

For our result, we need the following facts: Let D : diag λ_{1}, λ_{2}, λ_{3} with λ_{i} λ_{j} for i j be a diagonal matrix and let A GL 3, ℂ . If Ad_{A} D is a diagonal matrix, too, it has to be diag λ_{σ 1}, λ_{σ 2}, λ_{σ 3} for a permutation σ S_{3}. Furthermore, we can prove by a short calculation that Ad_{A} D D is equivalent to A being a diagonal matrix. From these facts we can conclude that if N^{k,l} is not N^{1,1}, the normalizer acts trivially on U 1 _{k,l} and we have:

Norm_{SU 3}U 1 _{k,l} ; ; { \atop }\begin{matrix}e^{iμ_{1}} & 0 & 0 \\ 0 & e^{iμ_{2}} & 0 \\ 0 & 0 & e^{iμ_{3}} \end{matrix} { \atop } ; ; ; ; ; e^{iμ_{1}}e^{iμ_{2}}e^{iμ_{3}} 1 ; ; U 1 ^{2} .

Next, we take a look at the N^{1,1}-case. Again, we first consider a diagonal matrix D diag λ_{1}, λ_{2}, λ_{3} , but now assume that λ_{1} λ_{2} λ_{3}. By similar arguments as above, we can prove that Ad_{A} D D if and only if

A { \atop }\begin{aligned}a_{11} & a_{12} \\ a_{21} & a_{22} \\ & & a_{33} \end{aligned} { \atop }for some a_{ij} ℂ .

Therefore,

Norm_{SU 3}U 1 _{1,1} ; ; { \atop }\begin{aligned}a_{11} & a_{12} \\ a_{21} & a_{22} \\ & & a_{33} \end{aligned} { \atop } ; ; ; ; ; a_{11}, a_{12}, a_{21}, a_{22}, a_{33} ℂ ; ; SU 3 .

In fact, we have:

Norm_{SU 3}U 1 _{1,1}  \begin{aligned}A \\ & det A^{1} \end{aligned}   ;  ;  ; A U 2  U 2 .

We call the subgroup of all A Norm_{SU 3}U 1 _{1,1} with det A 1 shortly SU 2 Norm_{SU 3} U 1 _{1,1} or the SU 2 -factor of Norm_{SU 3}U 1 _{1,1}, although U 2 is not isomorphic to SU 2 U 1 .

Let g be a SU 3 -invariant metric on N^{k,l} and let φ be the corresponding endomorphism of 𝔪. The adjoint action of any A Norm_{SU 3}U 1 _{k,l} on 𝔰𝔲 3 leaves 𝔪 invariant. We identify A with its action on 𝔪. In the generic case, we have A^{T} φA φ for all A in the normalizer. Therefore, the action of A does not change the metric and we cannot reduce the number of metrics which we have to consider in the generic case.

Next, we consider the N^{1,0}-case. If we choose A as an element of U 1 _{1,0}, its action of course leaves the metric invariant. If A is chosen as

A { \atop }\begin{matrix}e^{it} & 0 & 0 \\ 0 & e^{2it} & 0 \\ 0 & 0 & e^{it} \end{matrix} { \atop },

which is contained in Norm_{SU 3}U 1 _{1,0}, too, β_{1,5} β_{1,6}i is replaced by e^{6it} β_{1,5} β_{1,6}i . Since the action of A on each principal orbit induces an isometry, we can identify all the metrics which we obtain by this action. Our calculations therefore prove that it suffices to consider a family of metrics which depends on 5 instead of 6 parameters.

We finally consider the N^{1,1}-case. The subgroup SU 2 of Norm_{SU 3}U 1 _{1,1} acts as SO 3 on span e_{1}, e_{2}, e_{7} . Since we can diagonalize the upper left part of the matrix (5.33) by this action, we can reduce the set of metrics which we have to consider to a 7-parameter family.

The isotropy group K of the SU 3 -action on the singular orbit is greater than U 1 _{k,l}. We will see that the SU 3 -invariance of the metric on the singular orbit makes that metric diagonal in all cases but one. The only exception is if the principal orbit is N^{1,1} and K is generated by e_{8} and a x V_{1} V_{4} which is not a multiple of e_{1}, e_{2}, or e_{7}. The fact that the metric on SU 3 K is diagonal will often force the metric on the whole manifold to be diagonal, too. For this reason, we will mostly consider the differential equations for the diagonal case and only make some short remarks on the non-diagonal case. Another reason, why we focus on the diagonal case, is of course that our calculations are much more easier in that case.

Our next step is to deduce a sufficient condition for the holonomy reduction. Before we can express the torsion-freeness of a SU 3 -invariant Spin 7 -structure by a system of ordinary differential equations, we have to choose a frame for it. We start with the generic Aloff-Wallach spaces. Any invariant G_{2}-structure on N^{k,l} is determined by a single element ψ : 𝔪 Im 𝕆 of the fiber. We consider 𝔲 1 _{k,l} as a subalgebra of 𝔤𝔩 𝔪 . ψ can be chosen in such a way that ψ 𝔲 1 _{k,l} ψ^{1} is contained in the Cartan subalgebra of 𝔤_{2} which we have defined on page 22. Conversely, any ψ with this property determines an invariant G_{2}-structure. If we want the G_{2}-structure to be extendable to a chosen SO 7 -structure, we also have to require that ψ is orthogonal and orientation-preserving.

We will now construct an explicit ψ. The only imaginary octonions which are invariant with respect to the Cartan subalgebra of 𝔤_{2} are the multiples of i. Since the action of 𝔲 1 _{k,l} on 𝔪 leaves only e_{7} invariant, ψ has to map e_{7} to a multiple of i. The weights with which 𝔲 1 _{k,l} acts on 𝔪 satisfy the relation:

k l  1 2k l  1 k 2l .

By reversing the order of e_{3}, e_{4} and e_{5}, e_{6} , we also change the sign of the weights of V_{2} and V_{3}. After that change, we are able to take 𝔲 1 _{k,l} as a subalgebra of 𝔤_{2}. Since we can extend any G_{2}-structure on the principal orbit in a canonical way to a Spin 7 -structure on the whole manifold, the following f_{0}, …, f_{7} is a frame of an invariant Spin 7 -structure:

\begin{matrix}f_{0} : \frac{}{t} & f_{1} : \frac{1}{f} e_{7} & f_{2} : \frac{1}{a}e_{1} & f_{3} : \frac{1}{a}e_{2} \\ f_{4} : \frac{1}{b}e_{4} & f_{5} : \frac{1}{b}e_{3} & f_{6} : \frac{1}{c}e_{6} & f_{7} : \frac{1}{c}e_{5} \end{matrix}  (5.34)

The SU 3 -invariant four-form Ω, which is associated to this frame, is:

\begin{aligned}Ω & abcf e^{1367} abcf e^{1457} abcf e^{2357} abcf e^{2467} \\ & a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456} \\ & a^{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt \\ & abc e^{135} dt abc e^{146} dt abc e^{236} dt abc e^{245} dt \end{aligned}  (5.35)

Next, we express the condition dΩ 0 as a system of ordinary differential equations for the metric functions a, b, c, and f. By the usual methods we obtain:

\begin{matrix}\frac{a^{}}{a} & \frac{b^{2} c^{2} a^{2}}{abc} \frac{k l}{2Δ} \frac{f}{a^{2}} \\ \frac{b^{}}{b} & \frac{c^{2} a^{2} b^{2}}{abc} \frac{l}{2Δ} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{a^{2} b^{2} c^{2}}{abc} \frac{k}{2Δ} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & \frac{k l}{2Δ} \frac{f}{a^{2}} \frac{l}{2Δ} \frac{f}{b^{2}} \frac{k}{2Δ} \frac{f}{c^{2}} \end{matrix}  (5.36)

In the above system, we denote k^{2} lk l^{2} by Δ. Furthermore, we have replaced t by t for cosmetic reasons. This convention will be maintained throughout this section. The calculations we have made in order to derive the four-form (5.35) and the system (5.36) can be found in Appendix C.1.

Next, we consider the exceptional Aloff-Wallach space N^{1,0}. If we make the assumption that the restriction of the metric to any principal orbit is diagonal, we can choose the same frame and the same four-form Ω as above. Consequently, the equation dΩ 0 is equivalent to the system (5.36) with k 1, l 0, and Δ 1.

Finally, we consider N^{1,1} as principal orbit. We again assume that the restriction of the metric to any principal orbit is diagonal. For the same reasons as in the other cases, there is a natural choice of the frame f_{0}, …, f_{7} :

\begin{matrix}f_{0} : \frac{}{t} & f_{1} : \frac{1}{f} e_{7} & f_{2} : \frac{1}{a_{1}} e_{1} & f_{3} : \frac{1}{a_{2}} e_{2} \\ f_{4} : \frac{1}{b}e_{4} & f_{5} : \frac{1}{b}e_{3} & f_{6} : \frac{1}{c}e_{6} & f_{7} : \frac{1}{c}e_{5} \end{matrix}  (5.37)

For the four-form Ω, which is associated to this frame, we obtain:

\begin{aligned}Ω & a_{1}bcf e^{1367} a_{1}bcf e^{1457} a_{2}bcf e^{2357} a_{2}bcf e^{2467} \\ & a_{1}a_{2}b^{2} e^{1234} a_{1}a_{2}c^{2} e^{1256} b^{2}c^{2} e^{3456} \\ & a_{1}a_{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt \\ & a_{1}bc e^{135} dt a_{1}bc e^{146} dt a_{2}bc e^{236} dt a_{2}bc e^{245} dt \end{aligned}  (5.38)

The equation dΩ 0 is equivalent to:

\begin{matrix}\frac{a_{1}^{}}{a_{1}} & \frac{b^{2} c^{2} a_{1}^{2}}{a_{1}bc} 3\frac{a_{1}^{2} a_{2}^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{a_{2}^{}}{a_{2}} & \frac{b^{2} c^{2} a_{2}^{2}}{a_{2}bc} 3\frac{a_{2}^{2} a_{1}^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{b^{}}{b} & \frac{1}{2} \frac{a_{1}^{2} c^{2} b^{2}}{a_{1}bc} \frac{1}{2} \frac{a_{2}^{2} c^{2} b^{2}}{a_{2}bc} \frac{1}{6} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{1}{2} \frac{a_{1}^{2} b^{2} c^{2}}{a_{1}bc} \frac{1}{2} \frac{a_{2}^{2} b^{2} c^{2}}{a_{2}bc} \frac{1}{6} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & 3\frac{a_{1} a_{2} ^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \frac{1}{6} \frac{f}{b^{2}} \frac{1}{6} \frac{f}{c^{2}} \end{matrix}  (5.39)

In the above system, we again have replaced t by t. The detailed calculations which were necessary to obtain the four-form (5.38) and the system (5.39) can be found in Appendix C.2.

We consider the question if there are any SU 3 -invariant G_{2}-structures on the Aloff-Wallach spaces except those which we have constructed on the previous pages. Let g be an arbitrary invariant metric on an Aloff-Wallach space N^{k,l}. In Lemma 3.1.50 we have proven that the space of all invariant G_{2}-structures on N^{k,l} whose associated metric is g and whose orientation is fixed can be described by:

Norm_{SO 7}U 1 _{k,l} Norm_{G_{2}}U 1 _{k,l} .

In this situation, U 1 _{k,l} is identified with its representation on 𝔪 or Im 𝕆 respectively. We first investigate the problem on the Lie algebra level. The tangent space of the above coset space can be identified with:

Norm_{𝔰𝔬 7}𝔲 1 _{k,l} Norm_{𝔤_{2}}𝔲 1 _{k,l} .

The Lie algebra 𝔲 1 _{k,l} is generated by a z 𝔤𝔩 Im 𝕆 whose matrix representation with respect to the standard basis of Im 𝕆 is:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & k l \\ & k l & 0 \\ & & & 0 & 2k l \\ & & & 2k l & 0 \\ & & & & & 0 & k 2l \\ & & & & & k 2l & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

By similar arguments as on page 110, we see that both of the above normalizers coincide with the corresponding centralizers. Let 𝔤 be either 𝔤_{2} or 𝔰𝔬 7 and let κ be the Killing form of 𝔤. A x 𝔤 is contained in the normalizer Norm_{𝔤}𝔲 1 _{k,l} if and only if

x, z λz for a λ ℝ .

From this relation, it follows that

0 κ x, z, z κ x, z , z λ κ z, z .

The above equation can be satisfied for λ 0 only and thus we have

Norm_{𝔤}𝔲 1 _{k,l} x 𝔤 x, z 0 C_{𝔤} 𝔲 1 _{k,l} .

We are going to determine C_{𝔤} 𝔲 1 _{k,l}. Our considerations first will be done for the complexification 𝔤 ℂ, since this will simplify some of our arguments. Any x 𝔤 ℂ has a Cartan decomposition

x h _{α Φ} μ_{α}x_{α} with μ_{α} ℂ .

In the above formula, h is an element of the standard Cartan subalgebra 𝔥 of 𝔤 ℂ, which is the complexification of the subalgebra from page 22 or 110 respectively. Furthermore, Φ is the root system of 𝔤 ℂ and x_{α} is a suitable generator of the root space L_{α} of α. With this notation, the centralizer can be described as follows:

C_{𝔤 ℂ} 𝔲 1 _{k,l} ℂ x 𝔤 ℂ ; ; ; ; z, x _{α Φ} α z μ_{α}x_{α} 0 .

Let Φ^{}: α Φ α z 0 . The above formula can be simplified to:

C_{𝔤 ℂ} 𝔲 1 _{k,l} ℂ 𝔥 _{α Φ^{}}L_{α} .

We specialize to the case 𝔤 𝔰𝔬 7 . As usual, we will identify the Cartan subalgebra of 𝔰𝔬 7, ℂ and its dual space by the Killing form. In Section 5.2, we have chosen a basis θ_{1}, θ_{2}, θ_{3} of the Cartan subalgebra of 𝔰𝔬 7, ℂ . With respect to this basis, z becomes:

z k l θ_{1} 2k l θ_{2} k 2l θ_{3} .

In the above formula, we had to multiply the weights by which z acts on Im 𝕆 by 1. The reason for this are the signs we had chosen in Section 5.2 when we fixed a basis of the Cartan subalgebra of 𝔰𝔬 7 . Let α _{i 1}^{3} α_{i}θ_{i} be a weight of 𝔰𝔬 7, ℂ . The equation α z 0 is equivalent to:

k l α_{1} 2k l α_{2} k 2l α_{3} 0 .

We recall that the root system of 𝔰𝔬 7, ℂ is

θ_{i} 1 i 3 θ_{i} θ_{j} 1 i j 3 .

A root of the first kind can be contained in Φ^{}only if one of the coefficients k l, 2k l, k 2l vanishes. This is the case if and only if N^{k,l} N^{1,1}. In that situation, it is exactly one of those terms which vanishes. A root of the second kind can be in Φ^{}only if two of the above terms coincide up to the sign. In both of the exceptional cases, this happens exactly once. If N^{k,l} is generic, none of these phenomena do happen and Φ^{}is empty. We are now able to describe the normalizer in all of the three cases:

• Let N^{k,l} be a generic Aloff-Wallach space. Since the set Φ^{}is empty, Norm_{𝔰𝔬 7,ℂ} 𝔲 1 _{k,l} ℂ is the Cartan subalgebra of 𝔰𝔬 7, ℂ . The normalizer Norm_{𝔰𝔬 7}𝔲 1 _{k,l} therefore has to be isomorphic to 3𝔲 1 .

• If N^{k,l} N^{1,0}, the set Φ^{}consists of two roots α. Norm_{𝔰𝔬 7,ℂ} 𝔲 1 _{k,l} ℂ therefore has to be isomorphic to 𝔰𝔩 2, ℂ 2 𝔤𝔩 1, ℂ . Since Norm_{𝔰𝔬 7}𝔲 1 _{k,l} is the compact real form of Norm_{𝔰𝔬 7,ℂ} 𝔲 1 _{k,l} ℂ , it is isomorphic to 𝔰𝔲 2 2𝔲 1 . We are able to describe the embedding of the semisimple summand into 𝔰𝔬 7 explicitly: If k 1 and l 0, the matrix representation of z with respect to the basis i, j, k, kϵ, jϵ, ϵ, iϵ is:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & 1 \\ & 1 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \\ & & & & & 0 & 2 \\ & & & & & 2 & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

The Lie algebra 𝔰𝔲 2 acts irreducibly on span j, k, kϵ, jϵ by its complex two-dimensional representation. This action commutes with the action of z on Im 𝕆 . Therefore, the subalgebra 𝔰𝔲 2 𝔤𝔩 span j, k, kϵ, jϵ which we have constructed has to be contained in the normalizer. This subalgebra and the standard Cartan subalgebra of 𝔰𝔬 7 together generate all of Norm_{𝔰𝔬 7}𝔲 1 _{1,0}.

• If N^{k,l} N^{1,1}, the set Φ^{}consists of two pairs of roots α β . Norm_{𝔰𝔬 7,ℂ} 𝔲 1 _{k,l} ℂ therefore is the direct sum of a Lie algebra with Dynkin diagram A_{1} A_{1} and a onedimensional Lie algebra. Since A_{1} A_{1} is the Dynkin diagram of 𝔰𝔩 2, ℂ 𝔰𝔩 2, ℂ and Norm_{𝔰𝔬 7}𝔲 1 _{k,l} is the compact real form of Norm_{𝔰𝔬 7,ℂ} 𝔲 1 _{k,l} ℂ , Norm_{𝔰𝔬 7}𝔲 1 _{k,l} is isomorphic to 2𝔰𝔲 2 𝔲 1 . This normalizer can be described explicitly, too: If k l 1, the matrix representation of z with respect to the standard basis of Im 𝕆 is:

 ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ & & & 0 & 3 \\ & & & 3 & 0 \\ & & & & & 0 & 3 \\ & & & & & 3 & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;

𝔰𝔲 2 acts irreducibly on Im ℍ and on ℍϵ. Furthermore, the subalgebra of type 2𝔰𝔲 2 𝔤𝔩 Im 𝕆 which is defined by this action is contained in 𝔰𝔬 7 and commutes with z. Therefore, the algebra has to be contained in Norm_{𝔰𝔬 7}𝔲 1 _{1,1}, too. Analogously to the previous case, Norm_{𝔰𝔬 7}𝔲 1 _{1,1} is generated by the Lie algebra of type 2𝔰𝔲 2 which we have constructed and the standard Cartan subalgebra of 𝔰𝔬 7 .

We denote the Cartan subalgebra of 𝔤_{2} which we have defined on page 22 shortly by 2𝔲 1 . Since we have:

2𝔲 1 Norm_{𝔤_{2}}𝔲 1 _{k,l} 𝔤_{2} ,

the rank of Norm_{𝔤_{2}}𝔲 1 _{k,l} has to equal 2. By checking the subalgebras of Norm_{𝔰𝔬 7}𝔲 1 _{k,l} of rank two, we thus can determine Norm_{𝔤_{2}}𝔲 1 _{k,l}. Again, we consider the three cases separately:

• Let N^{k,l} be a generic Aloff-Wallach space. Under this assumption, we have

Norm_{𝔤_{2}}𝔲 1 _{k,l} Norm_{𝔰𝔬 7}𝔲 1 _{k,l} 𝔤_{2} 2𝔲 1 .

• Let N^{k,l} N^{1,0}. Without loss of generality, we assume that k 1, l 0. We have to check if the semisimple part 𝔨 of Norm_{𝔰𝔬 7}𝔲 1 _{1,0} is contained in 𝔤_{2}. If this is the case, it follows from rank Norm_{𝔤_{2}}𝔲 1 _{1,0} 2 that Norm_{𝔤_{2}}𝔲 1 _{1,0} 𝔰𝔲 2 𝔲 1 .

The intersection of 𝔨 and 2𝔲 1 is obviously one-dimensional and contained in 𝔤_{2}. It suffices to proof that there is another x 𝔨 𝔨 2𝔲 1 which is contained in 𝔤_{2}. The reason for this is that in this situation 𝔨 2𝔲 1 , x, and 𝔨 2𝔲 1 , x span all of 𝔨 𝔰𝔲 2 and thus 𝔨 Norm_{𝔤_{2}}𝔲 1 _{1,1}. The matrix

\begin{matrix}0 & i \\ i & 0 \end{matrix} 𝔰𝔲 2

corresponds to a x 𝔨 𝔰𝔬 7 , whose matrix representation with respect to the standard basis of Im 𝕆 is:

 ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;

As we have seen in Section 2.1, x is contained in 𝔤_{2} if and only if the three-form ω is annihilated by the pull-back of x. We have:

x^{}ω x^{}dx^{123} dx^{145} dx^{167} dx^{246} dx^{257} dx^{347} dx^{356}

0 dx^{163} dx^{127} 0 0 0

0 dx^{127} dx^{163}

dx^{646} 0 dx^{242}

dx^{657} 0 dx^{253}

dx^{747} 0 dx^{343}

dx^{756} 0 dx^{352} 0 .

Therefore, x 𝔤_{2} and we can conclude that Norm_{𝔤_{2}}𝔲 1 _{1,0} 𝔰𝔲 2 𝔲 1 .

• Finally, let N^{k,l} N^{1,1}. We take a look at the proof of Lemma 4.1, where we explicitly have described all subalgebras of 𝔤_{2}. It is easy to see that the subalgebra 𝔰𝔲 2 _{1,2} 𝔲 1 of 𝔤_{2} is also contained in Norm_{𝔰𝔬 7}𝔲 1 _{1,1}. Since rank Norm_{𝔤_{2}}𝔲 1 _{1,1} 2, the only possibilities for Norm_{𝔤_{2}}𝔲 1 _{1,1} are 𝔰𝔲 2 _{1,2} 𝔲 1 and 2𝔰𝔲 2 . By a short calculation, we can exclude the second possibility. Thus, we have Norm_{𝔤_{2}}𝔲 1 _{1,1} 𝔰𝔲 2 𝔲 1 .

The connected components of the coset space Norm_{SO 7}U 1 _{k,l} Norm_{G_{2}}U 1 _{k,l} are all diffeomorphic and the diffeomorphisms are given by the action of an element of Norm_{SO 7}U 1 _{k,l}. By passing from the Lie algebras to the corresponding connected Lie groups we are able to describe the connected components of that space. We will denote the identity component of a Lie group by the index ”e”. The quotient Norm_{SO 7}U 1 _{k,l} _{e} Norm_{G_{2}}U 1 _{k,l} _{e} covers any connected component of Norm_{SO 7}U 1 _{k,l} Norm_{G_{2}}U 1 _{k,l}. This fact will help us to describe a transitive, almost free group action on the connected components of Norm_{SO 7}U 1 _{k,l} Norm_{G_{2}}U 1 _{k,l}:

• If N^{k,l} is generic, we have:

Norm_{SO 7}U 1 _{k,l} _{e} Norm_{G_{2}}U 1 _{k,l} _{e} U 1 ^{3} U 1 ^{2} U 1 .

Let T be a one-dimensional connected Lie subgroup of the standard maximal torus of SO 7 . We choose T in such a way that G_{2} T is trivial. The action of T on a frame of a fixed homogeneous G_{2}-structure ω generates a family of G_{2}-structures, which is diffeomorphic to the circle. This family is a connected component of the space of all SU 3 -invariant G_{2}-structures on N^{k,l} which have the same associated metric and orientation as ω. One possible choice of T is:

T :  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; 1 1 0 0 1 cos θ sin θ sin θ cos θ 1 0 0 1  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; : T_{θ} ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; θ ℝ ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; (5.40)

• If N^{k,l} N^{1,0}, we have:

Norm_{SO 7}U 1 _{k,l} _{e} Norm_{G_{2}}U 1 _{k,l} _{e} U 2 U 1 U 2 U 1 .

It is easy to see that T Norm_{SO 7}U 1 _{k,l} and that T G_{2} is trivial. Therefore, the space of all SU 3 -invariant G_{2}-structures which have the same associated metric and orientation is in this case generated by the action of T, too.

• The third case which we have to consider is N^{k,l} N^{1,1}. First, we take a look at the situation on the level of Lie algebras. We recall that:

– Norm_{𝔰𝔬 7}𝔲 1 _{1,1} 2𝔰𝔲 2 𝔲 1

– Norm_{𝔤_{2}}𝔲 1 _{1,1} 𝔰𝔲 2 𝔲 1

We compare the abelian part of both algebras. The abelian summand of Norm_{𝔰𝔬 7}𝔲 1 _{1,1} has to be a subalgebra of the standard Cartan subalgebra of 𝔰𝔬 7 . Furthermore, it has to commute with 2𝔰𝔲 2 Norm_{𝔰𝔬 7}𝔲 1 _{1,1}. These conditions leave only one choice for the abelian summand open. It has to be generated by:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & 0 \\ & 0 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \\ & & & & & 0 & 1 \\ & & & & & 1 & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

The abelian part of Norm_{𝔤_{2}}𝔲 1 _{1,1} 𝔰𝔲 2 _{1,2} 𝔲 1 is generated by the same matrix. We therefore can cancel that summand and obtain:

Norm_{SO 7}U 1 _{1,1} _{e} Norm_{G_{2}}U 1 _{1,1} _{e} SO 3 SU 2 △SU 2 .

The group △SU 2 by which we divide is not the second factor of SO 3 SU 2 but the Lie subgroup of G_{2} which is associated to 𝔰𝔲 2 _{1,2}. △SU 2 is ”diagonally” embedded into SO 3 SU 2 . More precisely, it acts as SU 2 ℤ_{2} on the first and canonically on the second factor. We take a look at the orbit of SO 3 in SO 3 SU 2 △SU 2 . Since the intersection of SO 3 and △SU 2 is discrete, the orbit is a three-dimensional submanifold of SO 3 SU 2 △SU 2 . In particular, it is an open set. Since SO 3 is a closed subgroup of SO 3 SU 2 , the orbit is closed, too. SO 3 SU 2 △SU 2 is a connected space and we can conclude that SO 3 acts transitively on that space. All in all, we have proven that the action of SO 3 on Im ℍ generates any connected component of the space which we consider.

The SO 3 -action on span f_{1}, f_{2}, f_{3} , which we have considered above, is in general not the same as the action of SU 2 Norm_{SU 3}U 1 _{1,1} on span e_{1}, e_{2}, e_{7} . The reason for this simply is that the first action preserves the SU 3 -invariant metric which we have fixed, while the action of the normalizer preserves the normal metric q. Nevertheless, if some of the elements of f_{1}, f_{2}, f_{3} have the same length with respect to q, a subset of the G_{2}-structures which we consider can be generated by the action of a subgroup of Norm_{SU 3}U 1 _{1,1}. In that case, that subgroup even acts by isometries. Any element of SO 3 is conjugate to a matrix of type:

{ \atop }1 ; cos θ sin θ ; sin θ cos θ { \atop }.

Therefore, we will later on consider the G_{2}-structures which are generated by the action of

S : ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 1 cos θ sin θ sin θ cos θ 1 0 0 1 1 0 0 1 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; θ ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

on a fixed frame. We hope that the other invariant G_{2}-structures which have the same extension to a SO 7 -structure behave similar as these special ones. Since SNorm_{G_{2}}U 1 _{1,1} T Norm_{G_{2}}U 1 _{1,1}, we can replace the action of S by the action of T.

Until now, we have only determined the type of the connected components of Norm_{SO 7}U 1 _{k,l} Norm_{G_{2}}U 1 _{k,l}, but not their number. In the generic case, this number could be determined by similar methods as in Section 5.2. Unfortunately, those methods do not work in the two exceptional cases. We will briefly explain why our arguments would fail: In Section 5.2, we successfully have determined the number of connected components of the space of G_{2}-structures. In order to do this, we had to prove that Norm_{SO 7}U 1 ^{2} Norm_{SO 7}U 1 ^{3}, where U 1 ^{3} is the maximal torus of SO 7 . This allowed us to work with the Weyl group Norm_{SO 7}U 1 ^{3} U 1 ^{3}.

If N^{k,l} is exceptional, Norm_{SO 7}U 1 _{k,l} is not a subgroup of Norm_{SO 7}U 1 ^{3}, since the identity component of the second group is U 1 ^{3} and the identity component Norm_{SO 7}U 1 _{k,l} is a larger group. Therefore, we cannot rule out by the above methods that there are further homogeneous G_{2}-structures, which cannot be obtained by the group actions from the previous pages. It is possible that the additional G_{2}-structures could be extended to new examples of parallel cohomogeneity-one Spin 7 -structures. We will nevertheless not further discuss this issue. There are three reasons for our decision: First, the study of the Spin 7 -structures whose restriction to a principal orbit is one of the G_{2}-structures which we have constructed so far is already a rewarding project even if further examples existed. Second, it may be possible that we switch to another connected component of Norm_{SO 7}U 1 _{k,l} Norm_{G_{2}}U 1 _{k,l} if we change the sign of some of the functions a, a_{1}, a_{2}, b, c, and f. Therefore, we may already describe more than one or even all connected components by our ansatz for Ω. A third reason is that the results which we have found are sufficient to make statements on the holonomy of our metrics.

On the following pages, we take a look at the new cohomogeneity-one Spin 7 -structures whose existence follows from the considerations we have made. We first consider the case where N^{k,l} is generic or N^{1,0}. The action of the matrix T_{θ} from page 161 on the frame (5.34) yields the following more general frame:

f_{0} : \frac{}{t}  f_{1} : \frac{1}{f} e_{7}  f_{2} : \frac{1}{a}e_{1} f_{3} : \frac{1}{a}e_{2}
f_{4} : \frac{1}{b} sin θ e_{3} \frac{1}{b} cos θ e_{4} f_{5} : \frac{1}{b} cos θ e_{3} \frac{1}{b} sin θ e_{4} f_{6} : \frac{1}{c}e_{6} f_{7} : \frac{1}{c}e_{5}

(5.41)

With help of a short MAPLE program, we can determine the four-form Ω which is associated to this frame. We can calculate dΩ, too. Since we have made that calculations for θ 0 by hand, we can compare our results with the four-form (5.35) in order to check our program. The exterior derivative dΩ which we have calculated in Appendix C.1 can also be used for checking our results. We obtain:

Ω abcf cos θ e^{1367} abcf cos θ e^{1457} abcf cos θ e^{2357} abcf cos θ e^{2467}
abcf sin θ e^{1357} abcf sin θ e^{1467} abcf sin θ e^{2367} abcf sin θ e^{2457}
a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
a^{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt
abc cos θ e^{135} dt abc cos θ e^{146} dt abc cos θ e^{236} dt abc cos θ e^{245} dt
abc sin θ e^{136} dt abc sin θ e^{145} dt abc sin θ e^{235} dt abc sin θ e^{246} dt

The exterior derivative of ω is given by:

d ω 4abcf sin θ e^{12347} e^{12567} e^{34567} .

This term has to vanish in order to make Ω a closed form. We have d ω 0 if and only if θ πℤ. If θ 2πℤ, we again obtain the special form (5.35) which we have considered earlier in this section. Therefore, we only have to treat the case θ π. We insert θ π in the frame (5.41). The result coincides with (5.34), except that f_{4} and f_{5} have the opposite sign. This change of signs is equivalent to changing the sign of the function b. Since we have allowed positive and negative values for the function b, we do not have to consider the case θ π separately.

Let g be a cohomogeneity-one metric which has a singular orbit, satisfies (5.36), and whose principal orbit is a generic Aloff-Wallach space or N^{1,0}. Since any complete cohomogeneityone metric with holonomy Spin 7 which is not a Riemannian product has a singular orbit, we can assume that our first condition on g is justified. We will prove that the holonomy of g is all of Spin 7 . If the holonomy was not all of Spin 7 , it would be contained in G_{2} or SU 4 . In the first case, there would exist a parallel vector field X on the manifold. Let η be the dual of X. Since η has to be SU 3 -invariant, η α t e^{7} β t dt for some functions α and β. If X is parallel, η is closed. de^{7} and dt e^{7} are linearly independent and we can easily conclude from dη 0 that α 0. Since X has to be of constant length, it has to be a multiple of dt. In that situation, we have a^{}b^{}c^{}f^{}0 and the manifold is a Riemannian product. Since g is Ricci-flat, its restriction to the principal orbit is Ricci-flat, too, which is impossible. In the second case, it follows from Lemma 2.3.17 that in this situation there exists a one-parameter family of parallel Spin 7 -structures. Since we have proven that the set of all invariant coclosed G_{2}-structures on the principal orbit is discrete, this cannot be the case. Therefore, the holonomy of g has to be all of Spin 7 .

We proceed to the N^{1,1}-case. We assume that k l 1 and let T_{θ} act on the frame (5.37):

f_{0} : \frac{}{t}  f_{1} : \frac{1}{f} e_{7}  f_{2} : \frac{1}{a_{1}} e_{1} f_{3} : \frac{1}{a_{2}} e_{2}
f_{4} : \frac{1}{b} sin θ e_{3} \frac{1}{b} cos θ e_{4} f_{5} : \frac{1}{b} cos θ e_{3} \frac{1}{b} sin θ e_{4} f_{6} : \frac{1}{c}e_{6} f_{7} : \frac{1}{c}e_{5}

(5.42)

Again, we write a MAPLE program and obtain for the four-form Ω which corresponds to (5.42):

Ω a_{1}bcf cos θ e^{1367} a_{1}bcf cos θ e^{1457} a_{2}bcf cos θ e^{2357} a_{2}bcf cos θ e^{2467}
a_{1}bcf sin θ e^{1357} a_{1}bcf sin θ e^{1467} a_{2}bcf sin θ e^{2367} a_{2}bcf sin θ e^{2457}
a_{1}a_{2}b^{2} e^{1234} a_{1}a_{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
a_{1}a_{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt
a_{1}bc cos θ e^{135} dt a_{1}bc cos θ e^{146} dt
a_{2}bc cos θ e^{236} dt a_{2}bc cos θ e^{245} dt
a_{1}bc sin θ e^{136} dt a_{1}bc sin θ e^{145} dt
a_{2}bc sin θ e^{235} dt a_{2}bc sin θ e^{246} dt

(5.43)

The exterior derivative of this four-form is:

dΩ a_{1}^{}bcf sin θ a_{1}b^{}cf sin θ a_{1}bc^{}f sin θ a_{1}bcf^{}sin θ a_{1}bcfθ^{}cos θ

6a_{1}bc sin θ 6a_{2}bc sin θ e^{1357} dt a_{1}^{}bcf cos θ a_{1}b^{}cf cos θ a_{1}bc^{}f cos θ a_{1}bcf^{}cos θ a_{1}bcfθ^{}sin θ

6a_{1}bc cos θ 6a_{2}bc cos θ a_{1}a_{2}f b^{2}f c^{2}f e^{1367} dt a_{1}^{}bcf cos θ a_{1}b^{}cf cos θ a_{1}bc^{}f cos θ a_{1}bcf^{}cos θ a_{1}bcfθ^{}sin θ

6a_{1}bc cos θ 6a_{2}bc cos θ a_{1}a_{2}f b^{2}f c^{2}f e^{1457} dt a_{1}^{}bcf sin θ a_{1}b^{}cf sin θ a_{1}bc^{}f sin θ a_{1}bcf^{}sin θ a_{1}bcfθ^{}cos θ

6a_{1}bc sin θ 6a_{2}bc sin θ e^{1467} dt a_{2}^{}bcf cos θ a_{2}b^{}cf cos θ a_{2}bc^{}f cos θ a_{2}bcf^{}cos θ a_{2}bcfθ^{}sin θ

6a_{1}bc cos θ 6a_{2}bc cos θ a_{1}a_{2}f b^{2}f c^{2}f e^{2357} dt a_{2}^{}bcf sin θ a_{2}b^{}cf sin θ a_{2}bc^{}f sin θ a_{2}bcf^{}sin θ a_{2}bcfθ^{}cos θ

6a_{1}bc sin θ 6a_{2}bc sin θ e^{2367} dt a_{2}^{}bcf sin θ a_{2}b^{}cf sin θ a_{2}bc^{}f sin θ a_{2}bcf^{}sin θ a_{2}bcfθ^{}cos θ

6a_{1}bc sin θ 6a_{2}bc sin θ e^{2457} dt a_{2}^{}bcf cos θ a_{2}b^{}cf cos θ a_{2}bc^{}f cos θ a_{2}bcf^{}cos θ a_{2}bcfθ^{}sin θ

6a_{1}bc cos θ 6a_{2}bc cos θ a_{1}a_{2}f b^{2}f c^{2}f e^{2467} dt

a_{1}^{}a_{2}b^{2} a_{1}a_{2}^{}b^{2} 2a_{1}a_{2}bb^{}\frac{1}{3}a_{1}a_{2}f \frac{2}{3}b^{2}f 2a_{1}bc cos θ 2a_{2}bc cos θ e^{1234} dt

a_{1}^{}a_{2}c^{2} a_{1}a_{2}^{}c^{2} 2a_{1}a_{2}cc^{}\frac{1}{3}a_{1}a_{2}f \frac{2}{3}c^{2}f 2a_{1}bc cos θ 2a_{2}bc cos θ e^{1256} dt

2bb^{}c^{2} 2b^{2}cc^{}\frac{1}{3}b^{2}f \frac{1}{3}c^{2}f 2a_{1}bc cos θ 2a_{2}bc cos θ e^{3456} dt

2a_{1}bcf sin θ 2a_{2}bcf sin θ e^{12347}

2a_{1}bcf sin θ 2a_{2}bcf sin θ e^{12567}

2a_{1}bcf sin θ 2a_{2}bcf sin θ e^{34567}

We want d ω to vanish and thus either have to require that sin θ 0 or that a_{2} a_{1}. In the first case, we can make the same arguments as before and obtain no other examples as in the special case θ 0, which we have already considered in this section. We therefore analyze the equation dΩ 0 under the assumption that a_{2} a_{1}. In that situation, the condition that the coefficients of e^{1367} dt and e^{2357} dt vanish is equivalent to:

a_{1}^{}bcf cos θ a_{1}b^{}cf cos θ a_{1}bc^{}f cos θ a_{1}bcf^{}cos θ a_{1}bcfθ^{}sin θ

12a_{1}bc cos θ a_{1}^{2}f b^{2}f c^{2}f  0

a_{1}^{}bcf cos θ a_{1}b^{}cf cos θ a_{1}bc^{}f cos θ a_{1}bcf^{}cos θ a_{1}bcfθ^{}sin θ

12a_{1}bc cos θ a_{1}^{2}f b^{2}f c^{2}f  0

Since f has to be non-zero, we obtain a_{1}^{2} b^{2} c^{2} from the above two equations. With this additional relation, it is easy to see that dΩ 0 is equivalent to:

a_{1}^{}bcf sin θ a_{1}b^{}cf sin θ a_{1}bc^{}f sin θ a_{1}bcf^{}sin θ a_{1}bcfθ^{}cos θ 12a_{1}bc sin θ  0
a_{1}^{}bcf cos θ a_{1}b^{}cf cos θ a_{1}bc^{}f cos θ a_{1}bcf^{}cos θ a_{1}bcfθ^{}sin θ 12a_{1}bc cos θ  0
2a_{1}^{}a_{1}b^{2} 2a_{1}^{2}b^{}b \frac{1}{3}a_{1}^{2}f \frac{2}{3}b^{2}f  0
2a_{1}^{}a_{1}c^{2} 2a_{1}^{2}c^{}c \frac{1}{3}a_{1}^{2}f \frac{2}{3}c^{2}f  0
2b^{}bc^{2} 2b^{2}c^{}c \frac{1}{3}b^{2}f \frac{1}{3}c^{2}f  0

By dividing the above equations by suitable products of the metric functions, we obtain:

\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} sin θ θ^{}cos θ  12\frac{1}{f} sin θ
\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} cos θ θ^{}sin θ  12\frac{1}{f} cos θ
\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b}  \frac{1}{3} \frac{f}{a_{1}^{2}} \frac{1}{6} \frac{f}{b^{2}}
\frac{a_{1}^{}}{a_{1}} \frac{c^{}}{c}  \frac{1}{3} \frac{f}{a_{1}^{2}} \frac{1}{6} \frac{f}{c^{2}}
\frac{b^{}}{b} \frac{c^{}}{c}  \frac{1}{6} \frac{f}{b^{2}} \frac{1}{6} \frac{f}{c^{2}}

From the first two equations, we conclude that θ can be chosen as an arbitrary constant. Our system finally simplifies to:

\frac{a_{1}^{}}{a_{1}} \frac{1}{3} \frac{f}{a_{1}^{2}}  \frac{b^{}}{b} \frac{1}{6} \frac{f}{b^{2}}  \frac{c^{}}{c} \frac{1}{6} \frac{f}{c^{2}}  \frac{f^{}}{f} 12\frac{1}{f} \frac{1}{3} \frac{f}{a_{1}^{2}} \frac{1}{6} \frac{f}{b^{2}} \frac{1}{6} \frac{f}{c^{2}} .

If we consider the system (5.39) under the assumption that a_{2} a_{1} and a_{1}^{2} b^{2} c^{2}, we obtain the same four equations as above. There are two conclusions we can draw from this fact: First, we have proven that we obtain no new metrics by considering the more general Spin 7 - structures which are described by (5.42) and (5.43). Second, it follows that any solution of (5.39) which satisfies a_{2} a_{1} and a_{1}^{2} b^{2} c^{2} yields a metric whose holonomy is contained in SU 4 . The reason for this is that if the two relations hold, we can choose θ arbitrarily and have found a one-parameter family of parallel Spin 7 -structures with the same associated metric. We apply Lemma 2.3.17 and conclude that the holonomy is a subgroup of SU 4 . The existence of a parallel vector field can be excluded by similar arguments as in the previous case. The dual η of such a vector field has to coincide with α t e^{1} β t e^{2} γ t e^{7} δ t dt for some functions α, β, γ, and δ. Since dη 0 and de^{1}, de^{2}, de^{7}, dt e^{1}, dt e^{2}, dt e^{7} are all linearly independent, it follows that α, β, γ 0. Therefore, we have η dt, which can be excluded by the same arguments as in the previous case. We have proven that there exists no non-zero tangent vector which is left invariant by the holonomy group. The holonomy thus is either SU 4 , Sp 2 , or SU 2 SU 2 . It is possible to prove that there actually are solutions of (5.39) which obey a_{2} a_{1} and a_{1}^{2} b^{2} c^{2}. Metrics of that kind have been considered by Kanno and Yasui in [49].

We will make some short remarks on the question if the above case is the only one where the holonomy is Spin 7 . For the same reasons as above, we only have to decide if the holonomy is contained in SU 4 . We recall that the four-forms of type (5.43) are not the only possible invariant Spin 7 -structures, since there is a bigger family of invariant G_{2}-structures on N^{1,1} which is generated by a SO 3 - rather than a U 1 -action. If we had considered a more general frame which was obtained by the action of SO 3 Norm_{SO 7}U 1 _{1,1} on (5.37), we could have found other sufficient conditions for the holonomy to be contained in SU 4 . It even might be possible that another choice of the Spin 7 -structure yielded a diagonal cohomogeneity-one metric with holonomy Spin 7 which did not obey the system (5.39). Since the calculations for the more general frame would be much more extensive than the above ones, we have restricted ourselves to a special case. The great number of possible parallel Spin 7 -structures is also the reason why we will not determine the holonomy but only characterize it as a subgroup of a bigger group.

The results which we have obtained so far motivate why we only consider metrics which satisfy the systems (5.36) and (5.39). Unlike in Section 5.2 and 5.3, there is no easy method to solve those systems of differential equations except in special cases. Since the equations are highly non-linear, it is even hard to decide if the resulting metrics are complete.

In order to prove at least local existence results, we make a power series ansatz for (5.36) and (5.39) near a singular orbit. At least one of the metric functions has to converge to zero at the singular orbit. The fact that we divide by this function in the differential equations will cause some difficulties, which we have to solve. We will investigate which invariant metrics on the singular orbit can be extended to cohomogeneity-one metrics with holonomy Spin 7 and if there are any initial conditions of higher order which we can prescribe. Before we start this program, we first have to classify the possible singular orbits:

Lemma 5.4.2. Let U 1 be embedded into SU 3 as U 1 _{k,l}. Furthermore, let K be a connected, closed group with U 1 _{k,l} K SU 3 . The Lie algebra of K we denote by 𝔨. In this situation, 𝔨 and K can be found in the table below. Moreover, K U 1 _{k,l} and SU 3 K satisfy the following topological conditions:

𝔨  K  K U 1 _{k,l}  SU 3 K  Condition on k and l
2𝔲 1  U 1 ^{2} S^{1}  SU 3 U 1 ^{2}
𝔰𝔲 2  SU 2  S^{2}  S^{5}  k l k l 0
𝔰𝔲 2  SO 3  S^{2}  SU 3 SO 3 k l k l 0
𝔰𝔲 2 𝔲 1 U 2  S^{3} ℤ_{k l}  ℂℙ^{2}  k l k l 0
𝔰𝔲 2 𝔲 1 U 2  S^{3}  ℂℙ^{2}  k l k l 0
𝔰𝔲 2 𝔲 1 U 2  S^{2} S^{1}  ℂℙ^{2}  k l k l 0
𝔰𝔲 3  SU 3  N^{k,l} S^{7} Γ

In the above table, Γ denotes an arbitrary discrete subgroup of O 8 and the group ℤ_{k l}, by which we divide S^{3}, is explicitly described by (5.46).

Proof : The fact that 𝔨 has to be an 𝔲 1 _{k,l}-module, reduces the number of subspaces of 𝔰𝔲 3 which we have to consider. We will check for all 𝔲 1 _{k,l}-modules 𝔨 with 𝔲 1 _{k,l} 𝔨 𝔰𝔲 3 if they are closed under the Lie bracket and if K U 1 _{k,l} is covered by a sphere. If N^{k,l} is an exceptional Aloff-Wallach space, 𝔪 contains pairs U, U^{}of equivalent submodules. This will make our considerations more complicated than in the previous two sections, since there may be submodules of 𝔨 which are transversely embedded into U U^{}. Therefore, we often have to distinguish the different types of Aloff-Wallach spaces in the course of this proof. In order to find all the possibilities for 𝔨, we consider each of the possible values of dim 𝔨 separately:

1. 𝔨 𝔲 1 _{k,l} W, W one-dimensional: If N^{k,l} N^{1,1}, W has to be V_{4}, since this is the only one-dimensional submodule of 𝔪. In this case, 𝔨 is obviously a Lie algebra and isomorphic to 2𝔲 1 . Thus, we have K U 1 ^{2}, K U 1 _{k,l} S^{1}, and SU 3 K SU 3 U 1 ^{2}.

If N^{k,l} N^{1,1}, either V_{1}, V_{2}, or V_{3} is trivial. Without loss of generality, we assume that k l 1, which makes V_{1} trivial. In this situation, we can choose W as span x , where x V_{1} V_{4} 0 is arbitrary. Since x commutes with 𝔲 1 _{1,1}, 𝔨 is closed under the Lie bracket and abelian. Therefore, we again have: K U 1 ^{2}, K U 1 _{1,1} S^{1}, and SU 3 K SU 3 U 1 ^{2}.

2. 𝔨 𝔲 1 _{k,l} W, W two-dimensional: If N^{k,l} is a generic Aloff-Wallach space, the only two-dimensional submodules of 𝔪 are V_{1}, V_{2}, and V_{3}. We consider the case W V_{1}. V_{1} is spanned by e_{1} and e_{2} and we have:

e_{1}, e_{2}  { \atop }\begin{matrix}2i & 0 & 0 \\ 0 & 2i & 0 \\ 0 & 0 & 0 \end{matrix} { \atop }.  (5.44)

The commutator e_{1}, e_{2} is contained in 𝔲 1 _{k,l} V_{1} if and only if k 1 and l 1. Analogously, we can prove that the case W V_{2} is only possible for k 1 and l 0, and that the case W V_{3} is only possible for k 0 and l 1. Therefore, 𝔨 cannot be closed in the generic case.

Next, we consider the case N^{k,l} N^{1,0}. We assume without loss of generality that k 1 and l 1. Formula (5.44) together with the fact that V_{1} is a non-trivial 𝔲 1 _{1, 1} - module proves that 𝔨 is a Lie algebra, which is isomorphic to 𝔰𝔲 2 . In this situation, we have K SU 2 and K U 1 _{1, 1} S^{2} due to the Hopf fibration. Moreover, SU 3 K S^{5}.

Since the modules V_{2} and V_{3} are 𝔲 1 _{1, 1} -equivariantly isomorphic, there may exist a two-dimensional space W which is transversely embedded into V_{2} V_{3} such that 𝔲 1 _{1, 1} W is closed under the Lie bracket. The first of the exceptional cases therefore is not yet fully handled. We assume that such a W exists. Since 𝔨 is the Lie algebra of a compact Lie group, it is either isomorphic to 3𝔲 1 or to 𝔰𝔲 2 . The first case can be excluded, since the rank of 𝔰𝔲 3 is two. Therefore, 𝔨 has to be isomorphic to 𝔰𝔲 2 . There are up to conjugation two embeddings of 𝔰𝔲 2 into 𝔰𝔲 3 . The first embedding is induced by the inclusion of SO 3 into SU 3 . The second subalgebra of type 𝔰𝔲 2 is given by 𝔲 1 _{1, 1} V_{1} and can be identified with a subalgebra of 𝔤𝔩 2, ℂ . In the first case, 𝔨 is a conjugate of 𝔰𝔬 3 and has to contain 𝔲 1 _{1, 1} . One possible choice of 𝔨 is:

𝔨 A^{1} span e_{1}, e_{3}, e_{5} A

with

A : { \atop }\begin{matrix}\frac{1}{2} \overline{2} & \frac{1}{2} \overline{2} & 0 \\ \frac{1}{2} \overline{2} i & \frac{1}{2} \overline{2} i & 0 \\ 0 & 0 & 1 \end{matrix} { \atop }.

We obtain:

𝔨 span e_{8}, \frac{1}{2} \overline{2} e_{3} \frac{1}{2} \overline{2} e_{5}, \frac{1}{2} \overline{2} e_{4} \frac{1}{2} \overline{2} e_{6} .

If there is another 𝔨 which is also tangent to a Lie group isomorphic to SO 3 , it is conjugate to the above 𝔨 and also contains 𝔲 1 _{1, 1} as a Cartan subalgebra. We can assume that the matrix B by which we conjugate preserves 𝔲 1 _{1, 1} . In other words: B Norm_{SU 3}U 1 _{1, 1} . We already have proven that this normalizer is the standard maximal torus of SU 3 . The action of U 1 _{1, 1} by conjugation obviously preserves the algebra 𝔨 which we have chosen. By a short calculation, we see that the action of U 1 _{1,1} does not preserve 𝔨. We therefore have found a one-parameter family of Lie algebras which are conjugate to each other. The associated Lie group K of any Lie algebra 𝔨 in this family is isomorphic to SO 3 . By an inner automorphism of SU 3 , it can be mapped to the standard embedding of SO 3 into SU 3 . The group U 1 _{1, 1} is conjugate to the following subgroup of SO 3 :

\begin{aligned}A \\ & 1 \end{aligned} ; ; ; A SO 2 .

Therefore, we have K U 1 _{1, 1} S^{2} and the singular orbit is the symmetric space SU 3 SO 3 .

Next, we assume that the embedding of 𝔨 into 𝔰𝔲 3 is of the second type. In this case, 𝔨 is conjugate to the Lie algebra

\begin{aligned}x \\ & 0 \end{aligned}  ;  ;  ; x 𝔰𝔲 2

and contains 𝔲 1 _{1, 1} as a Cartan subalgebra. We can prove by the same arguments as above that 𝔨 is conjugate by an element of Norm_{SU 3}U 1 _{1, 1} to the above Lie algebra. Since that algebra is preserved by the normalizer, 𝔨 is in this situation unique. Moreover, it coincides with a Lie subalgebra of 𝔰𝔲 3 which we have already considered.

We finally have to investigate the case N^{k,l} N^{1,1}. If k l 1, V_{1} is trivial and V_{2}, V_{3} are equivalent modules. There are two possibilities for W: Either W V_{1} or W V_{2} V_{3}. We first assume that W V_{1}. Since e_{1}, e_{2} 𝔲 1 _{1, 1} 0 , 𝔨 is not closed under the Lie bracket. If W V_{2} V_{3}, 𝔨 would have to be a Lie algebra which is isomorphic to 𝔰𝔲 2 . Furthermore, 𝔲 1 _{1,1} would be a Cartan subalgebra of 𝔨. No matter how we embed 𝔰𝔲 2 into 𝔰𝔲 3 , its Cartan subalgebras are all conjugate to 𝔲 1 _{1, 1} . Therefore, we can exclude this case.

3. 𝔨 𝔲 1 _{k,l} W, W three-dimensional: We again consider the three different types of Aloff-Wallach spaces separately. First, we assume that N^{k,l} is a generic Aloff-Wallach space. In this situation, we have W V_{i} V_{4} for an i 1, 2, 3 . Since the direct sum 𝔲 1 _{k,l} V_{4} is a two-dimensional abelian Lie algebra, we will denote that space by 2𝔲 1 in the course of this proof. It is easy to see that V_{i}, V_{i} 2𝔲 1 and 2𝔲 1 , V_{i} V_{i}. Therefore, 𝔨 is closed under the Lie bracket. As a Lie algebra it is isomorphic to 𝔰𝔲 2 𝔲 1 𝔲 2 . We assume without loss of generality that i 1. The Lie group K is given by:

\begin{aligned}A \\ & det A^{1} \end{aligned} ; ; ; A U 2 .

K is isomorphic to U 2 . The group U 2 is not a direct product SU 2 U 1 , but it is covered by that product. We can therefore analyze the topology of K U 1 _{k,l} by the same method as in the proof of Lemma 3.1.12: We define a map π : SU 2 K U 1 _{k,l} by π h : hU 1 _{k,l}, where SU 2 is the Lie subgroup of SU 3 with Lie algebra 𝔲 1 _{1, 1} V_{1}. It is easy to see that π is a covering map. Its kernel is SU 2 U 1 _{k,l}. This intersection is, except for k 1, l 1,

 ;  ; { \atop }\begin{matrix}e^{2πi \frac{mk}{k l}} & 0 \\ 0 & e^{2πi \frac{ml}{k l}} \end{matrix} { \atop } ;  ;  ;  ;  ; m ℤ ;  ; .  (5.45)

Since k and l are coprime, the above group is in fact:

\begin{matrix}e^{2πi \frac{m}{k l}} & 0 \\ 0 & e^{2πi \frac{m}{k l}} \end{matrix}   ;  ;  ;  ;  ; m ℤ ;  ; .  (5.46)

SU 2 is diffeomorphic to S^{3} and the quotient K U 1 _{k,l} thus is a lens space which is usually denoted by L k l, 1 . We will nevertheless denote K U 1 _{k,l} by S^{3} ℤ_{k l}, since in this thesis ℤ_{k l} will always be the above discrete group. The singular orbit which we obtain for our choice of K is SU 3 S U 2 U 1 ℂℙ^{2} for all k and l.

Next, we consider the exceptional case k 1, l 1. If we choose W as V_{i} V_{4} with i 2, 3 , K again is isomorphic to U 2 . By conjugating SU 3 by a permutation matrix, we can replace K by the Lie subgroup of SU 3 whose Lie algebra is 2𝔲 1 V_{1} and U 1 _{1, 1} by U 1 _{0,1} or U 1 _{1,0} . Therefore, we obtain K U 1 _{1, 1} S^{3} for the original K. For W V_{1} V_{4}, we can prove that the kernel of the covering map from SU 2 U 1 _{1, 1} U 1 _{1,1} to K U 1 _{1, 1} which is defined by h_{1}U 1 _{1, 1} , h_{2} h_{1}h_{2}U 1 _{1, 1} is contained in U 1 _{1,1}. Therefore, we obtain K U 1 _{1, 1} SU 2 U 1 _{1, 1} U 1 _{1,1} ℤ_{2} S^{2} S^{1}. In both cases, we have as before SU 3 K ℂℙ^{2}. In order to finish the first exceptional case, we have to check if we can choose W W^{}V_{4} with W^{}V_{2} V_{3} but W^{}V_{2}, V_{3}. Since 2𝔲 1 is contained in 𝔨, W^{}has to be a 2𝔲 1 -module. The q-orthogonal complement of 2𝔲 1 in 𝔰𝔲 3 decomposes with respect to 2𝔲 1 into V_{1} V_{2} V_{3} and the three modules are pairwise inequivalent. Therefore, 𝔨 2𝔲 1 V_{i} with i 1, 2, 3 and the above possibilities for 𝔨 are the only ones.

We finally consider the case k l 1. In this case, 𝔨 necessarily has to be a Lie algebra which is isomorphic to 𝔰𝔲 2 𝔲 1 , since K is compact and rank 𝔨 2. We decompose 𝔨 into the direct sum 𝔥 W^{}of 𝔲 1 _{1,1}-modules, where 𝔥 is a Cartan subalgebra of 𝔨 and W^{}is the q-orthogonal complement of 𝔥. Since span e_{1}, e_{2}, e_{7} is the centralizer of 𝔲 1 _{1,1}, 𝔥 can be any of the following spaces:

𝔥 𝔲 1 _{1,1} span x with x span e_{1}, e_{2}, e_{7} 0 .

The 𝔲 1 _{1,1}-module W^{}is either a subspace of span e_{1}, e_{2}, e_{7} or of V_{2} V_{3}. In the first case, we have to choose W^{}as the complement of span x in span e_{1}, e_{2}, e_{7} . 𝔨 therefore is 2𝔲 1 V_{1}, which is isomorphic to 𝔰𝔲 2 𝔲 1 . As in the other cases, we have K U 1 _{1,1} S^{3} ℤ_{2} ℝℙ^{3} and SU 3 K ℂℙ^{2}. We turn to the case where W^{}V_{2} V_{3}. It is easy to see that there exists an

A  \begin{aligned}A^{} \\ & 1 \end{aligned}  ;  ;  ; A^{}SU 2  with A 𝔥 A^{1} 𝔲 1 _{1,1} 𝔲 1 _{1, 1} 2𝔲 1 .

We conjugate our decomposition of 𝔨 by A and obtain:

A 𝔨 A^{1} 2𝔲 1 AW^{}A^{1} .

Since AW^{}A^{1} is a 2𝔲 1 -module, we can prove by the same arguments as in the case k 1, l 1 that AW^{}A^{1} is either V_{2} or V_{3}. If A Id_{ℂ^{3}}, we obtain by the same arguments as in the other exceptional case that K U 1 _{1,1} S^{3} and SU 3 K ℂℙ^{2}. If A Id_{ℂ^{3}}, 𝔨 still is a Lie algebra with 𝔲 1 _{1,1} 𝔨 and our results on K U 1 _{1,1} and SU 3 K remain true. We therefore have found a continuous family of subalgebras which yield the same singular orbit and the same collapsing sphere.

4. 𝔨 𝔲 1 _{k,l} W, W four-dimensional: In this case, 𝔨 has to be a five-dimensional Lie algebra. Since for any root α of 𝔨, α is a root of 𝔨, too, we have dim 𝔨 rank𝔨 mod2 . Therefore, the rank of 𝔨 has to be odd. Since rank 𝔰𝔲 3 2, the only possibility for rank 𝔨 is in fact 1. For that reason, 𝔨 has to be either 𝔲 1 or 𝔰𝔲 2 . Since both of these Lie algebras are not five-dimensional, we can exclude this case.

5. 𝔨 𝔲 1 _{k,l} W, W five-dimensional: 𝔨 has to be six-dimensional and closed under the Lie bracket. The only six-dimensional Lie algebra of rank 2 which belongs to a compact Lie group is 𝔰𝔲 2 𝔰𝔲 2 . No matter how we embed the first summand into 𝔰𝔲 3 , we cannot construct another three-dimensional subspace of 𝔰𝔲 3 on which the first summand acts trivially. Therefore, this case can be excluded, too.

6. 𝔨 𝔲 1 _{k,l} W, W six-dimensional: We can prove by the same arguments as we have made for dim W 4 that this case is impossible.

7. 𝔨 𝔲 1 _{k,l} W, W seven-dimensional: In this case, 𝔨 is all of 𝔰𝔲 3 . The only thing left to be done in this proof is to exclude that N^{k,l} is a quotient of S^{7} by a discrete subgroup Γ O 8 . We have proven that for k, l 1, 1 the space N^{k,l} is a S^{3} ℤ_{k l}- bundle over ℂℙ^{2}. Under the assumption that N^{k,l} S^{7} Γ, we have the following exact sequence:

… π_{2} S^{7} Γ π_{2} ℂℙ^{2} π_{1} S^{3} ℤ_{k l} π_{1} S^{7} Γ π_{1} ℂℙ^{2} ….

If we make this exact sequence explicit, we obtain:

… 0 ℤ ℤ_{k l} Γ 0 ….

Since there is no injective group homomorphism from ℤ to ℤ_{k l}, the Aloff-Wallach space N^{k,l} cannot be covered by a sphere.

In the case k, l 1, 1 , we would have to replace S^{3} ℤ_{k l} by S^{2} S^{1} and our argument would fail. Since N^{1, 1} is diffeomorphic to N^{1,0}, that space nevertheless can be described as a S^{3}-bundle over ℂℙ^{2}. Therefore, our argument holds for all Aloff-Wallach spaces.

Remark 5.4.3. Let K and K^{}be Lie subgroups of SU 3 which satisfy U 1 _{k,l} K, K^{}and K^{}AKA^{1} for an A Norm_{SU 3}U 1 _{k,l}. In that situation, SU 3 K and SU 3 K^{}as well as K U 1 _{k,l} and K^{}U 1 _{k,l} are SU 3 -equivariantly diffeomorphic. We have already mentioned all of those cases in the above proof. We will nevertheless explicitly state them again, since different choices of K yield different initial conditions for our differential equations.

1. If N^{k,l} N^{1,1}, there are several groups K which satisfy U 1 _{k,l} K SU 3 and are isomorphic to U 1 ^{2}. These groups can be obtained by conjugation of the standard maximal torus of SU 3 by an arbitrary element of SU 2 Norm_{SU 3}U 1 _{k,l}.

2. If N^{k,l} N^{1, 1} , there is a one-parameter family of groups which are isomorphic to SO 3 and can be obtained by a U 1 -action on a fixed group. That U 1 -action coincides with the action of a subgroup of the normalizer Norm_{SU 3}U 1 _{k,l}.

3. If N^{k,l} N^{1,1}, we obtain by an action of SU 2 a further family of Lie algebras, which are all isomorphic to 𝔲 2 . That SU 2 -action coincides with the action of SU 2 Norm_{SU 3}U 1 _{k,l}.

For any N^{k,l}, there are three different Lie algebras 𝔨 which satisfy 𝔲 1 _{k,l} 𝔨 𝔰𝔲 3 and are isomorphic to 𝔲 2 , namely 2𝔲 1 V_{i} with i 1, 2, 3 . These algebras can be obtained from each other by conjugation by a permutation matrix. If we drop the condition k l 0 and consider all possible Aloff-Wallach spaces, we can restrict ourselves to the case 𝔨 2𝔲 1 V_{1}. Therefore, it was sufficient to include only that case, in which we have K U 1 _{k,l} S^{3} ℤ_{k l} in the table of the lemma.

Convention 5.4.4. We want to treat the three Lie algebras from Remark 5.4.3 in a uniform manner. Therefore, we will drop the convention k l 0 whenever we consider the case 𝔨 𝔲 2 . This will allow us to fix 𝔨 as 2𝔲 1 V_{1}. Since we can replace k, l by k, l , we can still assume that k l.

Our next aim is to classify the cohomogeneity-one metrics with at least one singular orbit which have an Aloff-Wallach space as principal orbit and whose holonomy is contained in Spin 7 . We will restrict ourselves to diagonal metrics. In the N^{1,1}-case, we will furthermore consider Spin 7 -structures with a frame which is given by (5.37) only. Therefore, we can work with the systems (5.36) and (5.39). There are several differences between those systems of ordinary differential equations and the systems (5.6) and (5.20) from Section 5.2 and 5.3. First, there are SU 3 -invariant metrics on the singular orbit which cannot be extended to a short-time solution of (5.36) or (5.39). Second, in some cases the solutions of those systems depend on additional free parameters of higher order. Third, our systems have, except in special cases, no explicit solution.

In order to classify the solutions of (5.36) and (5.39), we make an explicit power series ansatz. By this method, we find the possible initial metrics on the singular orbit and the free parameters of higher order. Furthermore, we can prove that the free parameters which we have found are the only ones and that the power series expansion exists up to infinite order. We want to know if the power series converges and satisfies the smoothness conditions from Theorem 3.2.18 for any choice of our parameters.

After having deduced the smoothness conditions, we can easily check them by symmetry arguments. Since the equations for the holonomy reduction are a special case of the cohomogeneityone Einstein condition, we can apply Theorem 3.2.24. It follows that our power series converges on a tubular neighborhood of the singular orbit. Moreover, we also obtain an existence result for local examples of cohomogeneity-one Einstein metrics. Since the free parameters for the metrics with special holonomy are a subset of the free parameters from Theorem 3.2.24, we have found a further, more conceptual explanation for the parameters which we have found. Theorem 3.2.24 also holds in the non-diagonal case. Therefore, some of our Einstein metrics are non-diagonal, too. In the cases where non-diagonal Einstein metrics exist, the existence of non-diagonal metrics whose holonomy is contained in Spin 7 cannot be excluded. Since our program is already an extensive one, we will not deal further with this issue.

Remark 5.4.5. In some of the cases which we will consider, Assumption 3.2.19 is not satisfied. Later in this section, we will give arguments why we can nevertheless apply Theorem 3.2.24. For those arguments, we need to know if the differential equations for the cohomogeneity-one Einstein condition respect the splitting S^{2} 𝔭 S^{2} 𝔭^{}. In order to show this and for reasons of completeness, we deduce the equations for diagonal metrics. We include the system for the generic Aloff-Wallach spaces and for N^{1,0} in the Appendix C.3. The system for N^{1,1} can be found in Appendix C.4. We also include the derivation of the equations in the appendix.

On the following pages, we carry out the program which we have outlined above for each of the possible singular orbits. We will have to distinguish between the three different types of Aloff-Wallach spaces. First, we consider the most simple case:

Principal orbit: A generic N^{k,l}, singular orbit: SU 3 U 1 ^{2}

As in the previous two sections, we assume that the singular orbit is at t 0. In Lemma 5.4.2, we have proven that the Lie algebra 2𝔲 1 of the isotropy group U 1 ^{2} is given by 𝔲 1 _{k,l} V_{4}. We therefore have f 0 0 as an initial condition. The functions a, b, and c are all non-zero at the singular orbit. Since f has to satisfy the equation

f^{}\frac{k l}{2Δ} \frac{f^{2}}{a^{2}} \frac{l}{2Δ} \frac{f^{2}}{b^{2}} \frac{k}{2Δ} \frac{f^{2}}{c^{2}}

from the system (5.36), we also have f^{}0 0. With help of a short induction proof, we see that the higher derivatives of f vanish at t 0, too. Any solution of the system (5.36) has to be analytic. Therefore, f vanishes for any value of t. The system (5.36) hence becomes:

\begin{matrix}\frac{a^{}}{a} & \frac{b^{2} c^{2} a^{2}}{abc} \\ \frac{b^{}}{b} & \frac{c^{2} a^{2} b^{2}}{abc} \\ \frac{c^{}}{c} & \frac{a^{2} b^{2} c^{2}}{abc} \end{matrix}  (5.47)

These equations also appear in the paper of Cleyton and Swann [20] and in the next chapter. They describe the reduction of the holonomy to a subgroup of G_{2} in the case of a cohomogeneity-one metric with principal orbit SU 3 U 1 ^{2}. Our considerations prove that the eight-dimensional metric (5.31) degenerates into a seven-dimensional one. Since we are mainly interested in eight-dimensional metrics, we will not further investigate the equations (5.47).

Our next aim is to classify the cohomogeneity-one Einstein metrics with a generic Aloff-Wallach space as principal orbit and SU 3 U 1 ^{2} as singular orbit. This classification is of interest of its own, but also has a further aim, since the modules which we have to determine for our classification will be needed for other purposes, too. Before we start our calculations, we have to describe the tangent and the normal space of the singular orbit as 2𝔲 1 -modules. As usual, we identify the tangent space of the singular orbit with the q-orthogonal complement 𝔭 of 2𝔲 1 𝔰𝔲 3 . 𝔭 decomposes into the direct sum V_{1} V_{2} V_{3} of 𝔲 1 _{k,l}-modules. It is easy to see that the summands V_{1}, V_{2}, and V_{3} are invariant with respect to 2𝔲 1 , too. We will describe these 2𝔲 1 -modules in more detail. 2𝔲 1 is spanned by e_{7} and 𝔲 1 _{k,l}. The basis element e_{7} is given by the matrix diag 2l k i, 2k l i, k l i . The two-dimensional real U 1 ^{2}-module on which diag exp 2l k iφ , exp 2k l iφ , exp k l iφ acts as diag exp 2l k irφ , exp 2k l irφ , exp k l irφ and diag exp kiφ , exp liφ , exp k l iφ acts as diag exp kisφ , exp lisφ , exp k l isφ we denote by 𝕍_{r,s}. Since 2l k and 2k l are not necessarily coprime if k and l are coprime, the weights of the U 1 ^{2}-action on 𝕍_{r,s} are r and s gcd 2l k, 2k l instead of r and s. We will nevertheless call r, s the weight of the U 1 ^{2}-action and we are able to calculate with r, s in the same way as we are used to. Unlike in Section 5.2 and 5.3, 𝕍_{0,0} does not denote the irreducible trivial module, which we will shortly denote by ℝ, but a two-dimensional reducible one. Since we have span e_{7} 𝔲 1 _{2l k, 2k l} , we can determine the weights of the action of span e_{7} on the V_{i} by the same method as we have used for the action of 𝔲 1 _{k,l}. We obtain:

\begin{matrix}V_{1} & 𝕍_{2l k 2k l ,k l}  𝕍_{3k 3l,k l} \\ V_{2} & 𝕍_{2 2l k 2k l,2k l}  𝕍_{3l,2k l} \\ V_{3} & 𝕍_{2l k 2 2k l ,k 2l}  𝕍_{3k,k 2l} \end{matrix}

The following two relations will be helpful to decompose S^{2} 𝔭 into irreducible 2𝔲 1 -submodules:

S^{2} 𝕍_{r,s}  𝕍_{2r,2s} ℝ
𝕍_{r_{1},s_{1}} 𝕍_{r_{2},s_{2}}  𝕍_{r_{1} r_{2},s_{1} s_{2}} 𝕍_{r_{1} r_{2},s_{1} s_{2}}

The two-dimensional space 𝕍_{r,s} can be 2𝔲 1 -equivariantly identified with ℂ. With this notation, the embedding of the trivial summand into S^{2} 𝕍_{r,s} is given by:

λ λ z ^{2} λ Re z Re z Im z Im z .

The projections onto the two summands of the decomposition of 𝕍_{r_{1},s_{1}} 𝕍_{r_{2},s_{2}} can be described by:

\begin{matrix}z_{1} z_{2} & z_{1} z_{2} \\ z_{1} z_{2} & z_{1} \overline{z_{2}} \end{matrix}

The isomorphism

𝕍_{r,s} 𝕍_{r, s} ,

which can be identified with the conjugation map, will be needed later on, too. With help of the above relations, we obtain:

S^{2} 𝔭 S^{2} 𝕍_{3k 3l,k l} 𝕍_{3l,2k l} 𝕍_{3k,k 2l}
S^{2} 𝕍_{3k 3l,k l} S^{2} 𝕍_{3l,2k l} S^{2} 𝕍_{3k,k 2l}
𝕍_{3k 3l,k l} 𝕍_{3l,2k l} 𝕍_{3k 3l,k l} 𝕍_{3k,k 2l} 𝕍_{3l,2k l} 𝕍_{3k,k 2l}
𝕍_{6k 6l,2k 2l} 𝕍_{6l,4k 2l} 𝕍_{6k,2k 4l} 3ℝ
𝕍_{3k 6l,3k} 𝕍_{3k, k 2l} 𝕍_{3l,2k l} 𝕍_{6k 3l, 3l}
𝕍_{3k 3l,3k 3l} 𝕍_{3k 3l,k l}

For our considerations, we have to determine the weight of the 2𝔲 1 -action on the normal space 𝔭^{}of the singular orbit. 𝔭^{}is a real two-dimensional 2𝔲 1 -module. The tangent vector \frac{}{t} of the geodesic γ which we have fixed is contained in 𝔭^{}, since γ intersects the singular orbit perpendicularly. U 1 _{k,l} is generated by e_{8} 2𝔲 1 . Since U 1 _{k,l} is the isotropy group of the SU 3 -action on the principal orbits, its action fixes \frac{}{t}. e_{8} therefore acts trivially on 𝔭^{}. e_{7} 2𝔲 1 generates the group U 1 _{2l k, 2k l} SU 3 , which we will shortly denote by U 1 ^{}. The weight of the action of U 1 ^{}on 𝔭^{}can be determined by an argument which we have used in Section 5.2. The orbit of the U 1 ^{}-action on a point p U 1 ^{2} U 1 _{k,l} is a loop in that space. Since π_{1} U 1 ^{2} U 1 _{k,l} ℤ, the homotopy class of the orbit corresponds to an integer. For geometric reasons, that integer coincides with the weight of the U 1 ^{}-action on 𝔭^{}. The group U 1 ^{}consists of the following 3 3-matrices:

; ; { \atop }\begin{matrix}e^{2l k iφ} & 0 & 0 \\ 0 & e^{2k l iφ} & 0 \\ 0 & 0 & e^{k l iφ} \end{matrix} { \atop }: A_{φ} ; ; ; ; ; φ ℝ ; ; .

Analogously, we have:

U 1 _{k,l} ; ; { \atop }\begin{matrix}e^{kiψ} & 0 & 0 \\ 0 & e^{liψ} & 0 \\ 0 & 0 & e^{k l iψ} \end{matrix} { \atop }: B_{ψ} ; ; ; ; ; ψ ℝ ; ; .

We want to describe the intersection of U 1 ^{}and U 1 _{k,l}, since its cardinality equals the winding number of the U 1 ^{}-orbit around U 1 ^{2} U 1 _{k,l}. Therefore, we search for the smallest positive φ such that there exists a ψ ℝ with A_{φ} B_{ψ}. It is easy to see that U 1 ^{}U 1 _{k,l} A_{jφ} j ℤ . The equation A_{φ} B_{ψ} is equivalent to:

\begin{matrix}2k l φ & lψ &  & 2πj_{1} \\ k l φ & k l ψ &  & 2πj_{2} \end{matrix}  (5.48)

for a pair j_{1}, j_{2} ℤ^{2}. We solve the above system for all j_{1}, j_{2} ℤ^{2} and obtain:

\begin{matrix}φ \\ ψ \end{matrix}  \frac{1}{2 k^{2} lk l^{2}}  \begin{matrix}k l 2πj_{1} & l 2πj_{2} \\ k l 2πj_{1} & 2k l 2πj_{2} \end{matrix} .

Since k l and l are coprime, we can choose j_{1} and j_{2} in such a way that φ becomes \frac{π}{k^{2} lk l^{2}} . This number obviously is the smallest possible value of φ . We let φ run from 0 to 2π no matter if 2l k and 2k l are coprime or not. Loosely speaking, U 1 ^{}thus intersects U 1 _{k,l} 2 k^{2} lk l^{2} times. Since the number of intersections coincides with the weight of the U 1 ^{}-action on 𝔭^{}, we have proven that in our notation:

𝔭^{}𝕍_{2 k^{2} lk l^{2} ,0} .

We are now able to decompose S^{m} 𝔭^{}into irreducible 2𝔲 1 -submodules and obtain:

S^{m} 𝔭^{}𝕍_{2m k^{2} lk l^{2} ,0} 𝕍_{2 m 2 k^{2} lk l^{2} ,0} … 𝕍_{4 k^{2} lk l^{2} ,0} ℝ  if m is even 𝕍_{2m k^{2} lk l^{2} ,0} 𝕍_{2 m 2 k^{2} lk l^{2} ,0} … 𝕍_{2 k^{2} lk l^{2} ,0}  if m is odd

The elements of S^{m} 𝔭^{}can be identified with polynomials of m^{th} order on the real twodimensional vector space ℂ. After this identification, the embedding of 𝕍_{2 m 2p k^{2} lk l^{2} ,0} (or ℝ) into S^{m} 𝔭^{}simply becomes w w z ^{2p} (or λ λ z ^{m} .

Before we can apply Theorem 3.2.24, we have to check if Assumption 3.2.19 is satisfied. This assumption states that 𝔭 and 𝔭^{}have no equivalent U 1 _{k,l}-submodules in common. Since U 1 _{k,l} acts trivially on 𝔭^{}, our assumption is violated if and only if any of the modules V_{1}, V_{2}, or V_{3} is trivial. As we have seen at the beginning of this section, this can only be the case if N^{k,l} N^{1,1}. Since we are now considering the generic case, Theorem 3.2.24 can be applied.

In order to determine the number of cohomogeneity-one Einstein metrics, we have to describe the spaces W_{2}^{v} : Hom_{U 1 ^{2}} S^{2} 𝔭^{}, S^{2} 𝔭^{}and W_{m}^{h} : Hom_{U 1 ^{2}} S^{m} 𝔭^{}, S^{2} 𝔭 . We start with the vertical case. S^{2} 𝔭^{}decomposes into two inequivalent irreducible 2𝔲 1 -submodules:

S^{2} 𝔭^{}𝕍_{4 k^{2} lk l^{2} ,0} ℝ .

The module 𝕍_{4 k^{2} lk l^{2} ,0} can be identified with ℂ which is equipped with a certain U 1 ^{2}- action. The U 1 ^{2}-equivariant maps from ℂ to itself are described by the multiplication by an arbitrary complex number. Since dim_{ℝ} ℂ 2, we have dim Hom_{U 1 ^{2}} 𝕍_{4 k^{2} lk l^{2} ,0} , 𝕍_{4 k^{2} lk l^{2} ,0} 2. Therefore, dim W_{2}^{v} 3. Since the vertical part of the metric is determined by f, there is only one initial condition for the vertical part which we can prescribe. For the same reasons as in Section 5.2 and 5.3, this initial condition describes the value of f^{}0 .

Next, we calculate the number of free parameters in the horizontal direction. Since the principal orbit is a generic Aloff-Wallach space, the weight of the U 1 _{k,l}-action on any of the submodules of S^{2} 𝔭 , except on 3ℝ, is non-zero. Therefore, the image of any U 1 ^{2}-equivariant homomorphism from S^{m} 𝔭^{}to S^{2} 𝔭 is contained in 3ℝ and we obtain:

dim W_{m}^{h}  3 if m is even 0 if m is odd

We recall that the number of free parameters of order m 2 in the horizontal direction is dim W_{m}^{h} dim W_{m 2}^{h} . The number of free parameters of 0^{th} (1^{st}) order is dim W_{0}^{h} (dim W_{1}^{h}). Therefore, we only have 3 parameters of 0^{th} order. This means that we can choose a 0 , b 0 , and c 0 freely. Since we have no choice for the first derivative of the metric, the initial shape operator has to vanish and the singular orbit has to be totally geodesic. All in all, we have proven that for any choice of f^{}0 and the Einstein constant, there is a unique extension of any initial metric on the singular orbit to a cohomogeneity-one Einstein metric. We turn to the next case:

Principal orbit: A N^{k,l} which is N^{1,0} , singular orbit: SU 3 U 1 ^{2}

In this situation, the only trivial U 1 _{k,l}-submodule of 𝔪 again is V_{4} and we have f 0 0 as initial condition. Since we restrict ourselves to diagonal metrics, we can work as before with the system 5.36 . Therefore, we can prove by the same arguments as in the previous case that f vanishes and the metric degenerates.

We apply the methods of Eschenburg and Wang [32] in order to describe the cohomogeneityone Einstein metrics with the orbit structure which we have chosen. Since none of the U 1 _{k,l}- modules V_{1}, V_{2}, and V_{3} is trivial, Assumption 3.2.19 is satisfied and we are in the situation of Theorem 3.2.24. If k 1 and l 0, the decompositions of S^{2} 𝔭 and S^{m} 𝔭^{}which we have found specialize to:

S^{2} 𝔭 𝕍_{6,2} 𝕍_{0,4} 𝕍_{6,2} 3ℝ
𝕍_{3,3} 𝕍_{3, 1} 𝕍_{0,2} 𝕍_{6,0}
𝕍_{3,3} 𝕍_{3,1}

and

S^{m} 𝔭^{}𝕍_{2m,0} 𝕍_{2 m 2 ,0} … 𝕍_{4,0} ℝ  if m is even 𝕍_{2m,0} 𝕍_{2 m 2 ,0} … 𝕍_{2,0}  if m is odd

S^{2} 𝔭 contains a summand which is isomorphic to 𝕍_{6,0}. Therefore, we have dim W_{m}^{h} 2 if m is odd and 3. We calculate dim W_{m}^{h} for all m ℕ_{0} and obtain:

dim W_{m}^{h} ; ; 3 if m is even 0 if m 1 2 if m 3 is odd

We conclude that there are three initial conditions of 0^{th} and two of 3^{rd} order which we can choose. Of course, there again is a further free initial condition for the vertical part of the metric, which describes the freedom in f^{}0 . Since the 𝕍_{6,0}-summand is contained in V_{1} V_{3}, the two parameters of 3^{rd} order describe how the initial metric, which is diagonal, changes into a non-diagonal one. If we had considered a system of differential equations which also described the non-diagonal metrics, we therefore might have obtained new examples of metrics with holonomy Spin 7 . We will nevertheless not carry out the calculations which were needed to check the existence of such examples.

Principal orbit: A N^{k,l} which is N^{1,1} , singular orbit: SU 3 U 1 ^{2}

Analogously to the previous case, we assume that the metric on any principal orbit is a diagonal metric of type (5.33). Furthermore, we assume that the Spin 7 -structure is given by the frame (5.37) and the four-form (5.38). We thus have to work with the system (5.39), which is more intricate than (5.36). Since the module V_{1} is in this situation trivial, we can choose the Lie algebra of the isotropy group U 1 ^{2} as 𝔲 1 _{k,l} span x with an arbitrary non-zero x V_{1} V_{4}. Therefore, we have further possibilities for the component of the metric which vanishes at t 0. Since we assume that the metric is diagonal, we can reduce the number of the possible x. The metric on the principal orbit N^{1,1} depends on a parameter t and will thus be denoted by g_{t}. We have lim_{t 0} g_{t} x, x 0. x is a linear combination αe_{1} βe_{2} γe_{3} for some coefficients α, β, γ ℝ. With the notation of (5.33), we have:

g_{t} x, x α^{2}a_{1} t ^{2} β^{2}a_{2} t ^{2} γ^{2}f t ^{2} .

This term converges to zero if and only if one of the three metric functions a_{1}, a_{2}, f converges to zero and the coefficients of the other two functions vanish. Therefore, we have x e_{1}, e_{2}, e_{7} and the only cases which we have to consider are a_{1} 0 0, a_{2} 0 0, and f 0 0. We will replace e_{7} by \frac{1}{3}e_{7}, since this will simplify some of our calculations. We will prove that the first two cases are equivalent to the third one. Let x \frac{1}{3}e_{7}. At the beginning of this section, we have seen that the SU 2 -factor of the normalizer Norm_{SU 3}U 1 _{1,1} acts on V_{1} V_{4} as SO 3 on ℝ^{3} and on V_{2} V_{3} as SU 2 on ℂ^{2}. Therefore, there exists a σ Norm_{SU 3}U 1 _{1,1} which interchanges x and \frac{1}{3}e_{7}. We denote the third element of e_{1}, e_{2}, \frac{1}{3}e_{7} by y. Since σ acts as an element of SO 3 on V_{1} V_{4} and preserves the normal metric q, its matrix representation with respect to the basis x, \frac{1}{3}e_{7}, y necessarily is:

{ \atop }\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} { \atop }

We will prove that there exists a τ Norm_{SU 3}U 1 _{1,1} which acts as σ on span e_{1}, e_{2}, \frac{1}{3}e_{7} and maps the metric g_{t} to another diagonal metric. Let ρ : SU 2 SO 3 be the canonical covering homomorphism. ρ h , where h is a unit quaternion, can be described by the linear map k hkh^{1} , which acts on Im ℍ ℝ^{3}. In order to find the τ Norm_{SU 3}U 1 _{1,1} which we are looking at, we first construct a τ^{}SU 2 such that ρ τ^{}is the above 3 3-matrix. That matrix is a rotation around the axis span 1, 1, 0 ^{T} by an angle of π. It is easy to see that τ^{}has to be \frac{1}{\overline{2}} i j SU 2 . Motivated by this observation, we define τ as \frac{1}{\overline{2}} x \frac{1}{3}e_{7} SU 2 . As usual, the group SU 2 is embedded into SU 3 by:

\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} { \atop }\begin{matrix}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{matrix} { \atop }

By a short calculation, we see that τ acts indeed as σ on span e_{1}, e_{2}, \frac{1}{3}e_{7} . Moreover, we can easily determine the action of τ on S^{2} span e_{3}, e_{4}, e_{5}, e_{6} . Since the sign of \frac{1}{\overline{2}} x \frac{1}{3}e_{7} gets canceled, we obtain the same action on S^{2} span e_{3}, e_{4}, e_{5}, e_{6} for both values of τ. For the following considerations, we assume that x e_{1}. For x e_{2}, we would obtain an analogous result. We examine the bilinear form whose matrix representation with respect to e_{3}, e_{4}, e_{5}, e_{6} is

 ;  ;  ; \begin{aligned}b^{2} & 0 \\ 0 & b^{2} \\ & & c^{2} & 0 \\ & & 0 & c^{2} \end{aligned}  ;  ;  ;

This form is mapped by the action of τ to:

 ;  ;  ; \begin{matrix}\frac{1}{2} b^{2} c^{2} & 0 & 0 & \frac{1}{2} b^{2} c^{2} \\ 0 & \frac{1}{2} b^{2} c^{2} & \frac{1}{2} b^{2} c^{2} & 0 \\ 0 & \frac{1}{2} b^{2} c^{2} & \frac{1}{2} b^{2} c^{2} & 0 \\ \frac{1}{2} b^{2} c^{2} & 0 & 0 & \frac{1}{2} b^{2} c^{2} \end{matrix}  ;  ;  ;

Therefore, the metric which we obtain by the action of τ is diagonal if and only if b^{2} c^{2}. We consider the system (5.39) with the initial conditions f 0 0 and b 0 ^{2} c 0 ^{2}. Furthermore, we assume that b^{2} t c^{2} t for all values of t. Although the system (5.39) is preserved after we interchange b and c, this assumption might not be automatically satisfied. The reason for this is that there may exist initial conditions of higher order which we can prescribe in such a way that b^{2} t c^{2} t becomes 0. The action of τ turns our initial value problem into a new one. Since τ Norm_{SU 3}U 1 _{1,1} transforms the metric into another diagonal metric, the new metric obeys according to Lemma 3.2.36 again the system (5.39) with the new initial conditions a_{1} 0 0 and b 0 ^{2} c 0 ^{2}. We consider the third equation of the system (5.39) under the assumption that we still have a_{1} 0 0 but the initial values of the other metric functions are arbitrary non-zero numbers. Since b 0 0, the right hand side of that equation has to converge for t 0 to a real number. In particular, the term \frac{1}{2} \frac{a_{1}^{2} c^{2} b^{2}}{a_{1}bc} from the right hand side has to converge to a real number, too. From this fact, it easily follows that b 0 ^{2} has to equal c 0 ^{2}. By the action of τ^{1} , this initial metric becomes another metric which satisfies f 0 0 and b 0 ^{2} c 0 ^{2}. Therefore, the diagonal metrics with the initial condition a_{1} 0 0 are isometric to those with f 0 0 and b^{2} 0 c^{2} 0 provided that there are no free parameters of higher order for b and c. Conversely, the metrics with f 0 0 and b 0 ^{2} c 0 ^{2} correspond in our situation to diagonal metrics with a_{1} 0 0 and those with b 0 ^{2} c 0 ^{2} correspond to non-diagonal metrics.

For the above reasons, we can restrict ourselves to the case f 0 0. This case we will now consider in detail. We take a look at the following three equations, which we have taken from (5.39):

a_{1}^{}\frac{b^{2} c^{2} a_{1}^{2}}{bc} 3\frac{a_{1}^{2} a_{2}^{2}}{a_{2}f} \frac{1}{3} \frac{f}{a_{2}}
a_{2}^{}\frac{b^{2} c^{2} a_{2}^{2}}{bc} 3\frac{a_{2}^{2} a_{1}^{2}}{a_{1}f} \frac{1}{3} \frac{f}{a_{1}}
f^{}3\frac{a_{1} a_{2} ^{2}}{a_{1}a_{2}}  \frac{1}{3} \frac{f^{2}}{a_{1}a_{2}}  \frac{1}{6} \frac{f^{2}}{b^{2}}  \frac{1}{6} \frac{f^{2}}{c^{2}}

Since we need f^{}0 0 in order to make the metric smooth at the singular orbit, we have a_{1} 0 a_{2} 0 0. The numerator of the term \frac{a_{1}^{2} a_{2}^{2}}{a_{2}f} has to vanish, since we do not want lim_{t 0} a_{i}^{}t . Therefore, we have a_{1} 0 a_{0} a_{2} 0 for a a_{0} ℝ 0 . We choose b 0 and c 0 as arbitrary real constants b_{0} and c_{0}. With these initial values, we make a power series ansatz for the system (5.39) and obtain:

\begin{matrix}a_{1} t &  & a_{0} &  & \frac{1}{2} \frac{a_{0}^{2} b_{0}^{2} c_{0}^{2}}{b_{0}c_{0}} t &  & \frac{1}{8} \frac{3a_{0}^{4} 2a_{0}^{2}b_{0}^{2} 2a_{0}^{2}c_{0}^{2} b_{0}^{4} 14b_{0}^{2}c_{0}^{2} c_{0}^{4}}{a_{0}b_{0}^{2}c_{0}^{2}} t^{2} &  & … \\ a_{2} t &  & a_{0} &  & \frac{1}{2} \frac{a_{0}^{2} b_{0}^{2} c_{0}^{2}}{b_{0}c_{0}} t &  & \frac{1}{8} \frac{3a_{0}^{4} 2a_{0}^{2}b_{0}^{2} 2a_{0}^{2}c_{0}^{2} b_{0}^{4} 14b_{0}^{2}c_{0}^{2} c_{0}^{4}}{a_{0}b_{0}^{2}c_{0}^{2}} t^{2} &  & … \\ b t &  & b_{0} &  & 0 t &  & \frac{1}{4} \frac{a_{0}^{4} 6a_{0}^{2}c_{0}^{2} b_{0}^{4} c_{0}^{4}}{a_{0}^{2}b_{0}c_{0}^{2}} t^{2} &  & … \\ c t &  & c_{0} &  & 0 t &  & \frac{1}{4} \frac{a_{0}^{4} 6a_{0}^{2}b_{0}^{2} b_{0}^{4} c_{0}^{4}}{a_{0}^{2}b_{0}^{2}c_{0}} t^{2} &  & … \\ f t &  & 0 &  & 12t &  & 0 t^{2} &  & … \end{matrix}  (5.49)

The next issue which we will discuss is how the smoothness conditions from Theorem 3.2.18 translate into conditions on the coefficients of the above power series. In order to answer this question, we have to consider the modules

W_{m} Hom_{U 1 ^{2}} S^{m} 𝔭^{}, S^{2} 𝔭 𝔭^{}.

Since 𝔭 𝕍_{6,0} 𝕍_{3,3} 𝕍_{3,3} and 𝔭^{}𝕍_{6,0} have one submodule in common, Assumption 3.2.19 is not satisfied and we do not have the splitting W_{m} W_{m}^{h} W_{m}^{v}. Since the metrics which we investigate are diagonal, they can be considered as elements of S^{2} 𝔭 S^{2} 𝔭^{}at a point of the singular orbit. Therefore, the smaller modules

W_{m}^{}: Hom_{U 1 ^{2}} S^{m} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}W_{m}^{h} W_{m}^{v}

are sufficient for our considerations. We describe W_{m}^{h} and W_{m}^{v} in detail. In our situation, we have:

S^{2} 𝔭 𝕍_{12,0} 𝕍_{6,6} 𝕍_{6,6} 3ℝ
𝕍_{9,3} 𝕍_{3, 3} 𝕍_{3,3} 𝕍_{9, 3}
𝕍_{0,6} 𝕍_{6,0}

For S^{m} 𝔭^{}, we obtain:

S^{m} 𝔭^{}𝕍_{6m,0} 𝕍_{6 m 2 ,0} … 𝕍_{12,0} ℝ  if m is even 𝕍_{6m,0} 𝕍_{6 m 2 ,0} … 𝕍_{6,0}  if m is odd

The dimensions of W_{m}^{h} and W_{m}^{v} can be calculated with help of Schur’s lemma:

dim W_{m}^{h} ; ; 3 if m 0 5 if m 2 and even 2 if m odd dim W_{m}^{v} ; ; 1 if m 0 3 if m 2 and even 0 if m odd

We have to discuss the geometric meaning of these numbers. First, we interpret the dimensions of the spaces W_{m}^{h}. W_{0}^{h} describes the SU 3 -invariant metrics on the singular orbit. By a short calculation, we see that any diagonal metric on the singular orbit which satisfies a_{1} 0 ^{2} a_{2} 0 ^{2} is SU 3 -invariant. Since dim W_{0}^{h} 3, there are no further invariant metrics on SU 3 U 1 ^{2}. If m is even and 2, there are two additional parameters. Since the 𝕍_{12,0}-summand, which is responsible for these parameters, is contained in S^{2} V_{1} , the m^{th} derivative of the restriction of the metric to V_{1} may depend on three parameters. One of the two additional parameters describes the m^{th} derivative of the off-diagonal function β_{1,2} and the other one describes the change of a_{1}^{2} a_{2}^{2} away from zero. Since we only consider diagonal metrics, we can ignore the degree of freedom for β_{1,2}^{m} 0 . The three parameters which were already there in the case m 0 are responsible for b^{2} ^{m} 0 , c^{2} ^{m} 0 , and the m^{th} derivative of the coefficient of e^{1} e^{1} e^{2} e^{2}. Our considerations prove that we can choose a_{1}^{2} ^{m} 0 , a_{2}^{2} ^{m} 0 , b^{2} ^{m} 0 , and c^{2} ^{m} 0 freely if m is even and 2. Next, we consider the case where m is odd. In that case, the m^{th} derivative of the metric depends on two parameters. The 𝕍_{6,0}-summand of S^{2} 𝔭 , which is responsible for these parameters, is contained in V_{2} V_{3}. Therefore, the two parameters describe how the off-diagonal functions β_{3,5} and β_{3,6} behave and we have no degree of freedom for the other coefficients. Since the restrictions on the coefficients are linear, a_{1}^{2} ^{m} 0 , a_{2}^{2} ^{m} 0 , b^{2} ^{m} 0 , and c^{2} ^{m} 0 have to vanish if m is odd. All in all, we have proven that a_{1}^{2}, a_{2}^{2}, b^{2}, and c^{2} are even functions. Since these functions are 0 at the singular orbit, the smoothness conditions on a_{1}, a_{2}, b, and c are that they all have to be even and that a_{1} 0 ^{2} a_{2} 0 ^{2}. The power series (5.49) does not seem to satisfy these conditions, since a_{1} and a_{2} are not even. We will deal with this problem later on.

Next, we explicitly describe the smoothness conditions for the vertical part of the metric. We can make use of the same arguments as in Section 5.2 and 5.3 and conclude that f has to be an odd function. The fact that the length of the collapsing circle U 1 ^{2} U 1 _{k,l} has to be 2πt O t^{2} for small values of t determines the value of f^{}0 . As in the previous two sections, these two conditions on f are necessary and sufficient for the smoothness of the vertical part. We calculate the value of f^{}0 which makes the metric smooth. The following map γ parameterizes U 1 ^{2} U 1 _{k,l}:

γ t : { \atop }\begin{matrix}e^{2l k it} & 0 & 0 \\ 0 & e^{2k l it} & 0 \\ 0 & 0 & e^{k l it} \end{matrix} { \atop }U 1 _{k,l}

We have already proven that the Lie group which is generated by e_{7} intersects U 1 _{k,l} for the first time at \frac{π}{k^{2} kl l^{2}} . Since we assume that k l 1, we have to define γ on 0, \frac{π}{3} in order to parameterize all of the circle. The length of the collapsing circle with radius t therefore is:

\frac{π}{3} ; 0 \overline{g e_{7}, e_{7}}ds \frac{π}{3} f t .

Since this length should equal 2πt O t^{2} , we have

\frac{f t}{t} 6 O t .

and conclude that f^{}0 has to be 6. From the power series ansatz (5.49), it follows that in our situation we actually have f^{}0 12. Therefore, the metrics with principal orbit N^{1,1} and singular orbit SU 3 U 1 ^{2} which solve (5.39) cannot be smoothly extended to the singular orbit. Nevertheless, there is a method to obtain smooth cohomogeneity-one metrics whose holonomy is contained in Spin 7 from the solutions of that system. Let

h : { \atop }\begin{matrix}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 1 \end{matrix} { \atop }SU 3 .

Furthermore, let K SU 3 be the group which is generated by U 1 _{1,1} and h. It is easy to see that K is a direct product of type U 1 ℤ_{2}. The action of h on 𝔪 we denote by a dot. We have:

\begin{matrix}h.e_{1} e_{1} & h.e_{2} e_{2} & h.e_{3} e_{4} & h.e_{4} e_{3} \\ h.e_{5} e_{6} & h.e_{6} e_{5} & h.e_{7} e_{7} \end{matrix}  (5.50)

These equations prove that the action of h preserves any diagonal metric of type (5.33). Therefore, the map φ : N^{1,1} N^{1,1} with φ gU 1 _{1,1} : hgh^{1} U 1 _{1,1} is an isometry. From this fact, it follows that N^{1,1} is a double cover of SU 3 K and that the covering map can be chosen as a local isometry. K is contained in the maximal torus of SU 3 which acts as a subgroup of G_{2} on 𝔪. h therefore preserves the G_{2}-structure on N^{1,1} and SU 3 K admits a SU 3 -invariant G_{2}-structure which is induced by the covering map. We divide the eightdimensional cohomogeneity-one manifold which we consider by the group which is generated by the action of h on all orbits. Since h preserves the G_{2}-structure, the Spin 7 -structure on our manifold turns into another Spin 7 -structure on the new space. The holonomy of the metric which we obtain this way therefore remains the same. The principal orbits of the old manifold turn into spaces of type SU 3 K. The singular orbit SU 3 U 1 ^{2} remains the same, since h is an element of U 1 ^{2} SU 3 . The action of h preserves the orientation. Therefore, the new space has a conical orbifold singularity at the singular orbit, which we will straighten out. It is easy to see that any circle which was wrapped r times around the origin of 𝔭^{}is wrapped 2r times around the origin of the new normal space. We reconsider the arguments which we have made and see that in this new situation we have to require f^{}0 12 instead of f^{}0 6 in order to make the metric smooth at the singular orbit SU 3 U 1 ^{2}. Since U 1 ^{2} now acts on 𝔭^{}as 𝕍_{12,0} rather than 𝕍_{6,0}, we have

dim W_{m}^{h}  3 if m even 2 if m odd

Since the values of dim W_{m}^{h} have changed, the smoothness conditions on the functions a_{1}, a_{2}, b, and c have changed, too. By similar arguments as for the principal orbit N^{1,1}, we can prove that for even m we still can choose b^{2} ^{m} 0 and c^{2} ^{m} 0 arbitrarily. The value of a_{1}^{2} ^{m} 0 has to equal a_{2}^{2} ^{m} 0 and can be chosen arbitrarily, too. If m is odd, there are two parameters on which the m^{th} derivative of the horizontal part of the metric may depend. These parameters correspond to the equivariant maps from 𝕍_{12,0} S^{2} V_{1} to 𝔭^{}. 𝕍_{12,0} is the trace-free part of S^{2} V_{1} . Therefore, b^{2} ^{m} 0 as well as c^{2} ^{m} 0 have to vanish and the two parameters describe the m^{th} derivatives of the coefficients of e^{1} e^{1} e^{2} e^{2} and e^{1} e^{2} e^{2} e^{1}. Since we assume that the metric is diagonal, we only have to consider the coefficient of e^{1} e^{1} e^{2} e^{2}. It follows that a_{1}^{2} ^{m} 0 has to coincide with a_{2}^{2} ^{m} 0 and can be chosen arbitrarily. Since a_{1} and a_{2} are analytic functions, we obtain a_{1}^{2} t a_{2}^{2} t as necessary smoothness condition. With help of a_{1} 0 a_{2} 0 , we conclude that a_{1} t a_{2} t . As in the previous case, b and c have to be even. The dimensions of W_{m}^{v} stay the same and we have the same conditions on f as before, except f^{}0 12.

On page 180, we have calculated the first summands of a formal power series solution of the system (5.39). Those summands satisfy the smoothness conditions which we have deduced above. Nevertheless, it is not a priori clear if there really exists a power series solution of (5.39) which is of infinite order and satisfies the conditions from Theorem 3.2.18. On the following pages, we will prove that there is indeed a unique power series solution of (5.39) for any choice of a_{1} 0 , b 0 , and c 0 . Since the system (5.39) is preserved if we replace a_{1} t , a_{2} t , b t , c t , f t by a_{2} t , a_{1} t , b t , c t , f t , that power series has to satisfy the smoothness conditions. Any formal power series which satisfies (5.39) and the smoothness conditions also satisfies the cohomogeneity-one Einstein condition. According to Eschenburg and Wang [32], the power series converges near the singular orbit if the cohomogeneity-one action obeys the condition which we have stated in 3.2.19. The result of Eschenburg and Wang on the convergence is also true in our situation, although we have not yet checked if a manifold with principal orbit SU 3 K and singular orbit SU 3 U 1 ^{2} satisfies Assumption 3.2.19. The reason for this is that the convergence result is proven with help of a Picard-iteration. Assumption 3.2.19 is only needed to secure that the metric is contained in S^{2} 𝔭 S^{2} 𝔭^{}and that the Picard iteration takes its values in the same space. We take a look at the cohomogeneity-one Einstein condition, which we have explicitly described in Appendix C.4. The differential equations which we have deduced in that appendix preserve the space S^{2} 𝔭 S^{2} 𝔭^{}. Therefore, the Picard-iteration cannot leave that space, too, and we can show by the same arguments as Eschenburg and Wang [32] that the power series converges. We thus will have proven the existence of local cohomogeneity-one metrics whose holonomy is contained in Spin 7 as soon as it is clear that a unique formal power series solution exists. The system (5.39) can be rewritten as:

\begin{matrix}a_{1}^{}a_{2}bcf & b^{2} c^{2} a_{1}^{2} a_{2}f 3 a_{1}^{2} a_{2}^{2} bc \frac{1}{3}bcf^{2} \\ a_{1}a_{2}^{}bcf & b^{2} c^{2} a_{2}^{2} a_{1}f 3 a_{2}^{2} a_{1}^{2} bc \frac{1}{3}bcf^{2} \\ b^{} & \frac{1}{2} \frac{a_{1}^{2} c^{2} b^{2}}{a_{1}c} \frac{1}{2} \frac{a_{2}^{2} c^{2} b^{2}}{a_{2}c} \frac{1}{6} \frac{f}{b} \\ c^{} & \frac{1}{2} \frac{a_{1}^{2} b^{2} c^{2}}{a_{1}b} \frac{1}{2} \frac{a_{2}^{2} b^{2} c^{2}}{a_{2}b} \frac{1}{6} \frac{f}{c} \\ f^{} & 3\frac{a_{1} a_{2} ^{2}}{a_{1}a_{2}} \frac{1}{3} \frac{f^{2}}{a_{1}a_{2}} \frac{1}{6} \frac{f^{2}}{b^{2}} \frac{1}{6} \frac{f^{2}}{c^{2}} \end{matrix}

By considering the m 1 ^{st} derivatives of the last three equations at t 0, we see that b^{m} 0 , c^{m} 0 , and f^{m} 0 are uniquely determined by the lower derivatives of a_{1}, a_{2}, b, c, and f. It remains to prove that a_{1}^{m} 0 and a_{2}^{m} 0 are uniquely determined, too. We consider the m^{th} derivatives of the first two equations at t 0. We will put all terms which contain any m^{th} derivative of a function on the left hand side. The other terms will be placed on the right hand side. Since any product which contains f 0 vanishes, only few terms survive and we obtain:

ma_{1}^{m} 0 a_{2} 0 b 0 c 0 f^{}0 a_{1}^{}0 a_{2} 0 b 0 c 0 f^{m} 0
b 0 ^{2} c 0 ^{2} a_{1} 0 ^{2} a_{2} 0 f^{m} 0
3 2a_{1}^{m} 0 a_{1} 0 2a_{2}^{m} 0 a_{2} 0 b 0 c 0
3 a_{1}^{2} 0 a_{2}^{2} 0 b^{m} 0 c 0 b 0 c^{m} 0 R_{1}

ma_{1} 0 a_{2}^{m} 0 b 0 c 0 f^{}0 a_{1} 0 a_{2}^{}0 b 0 c 0 f^{m} 0
b 0 ^{2} c 0 ^{2} a_{2} 0 ^{2} a_{1} 0 f^{m} 0
3 2a_{2}^{m} 0 a_{2} 0 2a_{1}^{m} 0 a_{1} 0 b 0 c 0
3 a_{2}^{2} 0 a_{1}^{2} 0 b^{m} 0 c 0 b 0 c^{m} 0 R_{2}

where R_{1} and R_{2} are polynomials which depend on the lower derivatives of our functions only. The reason for the factor m in the above equations is that the term a_{1}^{m} 0 f^{}0 appears m times if we apply the product rule to a_{1}^{}f ^{m} 0 . Since we have already calculated a power series expansion up to second order, we can assume that m 2. The above two equations are thus linear with respect to the m^{th} derivatives of our functions. b^{m} 0 , c^{m} 0 , and f^{m} 0 can be expressed in terms of the lower derivatives. Therefore, we can also put them on the right hand side, too. We define a_{0} : a_{1} 0 a_{2} 0 , b_{0} : b 0 , and c_{0} : c 0 . Since f^{}0 12, the above equations simplify to:

\begin{matrix}12m 6 a_{0}b_{0}c_{0} a_{1}^{m} 0 6 a_{0}b_{0}c_{0} a_{2}^{m} 0 & { \atop R_{1}} \\ 6 a_{0}b_{0}c_{0} a_{1}^{m} 0 12m 6 a_{0}b_{0}c_{0} a_{2}^{m} 0 & { \atop R_{2}} \end{matrix}

where { \atop R_{1}} and { \atop R_{2}} also are polynomials which depend on the lower derivatives of a_{1}, a_{2}, b, c, and f only. Since the above system has always a unique solution a_{1}^{m} 0 , a_{2}^{m} 0 , we have proven the existence of a unique power series which satisfies (5.39).

Before we turn to the next case, we will discuss how many cohomogeneity-one Einstein metrics exist in our situation. If the principal orbit is N^{1,1}, Assumption 3.2.19 is not satisfied and we cannot make a definitive statement on the number of Einstein metrics. The main problem is that we do not automatically know that if the metric g is in S^{2} 𝔭 S^{2} 𝔭^{}, the differential equations for the cohomogeneity-one Einstein condition respect that splitting, too. We will nevertheless apply Theorem 3.2.24 to our situation under the assumption that g S^{2} 𝔭 S^{2} 𝔭^{}, which is equivalent to β_{1,7} β_{2,7} 0, and state our result as a conjecture. If the principal orbit is N^{1,1} and the singular orbit is SU 3 U 1 ^{2}, we suppose that there are

• 3 parameters of 0^{th} order in the horizontal direction, which correspond to the initial values a_{1} 0 , b 0 , and c 0 ,

• 2 parameters of 1^{st} order in the horizontal direction, which correspond to β_{3,5}^{}0 and β_{3,6}^{}0 , and

• 2 parameters of 2^{nd} order in the horizontal direction, which correspond to β_{1,2}^{}0 and a_{1} a_{2} ^{}0 .

As usual, f^{}0 and the Einstein constant can be chosen arbitrarily, too. Since those metrics satisfy f^{}0 6, none of them is described by a power series of type (5.49).

If we restrict ourselves to diagonal metrics, the metric and the differential equations both live inside the subspace S^{2} 𝔭 S^{2} 𝔭^{}S^{2} 𝔭 𝔭^{}. Moreover, the differential equations leave any diagonal metric diagonal. We thus can go through the proof of Eschenburg and Wang [32] and see that all of their arguments hold true in our situation, too. If we omit the non-diagonal free parameters from the above list, we therefore have found all the free parameters for the diagonal case.

We will check if Assumption 3.2.19 is satisfied for SU 3 K as principal orbit. U 1 _{1,1} acts trivially on the normal space of the old cohomogeneity-one manifold. The normal space 𝔭^{}of our new space with principal orbit SU 3 K is obtained by taking the quotient with respect to the action of h. Therefore, h and K both act trivially on 𝔭^{}. We take a look at (5.50) and see that h acts non-trivially on V_{1}, V_{2}, and V_{3}. Assumption 3.2.19 thus is satisfied. If the principal orbit is SU 3 K, we can therefore freely choose • 3 parameters of 0^{th} order in the horizontal direction, which correspond to a_{1} 0 , b 0 ,

and c 0 , and

• 2 parameters of 1^{st} order in the horizontal direction, which correspond to β_{1,2}^{}0 and a_{1} a_{2} ^{}0 .

Again, we can choose f^{}0 and the Einstein constant freely. Since there is a free parameter of first order which describes how the diagonal metric on the singular orbit changes into a non-diagonal one, it might be possible to construct further metrics with special holonomy if we allow β_{1,2} to be 0.

Principal orbit: A N^{k,l} which is N^{1,0} , singular orbit: S^{5}

For reasons of simplicity, we assume that k 1 and l 1. Any SU 3 -invariant metric on N^{1, 1} is described by a matrix which is similar to (5.32). The only difference is that the off-diagonal coefficients are contained in V_{2} V_{3} V_{3} V_{2} instead of V_{1} V_{3} V_{3} V_{1} and hence will be denoted by β_{3,5} and β_{3,6} instead of β_{1,5} and β_{1,6}. Therefore, we still speak of ”metrics of type (5.32)”. In our situation, we have a 0 0 as initial condition, since V_{1} 𝔨. We only consider diagonal metrics and thus can work with the system (5.36). For k 1 and l 1, this system becomes:

\begin{matrix}a^{} & \frac{b^{2} c^{2} a^{2}}{bc} \\ \frac{b^{}}{b} & \frac{c^{2} a^{2} b^{2}}{abc} \frac{1}{2} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{a^{2} b^{2} c^{2}}{abc} \frac{1}{2} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & \frac{1}{2} \frac{f}{b^{2}} \frac{1}{2} \frac{f}{c^{2}} \end{matrix}  (5.51)

We want that the left hand sides of the second and the third equation converge to a real number for t 0. On the right hand side of these equations, we divide by a, which converges to zero. Therefore, we need to have b 0 c 0 as initial condition. If we replace c by c and f by f in the above system, the whole right hand side changes its sign. This change is the same as if we had replaced t by t. Since this replacement is nothing more than a change of the orientation, we do not need to consider the case b 0 c 0 separately. Therefore, we assume from now on that b 0 c 0 . We shortly denote b 0 and c 0 by b_{0} and f 0 by f_{0}. By a power series ansatz, we obtain:

\begin{matrix}a t &  & 0 &  & 2t &  & 0 t^{2} &  & \frac{1}{27} \frac{36b_{0}^{2} f_{0}^{2}}{b_{0}^{4}} t^{3} &  & … \\ b t &  & b_{0} &  & \frac{1}{6} \frac{f_{0}}{b_{0}} t &  & \frac{1}{72} \frac{72b_{0}^{2} 5f_{0}^{2}}{b_{0}^{3}} t^{2} &  & \frac{1}{6480} \frac{f_{0} 504b_{0}^{2} 167f_{0}^{2}}{b_{0}^{5}} t^{3} &  & … \\ c t &  & b_{0} &  & \frac{1}{6} \frac{f_{0}}{b_{0}} t &  & \frac{1}{72} \frac{72b_{0}^{2} 5f_{0}^{2}}{b_{0}^{3}} t^{2} &  & \frac{1}{6480} \frac{f_{0} 504b_{0}^{2} 167f_{0}^{2}}{b_{0}^{5}} t^{3} &  & … \\ f t &  & f_{0} &  & 0 t &  & \frac{1}{6} \frac{f_{0}^{3}}{b_{0}^{4}} t^{2} &  & 0 t^{3} &  & … \end{matrix}  (5.52)

Before we discuss if our initial value problem has smooth solutions, we have to describe the smoothness conditions for a cohomogeneity-one metric with principal orbit N^{1, 1} and singular orbit S^{5} explicitly. The isotropy group of the SU 3 -action on the singular orbit is isomorphic to SU 2 . In order to describe the smoothness conditions, we therefore have to investigate several real SU 2 -modules.

SU 2 acts irreducibly on the three-dimensional space 𝔭^{}, since its orbits are spheres. 𝔭^{} therefore can be identified with ℝ^{3}, on which SU 2 acts as SO 3 by matrix multiplication. We will give a short introduction to the representation theory of SU 2 , before we decompose the symmetric powers of 𝔭^{}into irreducible SU 2 -submodules. The complex two-dimensional representation of SU 2 we denote by ℂ^{2}. On the following pages, we will consider ℂ^{2} not as a complex two-dimensional but as a real four-dimensional vector space. We denote the space of all homogeneous polynomials of m^{th} order on ℂ^{2} by S_{ℝ}^{m} ℂ^{2} , where the index ”ℝ” indicates that ℂ^{2} is considered as real. Our first aim is to decompose the complex vector space S_{ℝ}^{m} ℂ^{2} ℂ of all homogeneous polynomials of m^{th} order with complex coefficients. Let s 0, …, m be arbitrary. S_{ℝ}^{m} ℂ^{2} ℂ obviously contains the space of all homogeneous polynomials which depend on s complex and m s conjugate complex variables as a submodule. Moreover, it is the direct sum of all those modules. The submodules of S_{ℝ}^{m} ℂ^{2} ℂ which we have introduced are not necessarily irreducible. We can split each of them into the space of those polynomials which are a multiple of z_{1} \overline{z_{1}} z_{2} \overline{z_{2}} and its complement. If m and m s are both 1, both of these summands are non-trivial. If m and m s are both 2, we can repeat this process. A polynomial P with P z_{1}, \overline{z_{1}}, … P z_{2}, \overline{z}_{2}, … 0 we call trace-free. The SU 2 -module which consists of all homogeneous trace-free polynomials which depend on s complex and m s conjugate complex variables we denote by 𝕍_{s,\overline{m s}}^{}. It is possible to prove that all 𝕍_{s,\overline{m s}}^{}are irreducible. The above considerations prove the following decomposition formula:

S_{ℝ}^{m} ℂ^{2} ℂ _{p 0}^{\frac{m}{2}} _{s 0}^{m 2p} 𝕍_{s,\overline{m 2p s}}^{}.

Any complex irreducible SU 2 -module is equivalent to S_{ℂ}^{p} ℂ^{2} , where p ℕ_{0} and ℂ^{2} now is considered as a complex vector space. In Chapter 4, we have denoted that module by 𝕍_{p}^{ℂ}. By a short calculation, we see that dim 𝕍_{s,\overline{m 2p s}}^{}m 2p 1. The module 𝕍_{s,\overline{m 2p s}}^{} therefore has to be equivalent to 𝕍_{m 2p}^{ℂ} and our decomposition formula can be rewritten as:

S_{ℝ}^{m} ℂ^{2} ℂ _{p 0}^{\frac{m}{2}} m 2p 1 𝕍_{m 2p}^{ℂ} ,

where m 2p 1 𝕍_{m 2p}^{ℂ} denotes the direct sum of m 2p 1 copies of 𝕍_{m 2p}^{ℂ} . For our considerations, the space S_{ℝ}^{m} ℂ^{2} of all homogeneous polynomials of m^{th} order with real coefficients is more important than S_{ℝ}^{m} ℂ^{2} ℂ. It is easy to see that S_{ℝ}^{m} ℂ^{2} consists of exactly those elements of S_{ℝ}^{m} ℂ^{2} ℂ which are invariant with respect to the conjugation map. The conjugation map maps 𝕍_{s,\overline{m 2p s}}^{}into 𝕍_{m 2p s,\overline{s}}^{}and vice versa. If m 2p is even and s \frac{m 2p}{2} , the subspace of 𝕍_{s,\overline{m 2p s}}^{}𝕍_{m 2p s,\overline{s}}^{}which is invariant under conjugation decomposes into two real submodules of the same dimension. As in Chapter 4, we denote the real irreducible SU 2 -module with weight m 2p by 𝕍_{m 2p}^{ℝ} . Since both of the above real submodules are irreducible, they have to be equivalent to 𝕍_{m 2p}^{ℝ} . In the case s \frac{m 2p}{2} , we obtain only one real module which is of the same type. If m 2p is odd, the subspace of 𝕍_{s,\overline{m 2p s}}^{}𝕍_{m 2p s,\overline{s}}^{}which is invariant under conjugation is irreducible. Since its real dimension is 2 m 2p 1 , it has to be equivalent to 𝕍_{m 2p}^{ℂ} . We therefore have the following decomposition formula:

S_{ℝ}^{m} ℂ^{2}   ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; _{p 0}^{\frac{m}{2}} 2p 1 𝕍_{2p}^{ℝ}  if m is even _{p 0}^{\frac{m 1}{2}} p 1 𝕍_{2p 1}^{ℂ} if m is odd

We are now able to deduce a decomposition formula for S^{m} 𝔭^{}. 𝔭^{}is equivalent to the SU 2 - module 𝕍_{2}^{ℝ}. That module can also be considered as the submodule of S_{ℝ}^{2} ℂ^{2} which consists of all homogeneous polynomials which depend on one complex and one conjugate complex variable. S^{m} 𝕍_{2}^{ℝ} therefore consists of certain polynomials of 2m^{th} order which depend on m complex and m conjugate complex variables. The only way of obtaining submodules of S^{m} 𝕍_{2}^{ℝ} therefore is to take the trace and we have:

S^{m} 𝕍_{2}^{ℝ} _{p 0}^{\frac{m}{2}}𝕍_{m 2p,m 2p}^{}_{p 0}^{\frac{m}{2}}𝕍_{2m 4p}^{ℝ} .

Next, we consider S^{2} 𝔭 as a SU 2 -module. SU 2 acts irreducibly on span e_{3}, e_{4}, e_{5}, e_{6} and trivially on span e_{7} . We therefore have

𝔭 𝕍_{1}^{ℂ} 𝕍_{0}^{ℝ}

and

\begin{aligned}S^{2} 𝔭 & S^{2} 𝕍_{1}^{ℂ} 𝕍_{1}^{ℂ} 𝕍_{0}^{ℝ} S^{2} 𝕍_{0}^{ℝ} \\ & 3𝕍_{2}^{ℝ} 𝕍_{1}^{ℂ} 2𝕍_{0}^{ℝ} . \end{aligned}

We will describe the spaces W_{m}^{h} and W_{m}^{v} in detail. First, we consider the horizontal part of the metric. It easily follows from Schur’s lemma that

dim W_{m}^{h}  2 if m is even 3 if m is odd

These dimensions we have to interpret. We start with the case m 0. It is easy to see that any SU 2 -invariant metric on the singular orbit has to be diagonal and satisfy b 0 ^{2} c 0 ^{2}. Therefore, the equation dim W_{0}^{h} 2 simply means that we can choose b_{0} and f_{0} freely and that there are no further restrictions on the initial values of the equations (5.51). We search for a geometric interpretation of the dimensions of the other W_{m}^{h}. For the following considerations, we denote the coefficients of the power series for a, b, c, and f by a_{m}, b_{m}, c_{m}, and f_{m}. We take a look at the power series (5.52) and see that the coefficients of low order obey the following relations:

b_{m} c_{m}  if m is even
b_{m} c_{m} and f_{m} 0 if m is odd

Since b, c, and f are analytic, these relations are equivalent to:

\begin{matrix}b t &  & c t \\ f t &  & f t \end{matrix}  (5.53)

We will prove that the horizontal part of an analytic diagonal metric of type (5.32) has a smooth extension to the singular orbit if and only if it satisfies (5.53). First, we prove the ”only if”-part of this statement. We identify the isotropy group SU 2 of the SU 3 -action on the singular orbit with Sp 1 . Let R_{j} be the matrix in Sp 1 which corresponds to the right-multiplication by j ℍ. This matrix acts on span e_{3}, e_{4}, e_{5}, e_{6}, e_{7} as

 ;  ;  ;  ;  ; \begin{aligned}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ & & & & 1 \end{aligned}  ;  ;  ;  ;  ;

On 𝔭^{}, R_{j} acts as an element of SO 3 . Since R_{j} is a rotation around an angle of \frac{π}{2} , it has to act on 𝔭^{}as a rotation around an angle of π. R_{i} U 1 _{1, 1} leaves \frac{}{t} invariant and thus acts on 𝔭^{}as a rotation around the axis span \frac{}{t} . R_{j} therefore has to act as a rotation around an axis which is perpendicular to span \frac{}{t} . This observation proves that R_{j} maps \frac{}{t} into its negative. Since the action of R_{j} has to leave the metric invariant, we obtain the following two relations:

\begin{matrix}g_{t} e_{3}, e_{3} & & g_{t} e_{5}, e_{5} \\ g_{t} e_{7}, e_{7} & & g_{t} e_{7}, e_{7} \end{matrix}

These equations can be rewritten as:

\begin{matrix}b^{2} t & & c^{2} t \\ f^{2} t & & f^{2} t \end{matrix}

Since we have b 0 c 0 0 and f 0 0, (5.53) has to be satisfied and we have proven the ”only if”-part of our statement. In order to prove the ”if”-part of our statement, we assume that g is an analytic diagonal metric which satisfies (5.53). We choose an arbitrary unit quaternion h and investigate the action of R_{h} on our metric. After we have done a calculation, which is similar to the above calculation, we see that R_{h} preserves the metric. We consider the action of R_{h} on the power series for b, c, and f and see that the invariance with respect to SU 2 which we have proven is the same as the SU 2 -equivariance in the statement of Theorem 3.2.18. Our metric thus is smooth and we have proven the ”if”-part, too.

If m is odd, there is only one degree of freedom for the m^{th} derivative of a metric which satisfies (5.53). Since in this case dim W_{m}^{h} 3, the other two parameters have to correspond to the off-diagonal coefficients of the metric.

We now consider the vertical part of the metric. With help of the decomposition formula for S^{m} 𝕍_{2}^{ℝ} , we obtain:

S^{2} 𝔭^{}𝕍_{4}^{ℝ} 𝕍_{0}^{ℝ} .

Since we have dim Hom_{SU 2} 𝕍_{4}^{ℝ}, 𝕍_{4}^{ℝ} 1, it follows that

dim W_{m}^{v} ; ; 1 if m 0 0 if m is odd 2 if m 2 and even

Since the action of U 1 _{1, 1} on the tangent space V_{1} of the collapsing sphere is irreducible, any invariant metric on the sphere has to be a multiple of the standard metric with sectional curvature 1. The vertical part of the metric therefore is solely described by the function a. By the usual arguments, we can prove that the vertical part is smooth if and only if a is odd and the length of any great circle of the collapsing sphere is 2πt O t^{2} for small t. The Lie group which is generated by e_{1} intersects U 1 _{1, 1} twice. Therefore, the length of the great circle is \overline{g_{t} e_{1}, e_{1}}π, which equals a t π. By the same argument as for the S^{1}-collapse, it follows that a^{}0 2. The first coefficients of the power series (5.52) thus satisfy all of the smoothness conditions. Moreover, it is possible to prove that the solutions of (5.51) satisfy those conditions, too. We again choose b_{0} and f_{0} arbitrarily. From the first equation of (5.51), we immediately obtain a^{}0 2. If we replace a by a, b by c, and c by b, the right hand side of (5.51) changes its sign. If we further replace t by t, we again obtain the original sign. Therefore, a t , b t , c t , f t and a t , c t , b t , f t are solutions of the same initial value problem. We will prove that the system (5.51) has for any choice of b_{0} and f_{0} a unique power series solution. From that uniqueness result, it follows that the power series satisfies the equations a t a t , b t c t , c t b t , and f t f t which are the remaining smoothness conditions. Any power series which solves (5.51) and is smooth at the singular orbit also satisfies the cohomogeneity-one Einstein condition. We can conclude with help of Theorem 3.2.24 that the power series converges. Since U 1 _{1, 1} acts trivially on e_{7} and on a one-dimensional subspace of 𝔭^{}, Assumption 3.2.19 is, as in the previous case, not satisfied. The metric g is nevertheless contained in S^{2} 𝔭 S^{2} 𝔭^{}, since we always have g e_{7}, \frac{}{t} 0. We thus can work around this problem by the same argument as before. After we have shown the existence and uniqueness of our power series solution, we finally have proven the local existence and smoothness of our metrics. In order to do begin with our proof, we rewrite the system (5.51) as:

\begin{matrix}a^{}bc & b^{2} c^{2} a^{2} \\ ab^{}bc & c^{2} a^{2} b^{2} b \frac{1}{2}acf \\ abc^{}c & a^{2} b^{2} c^{2} c \frac{1}{2}abf \\ b^{2}c^{2}f^{} & \frac{1}{2}c^{2}f^{2} \frac{1}{2}b^{2}f^{2} \end{matrix}  (5.54)

We consider the m^{th} derivatives of the above equations at t 0 and put the terms which contain the m 1 ^{st} derivative of a, b, c, or f on the left hand side. The other terms we put on the right hand side of our equations. After dividing by m 1 !, we obtain:

\begin{matrix}a_{m 1} b_{0}c_{0} & R_{1} \\ a_{0}b_{m 1} b_{0}c_{0} & R_{2} \\ a_{0}b_{0}c_{m 1} c_{0} & R_{3} \\ f_{m 1} b_{0}^{2}c_{0}^{2} & R_{4} \end{matrix}

where R_{1}, …, R_{4} are polynomials which depend only on a_{0}, …, f_{m}. Since a_{0} 0, the second and the third equation contain no information on b_{m 1} or c_{m 1} . We therefore have to consider the m 1 ^{st} derivatives of the second and third equation of (5.54) and obtain:

a_{m 1} b_{1}b_{0}c_{0} m 1 a_{1}b_{m 1} b_{0}c_{0} m 1 a_{0}b_{m 1} b_{1}c_{0} m 1 a_{0}b_{m 1} b_{0}c_{1} a_{0}b_{1}b_{m 1} c_{0} a_{0}b_{1}b_{0}c_{m 1}

2c_{0}c_{m 1} 2a_{0}a_{m 1} 2b_{0}b_{m 1} b_{0} c_{0}^{2} a_{0}^{2} b_{0}^{2} b_{m 1}

\frac{1}{2}a_{m 1} c_{0}f_{0} \frac{1}{2}a_{0}c_{m 1} f_{0} \frac{1}{2}a_{0}c_{0}f_{m 1} { \atop R_{2}}

a_{m 1} b_{0}c_{1}c_{0} a_{0}b_{m 1} c_{1}c_{0} m 1 a_{1}b_{0}c_{m 1} c_{0} m 1 a_{0}b_{1}c_{m 1} c_{0} m 1 a_{0}b_{0}c_{m 1} c_{1} a_{0}b_{0}c_{1}c_{m 1}

2a_{0}a_{m 1} 2b_{0}b_{m 1} 2c_{0}c_{m 1} c_{0} a_{0}^{2} b_{0}^{2} c_{0}^{2} c_{m 1}

\frac{1}{2}a_{m 1} b_{0}f_{0} \frac{1}{2}a_{0}b_{m 1} f_{0} \frac{1}{2}a_{0}b_{0}f_{m 1} { \atop R_{3}}

where { \atop R_{2}} and { \atop R_{3}} also are polynomials which depend only on a_{0}, …, f_{m}. As in the previous case, we assume that m 2 and the above system becomes linear with respect to a_{m 1} , …, f_{m 1} . By setting a_{0} 0, c_{0} b_{0}, a_{1} 2, b_{1} \frac{1}{6} \frac{f_{0}}{b_{0}} , and c_{1} \frac{1}{6} \frac{f_{0}}{b_{0}} , we obtain the following linear system for a_{m 1} , …, f_{m 1} :

b_{0}^{2}a_{m 1}  R_{1}
\frac{1}{6}b_{0}f_{0}a_{m 1} 2 m 1 b_{0}^{2}b_{m 1}  2b_{0}^{2}c_{m 1} 2b_{0}^{2}b_{m 1} \frac{1}{2}b_{0}f_{0}a_{m 1} { \atop R_{2}}
\frac{1}{6}b_{0}f_{0}a_{m 1} 2 m 1 b_{0}^{2}c_{m 1}  2b_{0}^{2}b_{m 1} 2b_{0}^{2}c_{m 1} \frac{1}{2}b_{0}f_{0}a_{m 1} { \atop R_{3}}
b_{0}^{4}f_{m 1}  R_{4}

which is equivalent to

; ; ; \begin{matrix}b_{0}^{2} & 0 & 0 & 0 \\ \frac{1}{3}b_{0}f_{0} & 2 m 2 b_{0}^{2} & 2b_{0}^{2} & 0 \\ \frac{1}{3}b_{0}f_{0} & 2b_{0}^{2} & 2 m 2 b_{0}^{2} & 0 \\ 0 & 0 & 0 & b_{0}^{4} \end{matrix} ; ; ; ; ; ; \begin{matrix}a_{m 1} \\ b_{m 1} \\ c_{m 1} \\ f_{m 1} \end{matrix} ; ; ; ; ; ; R_{1} ; { \atop R_{2}} ; { \atop R_{3}} ; R_{4} ; ; ;

The above matrix obviously has always rank 4. Therefore, there exists a unique formal power series which solves our initial value problem. All in all, we have proven the existence of a family of non-homothetic metrics with holonomy Spin 7 which depend on one parameter. Since there are non-diagonal smooth metrics with principal orbit N^{1, 1} and singular orbit S^{5}, there also may exist further non-diagonal examples of metrics with holonomy Spin 7 .

We finally classify the cohomogeneity-one Einstein metrics with principal orbit N^{1, 1} and singular orbit S^{5}. Although Assumption 3.2.19 is not satisfied, we can apply Theorem 3.2.24 to our situation. The reason for this is as above that since g e_{7}, \frac{}{t} 0, the metric is always contained in S^{2} 𝔭 S^{2} 𝔭^{}. Moreover, we as usual have Ric e_{7}, \frac{}{t} 0 and therefore Ric S^{2} 𝔭 S^{2} 𝔭^{}. We thus obtain the following numbers of initial conditions which we can prescribe:

1. 2 initial conditions of 0^{th} order in the horizontal direction, which describe the values of b 0 ( c 0 ) and f 0 ,

2. 3 initial conditions of 1^{st} order in the horizontal direction, which describe the values of b c ^{}0 , β_{3,5}^{}0 , and β_{3,6}^{}0 .

3. 1 initial condition in the vertical direction, which describes the value of a^{}0 .

As usual, the Einstein constant can also be fixed at an arbitrary value.

Principal orbit: A N^{k,l} which is N^{1,0} , singular orbit: SU 3 SO 3

As in the previous case, we assume that k 1, l 1 and denote the isotropy group of the SU 3 -action on the singular orbit by K and its Lie algebra by 𝔨. We have seen that there exists a one-parameter family of Lie groups K with U 1 _{1, 1} K SU 3 which are all isomorphic to SO 3 . That family is generated by the action of U 1 _{1,1}, which is a subset of Norm_{SU 3}U 1 _{1, 1} . The action of the normalizer preserves any diagonal metric. Therefore, we have to consider one special choice of K only. In Lemma 5.4.2, we have chosen 𝔨 as:

𝔨 span e_{8}, \frac{1}{2} \overline{2} e_{3} \frac{1}{2} \overline{2} e_{5}, \frac{1}{2} \overline{2} e_{4} \frac{1}{2} \overline{2} e_{6} .

The restriction of the metric to 𝔨 has to vanish at the singular orbit. Since we only consider diagonal metrics, we have e_{3} e_{5} ^{2} b^{2} 0 c^{2} 0 0. This equation forces b 0 and c 0 to be 0. In this situation, the metric vanishes on all of span e_{3}, e_{4}, e_{5}, e_{6} and the singular orbit is a space which is different from SU 3 SO 3 . In order to construct cohomogeneityone metrics with singular orbit SU 3 SO 3 , we have to consider non-diagonal metrics or change K by an element k SU 3 which is not contained in U 1 _{1, 1} . We choose the second ansatz. It is easy to see that if k Norm_{SU 3}U 1 _{1, 1} and the metric is diagonal, we are again in the same situation as above. Therefore, we will work with a k SU 3 which does not preserve U 1 _{1, 1} . Nevertheless, the space SU 3 kU 1 _{1, 1} k^{1} is still SU 3 -equivariantly diffeomorphic to N^{1, 1} . By conjugation by a suitable k SU 3 , we can map U 1 _{1, 1} to the group

; ; { \atop }cos φ sin φ ; sin φ cos φ ; 1 { \atop } ; ; ; ; ; φ ℝ ; ; ,

which we shortly denote by SO 2 . K can be mapped by the same element to the standard embedding of SO 3 into SU 3 . In this situation, the isotropy group of the SU 3 -action on the principal orbit is generated by e_{1}. The tangent space of the principal orbit SU 3 SO 2 can be identified with 𝔪 span e_{2}, …, e_{8} . It is easily possible to calculate the action of SO 2 on 𝔪. As a SO 2 -module, 𝔪 splits into the following irreducible submodules:

V_{0}^{}: span e_{7}
V_{1}^{}: span e_{3}, e_{5}
V_{2}^{}: span e_{4}, e_{6}
V_{3}^{}: span e_{2}, e_{8}

Since V_{1}^{}and V_{2}^{}are equivalent modules, there exist metrics on SU 3 SO 2 which are non-diagonal with respect to any basis of 𝔪 which is adapted to the above splitting. Since SU 3 SO 2 is SU 3 -equivariantly diffeomorphic to the exceptional Aloff-Wallach space N^{1, 1} , this is not surprising. In order to simplify our calculations, we fix the basis e_{7}, e_{3}, e_{5}, e_{4}, e_{6}, e_{2}, e_{8} of 𝔪 and consider only metrics which are diagonal with respect to this basis. The normalizer Norm_{SU 3}SO 2 is generated by e_{1} and e_{7}. V_{1}^{}is not invariant under the action of e_{7}. Therefore, Norm_{SU 3}SO 2 maps diagonal into non-diagonal metrics and vice versa. If we consider neither the conjugates of K with respect to Norm_{SU 3}SO 2 nor nondiagonal metrics, there will be metrics with singular orbit SU 3 SO 3 which will not be explored. For reasons of brevity, we nevertheless will not investigate those cases. The matrix representation of any of the diagonal metrics has the following form:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}f^{2} \\ & a^{2} & 0 \\ & 0 & a^{2} \\ & & & b^{2} & 0 \\ & & & 0 & b^{2} \\ & & & & & c^{2} & 0 \\ & & & & & 0 & c^{2} \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

where a, b, c, f ℝ 0 . The Lie algebra 𝔨 is spanned by e_{1}, e_{3}, e_{5} . Near the singular orbit, a therefore has to converge to zero. Conversely, any cohomogeneity-one metric of the above type with a 0 and b, c, f 0 on some orbit has SU 3 SO 3 as a singular orbit. With help of our new ansatz, we thus can construct a big class of diagonal cohomogeneity-one metrics with the required orbit structure.

By interchanging e_{2} and e_{8}, we obtain a new basis of 𝔪. The matrix representation of ad_{e_{1}} _{𝔪} with respect to this new basis is:

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{aligned}0 \\ & 0 & 1 \\ & 1 & 0 \\ & & & 0 & 1 \\ & & & 1 & 0 \\ & & & & & 0 & 2 \\ & & & & & 2 & 0 \end{aligned}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

The above matrix is contained in 𝔤_{2}. Therefore, we have found the following frame of a cohomogeneity-one Spin 7 -structure with principal orbit SU 3 SO 2 :

\begin{matrix}f_{0} : \frac{}{t} & f_{1} : \frac{1}{f} e_{7} & f_{2} : \frac{1}{a}e_{3} & f_{3} : \frac{1}{a}e_{5} \\ f_{4} : \frac{1}{b}e_{4} & f_{5} : \frac{1}{b}e_{6} & f_{6} : \frac{1}{c}e_{8} & f_{7} : \frac{1}{c}e_{2} \end{matrix}

There exists a U 1 -action on the tangent space which generates a one-parameter family of SU 3 -invariant Spin 7 -structures with the same associated metric. A possible frame of those structures is:

f_{0} : \frac{}{t}  f_{1} : \frac{1}{f} e_{7}
f_{2} : \frac{1}{a} cos θ e_{3} \frac{1}{a} sin θ e_{5} f_{3} : \frac{1}{a} sin θ e_{3} \frac{1}{a} cos θ e_{5}
f_{4} : \frac{1}{b} cos θ e_{4} \frac{1}{b} sin θ e_{6} f_{5} : \frac{1}{b} sin θ e_{4} \frac{1}{b} cos θ e_{6}
f_{6} : \frac{1}{c} cos θ e_{8} \frac{1}{c} sin θ e_{2} f_{7} : \frac{1}{c} sin θ e_{8} \frac{1}{c} cos θ e_{2}

(5.55)

Analogously to the case N^{1, 1} , the U 1 -action generates any connected component of the space of all SU 3 -invariant G_{2}-structures which have the same associated metric. We calculate the three-form ω and the four-form ω which are associated to the above frame. By a straight-forward but lengthy calculation, we obtain:

ω a^{2}f e^{357} b^{2}f e^{467} c^{2}f e^{278}
abc sin θ e^{234} abc cos θ e^{236} abc cos θ e^{245} abc sin θ e^{256}
abc cos θ e^{348} abc sin θ e^{368} abc sin θ e^{458} abc cos θ e^{568}

(5.56)

and

ω abcf cos θ e^{2347} abcf sin θ e^{2367} abcf sin θ e^{2457} abcf cos θ e^{2567}
abcf sin θ e^{3478} abcf cos θ e^{3678} abcf cos θ e^{4578} abcf sin θ e^{5678}
a^{2}b^{2} e^{3456} a^{2}c^{2}e^{2358} b^{2}c^{2}e^{2468}

(5.57)

The Spin 7 -structure is parallel only if d_{SU 3 SO 2} ω 0, where d_{SU 3 SO 2} denotes the exterior differential on the principal orbit. With help of equation (3.4), we calculate d_{SU 3 SO 2} ω _{eSO 2} e_{3}^{}, e_{4}^{}, e_{5}^{}, e_{6}^{}, e_{7}^{}and obtain:

d_{SU 3 SO 2} ω _{eSO 2} e_{3}^{}, e_{4}^{}, e_{5}^{}, e_{6}^{}, e_{7}^{}

ω_{eSO 2} e_{3}, e_{4} ^{}, e_{5}^{}, e_{6}^{}, e_{7}^{}ω_{eSO 2} e_{3}, e_{5} ^{}, e_{4}^{}, e_{6}^{}, e_{7}^{}

ω_{eSO 2} e_{3}, e_{6} ^{}, e_{4}^{}, e_{5}^{}, e_{7}^{}ω_{eSO 2} e_{3}, e_{7} ^{}, e_{4}^{}, e_{5}^{}, e_{6}^{}

ω_{eSO 2} e_{4}, e_{5} ^{}, e_{3}^{}, e_{6}^{}, e_{7}^{}ω_{eSO 2} e_{4}, e_{6} ^{}, e_{3}^{}, e_{5}^{}, e_{7}^{}

ω_{eSO 2} e_{4}, e_{7} ^{}, e_{3}^{}, e_{5}^{}, e_{6}^{}ω_{eSO 2} e_{5}, e_{6} ^{}, e_{3}^{}, e_{4}^{}, e_{7}^{}

ω_{eSO 2} e_{5}, e_{7} ^{}, e_{3}^{}, e_{4}^{}, e_{6}^{}ω_{eSO 2} e_{6}, e_{7} ^{}, e_{3}^{}, e_{4}^{}, e_{5}^{}

ω_{eSO 2} e_{7}^{}e_{8}^{}, e_{5}^{}, e_{6}^{}, e_{7}^{}ω_{eSO 2} e_{1}^{}, e_{4}^{}, e_{6}^{}, e_{7}^{}

ω_{eSO 2} e_{2}^{}, e_{4}^{}, e_{5}^{}, e_{7}^{}3 ω_{eSO 2} e_{4}^{}, e_{4}^{}, e_{5}^{}, e_{6}^{}

ω_{eSO 2} e_{2}^{}, e_{3}^{}, e_{6}^{}, e_{7}^{}ω_{eSO 2} e_{1}^{}, e_{3}^{}, e_{5}^{}, e_{7}^{}

3 ω_{eSO 2} e_{3}^{}, e_{3}^{}, e_{5}^{}, e_{6}^{}ω_{eSO 2} e_{7}^{}e_{8}^{}, e_{3}^{}, e_{4}^{}, e_{7}^{}

3 ω_{eSO 2} e_{6}^{}, e_{3}^{}, e_{4}^{}, e_{6}^{}3 ω_{eSO 2} e_{5}^{}, e_{3}^{}, e_{4}^{}, e_{5}^{}

4abcf sin θ

θ therefore has to be either 0 or π. The frames for θ 0 and θ π coincide up to a change of the signs of a, b, and c. We thus can assume that θ 0. In that situation, some of the coefficients of ω vanish and d_{SU 3 SO 2} ω can more easily be calculated:

d_{SU 3 SO 2} ω d_{SU 3 SO 2} abcf e^{2347} abcf e^{2567} abcf e^{3678} abcf e^{4578}
a^{2}b^{2} e^{3456} a^{2}c^{2}e^{2358} b^{2}c^{2}e^{2468}
abcf e^{36} e^{45} e^{347} e^{2} e^{26} 3e^{47} e^{48} e^{47}
e^{23} e^{25} 3e^{37} e^{38} e^{7} e^{234} e^{34} e^{56}
abcf e^{36} e^{45} e^{567} e^{2} e^{24} 3e^{67} e^{68} e^{67}
e^{25} e^{23} 3e^{57} e^{58} e^{7} e^{256} e^{34} e^{56}
abcf e^{26} 3e^{47} e^{48} e^{678} e^{3} e^{23} 3e^{57} e^{58} e^{78}
e^{36} e^{34} e^{56} e^{8} e^{367} e^{34} e^{56}
abcf e^{25} 3e^{37} e^{38} e^{578} e^{4} e^{24} 3e^{67} e^{68} e^{78}
e^{45} e^{34} e^{56} e^{8} e^{457} e^{34} e^{56}
a^{2}b^{2} e^{26} 3e^{47} e^{48} e^{456} e^{3} e^{25} 3e^{37} e^{38} e^{56}
e^{34} e^{24} 3e^{67} e^{68} e^{6} e^{345} e^{23} 3e^{57} e^{58}
a^{2}c^{2} e^{36} e^{45} e^{358} e^{2} e^{26} 3e^{47} e^{48} e^{58}
e^{23} e^{24} 3e^{67} e^{68} e^{8} e^{235} e^{34} e^{56}
b^{2}c^{2} e^{36} e^{45} e^{468} e^{2} e^{25} 3e^{37} e^{38} e^{68}
e^{24} e^{23} 3e^{57} e^{58} e^{8} e^{246} e^{34} e^{56}

3b^{2}c^{2} 3a^{2}c^{2} e^{23678} 3b^{2}c^{2} 3a^{2}c^{2} e^{24578}

Since the metric on the principal orbit has to be positive definite, we have c 0. Therefore, the equation d_{SU 3 SO 2} ω 0 is satisfied if and only if a b. Let d be the exterior differential on the eight-dimensional cohomogeneity-one manifold which we consider. We calculate dΩ _{eSO 2} e_{2}^{}, e_{3}^{}, e_{4}^{}, e_{8}^{}, \frac{}{t} under the assumption that θ 0 and a b:

dΩ _{eSO 2} e_{2}^{}, e_{3}^{}, e_{4}^{}, e_{8}^{}, \frac{}{t}
\frac{}{t} ω_{eSO 2} e_{2}^{}, e_{3}^{}, e_{4}^{}, e_{8}^{}d_{SU 3 SO 2}ω _{eSO 2} e_{2}^{}, e_{3}^{}, e_{4}^{}, e_{8}^{}
ω_{eSO 2} e_{2}, e_{3} ^{}, e_{4}^{}, e_{8}^{}ω_{eSO 2} e_{2}, e_{4} ^{}, e_{3}^{}, e_{8}^{}ω_{eSO 2} e_{2}, e_{8} ^{}, e_{3}^{}, e_{4}^{}
ω_{eSO 2} e_{3}, e_{4} ^{}, e_{2}^{}, e_{8}^{}ω_{eSO 2} e_{3}, e_{8} ^{}, e_{2}^{}, e_{4}^{}ω_{eSO 2} e_{4}, e_{8} ^{}, e_{2}^{}, e_{3}^{}
ω_{eSO 2} e_{6}^{}, e_{4}^{}, e_{8}^{}ω_{eSO 2} e_{5}^{}, e_{3}^{}, e_{8}^{}ω_{eSO 2} 2e_{1}^{}, e_{3}^{}, e_{4}^{}
ω_{eSO 2} e_{8}^{}e_{7}^{}, e_{2}^{}, e_{8}^{}ω_{eSO 2} e_{4}^{}, e_{2}^{}, e_{4}^{}ω_{eSO 2} e_{3}^{}, e_{2}^{}, e_{3}^{}
c^{2}f

The above calculation proves that a Spin 7 -structure which has a frame of type (5.55) can be parallel only if c 0 or f 0. Since the metric on the principal orbit has to be positive definite, none of these conditions can be satisfied. Nevertheless, the G_{2}-structure ω on the principal orbit is cosymplectic if θ 0 and a b. In Theorem 3.2.31, we have proven that ω can be extended to a parallel Spin 7 -structure on a neighborhood of the principal orbit if it is cosymplectic. Therefore, there have to exist other G_{2}-structures on SU 3 SO 2 which cannot be described by a frame of type f_{1}, …, f_{7} . The G_{2}-structures on the principal orbits have to remain in the same connected component of the space of all invariant G_{2}-structures on SU 3 SO 2 as ω. The only G_{2}-structures which satisfy these conditions are those which are connected to ω and whose associated metric is non-diagonal with respect to our basis. All in all, we have proven that any diagonal metric which is equipped with a G_{2}-structure of type (5.56) is immediately changed by the equation dΩ 0 into a non-diagonal one.

The tangent space of the singular orbit SU 3 SO 3 splits with respect to the action of SO 2 into V_{0}^{}V_{2}^{}V_{3}^{}. Since V_{1}^{}and V_{2}^{}are equivalent SO 2 -modules, Assumption 3.2.19 is not satisfied. The modules V_{0}^{}, V_{2}^{}, and V_{3}^{}are pairwise inequivalent and we thus can prove with help of Schur’s lemma that any SU 3 -invariant metric on the singular orbit is diagonal. We will investigate how a diagonal metric on the singular orbit can smoothly be extended into a non-diagonal cohomogeneity-one metric. In order to answer this question, we work with Theorem 3.2.18 and thus have to decompose certain SO 3 -modules. Since 3.2.19 is not satisfied, we have to consider the space W_{m} Hom_{SO 3} S^{m} 𝔭^{}, S^{2} 𝔭 𝔭^{} instead of the smaller space Hom_{SO 3} S^{m} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}. The normal space 𝔭^{}of the singular orbit is a three-dimensional irreducible 𝔰𝔲 2 -module and therefore equivalent to 𝕍_{2}^{ℝ}. Since the action of SO 3 on 𝔤𝔩 3, ℂ by conjugation leaves only the multiples of the identity matrix invariant, the subspace 𝔭 of 𝔰𝔲 3 contains no trivial submodule. By considering the low-dimensional 𝔰𝔲 2 -modules, we see that 𝔭 in fact has to be irreducible. The only fivedimensional irreducible 𝔰𝔲 2 -module is 𝕍_{4}^{ℝ} and we thus have proven that 𝔭 𝕍_{4}^{ℝ}. It is possible to prove with help of the Clebsch-Gordan formula that the space S^{2} 𝔭 decomposes as 𝕍_{8}^{ℝ} 𝕍_{4}^{ℝ} 𝕍_{0}^{ℝ}. The tensor product 𝔭 𝔭^{}splits into 𝕍_{6}^{ℝ} 𝕍_{4}^{ℝ} 𝕍_{2}^{ℝ} and we have:

S^{2} 𝔭 𝔭^{}S^{2} 𝔭 𝔭 𝔭^{}S^{2} 𝔭^{}
𝕍_{8}^{ℝ} 𝕍_{4}^{ℝ} 𝕍_{0}^{ℝ} 𝕍_{6}^{ℝ} 𝕍_{4}^{ℝ} 𝕍_{2}^{ℝ} 𝕍_{4}^{ℝ} 𝕍_{0}^{ℝ}
𝕍_{8}^{ℝ} 𝕍_{6}^{ℝ} 3𝕍_{4}^{ℝ} 𝕍_{2}^{ℝ} 2𝕍_{0}^{ℝ}

The module S^{m} 𝔭^{}decomposes in the same way as in the case where the singular orbit is S^{5}. We therefore obtain for the dimensions of W_{m}:

dim W_{m}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; 2 if m 0 1 if m 1 5 if m 2 2 if m 3 and odd 6 if m 4 and even  (5.58)

The space of all diagonal SU 3 -invariant metrics on N^{1, 1} has only four dimensions. The metric on the singular orbit therefore can smoothly change into an invariant, non-diagonal metric on N^{1, 1} . Our observation is even true if we take into account that the vertical part of the metric in fact depends on one parameter instead of dim Hom_{SO 3} S^{2m 2} 𝔭^{}, S^{2} 𝔭^{}2 parameters. This is a hint that parallel cohomogeneity-one Spin 7 -structures with principal orbit N^{1, 1} and singular orbit SU 3 SO 3 may indeed exist. In order to construct those parallel Spin 7 -structures, we have to consider a more general equation for the holonomy reduction which also describes non-diagonal metrics. Although we will not carry out that calculation, it nevertheless seems to be a promising endeavor.

As we have already remarked, Assumption 3.2.19 is in our situation not satisfied. Furthermore, we have not discussed the geometric meaning of the dimensions of W_{m}. Therefore, we will not further consider the issue of cohomogeneity-one Einstein metrics with principal orbit N^{1, 1} and singular orbit SU 3 SO 3 .

Principal orbit: An arbitrary Aloff-Wallach space N^{k,l}, singular orbit: ℂℙ^{2}

We will consider the three different types of Aloff-Wallach spaces simultaneously. The reason for this is that many arguments which work for one of these cases will also work for the other ones. The Lie algebra 𝔨 of the isotropy group of the SU 3 -action on ℂℙ^{2} is isomorphic to 𝔲 2 . If the principal orbit is a generic Aloff-Wallach space or N^{1,0}, 𝔨 is either 𝔲 1 _{k,l} V_{1} V_{4}, 𝔲 1 _{k,l} V_{2} V_{4}, or 𝔲 1 _{k,l} V_{3} V_{4}. As we have remarked in 5.4.4, it suffices to work with the case 𝔨 𝔲 1 _{k,l} V_{1} V_{4} if we consider all pairs k, l of coprime integers with k l. The pair 1, 1 has to be excluded from our considerations, since in that situation the space K U 1 _{k,l} is diffeomorphic to S^{2} S^{1} and not to a quotient of S^{3} by a discrete group. If k l 1, the action of SU 2 Norm_{SU 3}U 1 _{1,1} on 𝔲 1 _{1,1} V_{2} V_{4}, generates a family of Lie algebras which are all isomorphic to 𝔲 2 . It is easy to see that the action of the normalizer on 𝔲 1 _{1,1} V_{3} V_{4} generates the same family of Lie algebras. The algebra 𝔲 1 _{1,1} V_{1} V_{4} is the Lie algebra of Norm_{SU 3}U 1 _{1,1} and is therefore left invariant by the action of the normalizer. According to Lemma 5.4.2, the above 𝔨 are the only Lie algebras which are isomorphic to 𝔲 2 and satisfy 𝔲 1 _{k,l} 𝔨 𝔰𝔲 3 . For our considerations, we will always assume that the metric is diagonal. In this situation, we can easily show that even if k l 1, 𝔨 has to be one of the three subalgebras which are also there in the generic case.

Since we can obtain 𝔲 1 _{1,1} V_{3} V_{4} from 𝔲 1 _{1,1} V_{2} V_{4} by the action of the normalizer, we will restrict ourselves to the case 𝔨 𝔲 1 _{k,l} V_{i} V_{4} with i 1, 2 if k l 1. All in all, we have to consider the following three initial values problems:

1. The system (5.36) with a 0 f 0 0,

2. The system (5.39) with a_{1} 0 a_{2} 0 f 0 0,

3. The system (5.39) with b 0 f 0 0.

In the last of these cases, we have K U 1 _{k,l} S^{3} ℤ_{1 2} S^{3} and the space on which SU 3 acts with cohomogeneity one is a manifold. We make a power series ansatz for all of these three initial value problems and start with the first one. It follows from the second and third equation of (5.36) that we necessarily have b 0 ^{2} c 0 ^{2} if a 0 f 0 0. If a t , b t , c t , f t is a solution of (5.36), a t , b t , c t , f t is a solution of (5.36), too. For the same reasons as in the case where the singular orbit is S^{5}, we can therefore assume that b 0 c 0 : b_{0} and do not have to consider the case b 0 c 0 separately. With help of a MAPLE program, we see that the Taylor expansion of any solution of our initial value problem has to begin with

\begin{aligned}a t &  & 0 &  & t &  & 0 t^{2} &  & \frac{1}{24b_{0}^{2}} \frac{12Δ q k l}{Δ} t^{3} &  & 0 t^{4} & … \\ b t &  & b_{0} &  & 0 t &  & \frac{1}{6b_{0}} \frac{4k 5l}{k l} t^{2} &  & 0 t^{3} \\ & & &  & \frac{1}{288b_{0}^{3}} \frac{104k^{2} 224kl 140l^{2} Δ q k^{3} k^{2}l kl^{2} l^{3}}{Δ^{2}}  t^{4} & … \\ c t &  & b_{0} &  & 0 t &  & \frac{1}{6b_{0}} \frac{5k 4l}{k l} t^{2} &  & 0 t^{3} \\ & & &  & \frac{1}{288b_{0}^{3}} \frac{140k^{2} 224kl 104l^{2} Δ q k^{3} k^{2}l kl^{2} l^{3}}{Δ^{2}} t^{4} & … \\ f t &  & 0 &  & \frac{2Δ}{k l} t &  & 0 t^{2} &  & \frac{q}{6b_{0}^{2}} t^{3} &  & 0 t^{4} & … \end{aligned}  (5.59)

The parameter q of third order which appears in the above power series can be chosen arbitrarily. Before we discuss the meaning of q, we take a look at the other two initial value problems. Analogously to the previous case, any solution of (5.39) with a_{1} 0 a_{2} 0 f 0 has to satisfy b 0 ^{2} c 0 ^{2}. Since the system (5.39) is preserved if we replace c by c, f by f, and t by t we again can assume that b 0 c 0 : b_{0}. We obtain the following Taylor expansion for our functions:

\begin{matrix}a_{1} t & & 0 & & t & & 0 t^{2} & & _{\frac{q_{1}}{6b_{0}^{2}}} t^{3} & & 0 t^{4} & & \frac{2q_{2}^{1} 3q_{1}q_{2} 3q_{2}^{2} 3q_{1} 18q_{2}}{60b_{0}^{4}} t^{5} & … \\ a_{2} t & & 0 & & t & & 0 t^{2} & & _{\frac{q_{2}}{6b_{0}^{2}}} t^{3} & & 0 t^{4} & & \frac{3q_{1}^{2} 3q_{1}q_{2} 2q_{2}^{2} 18q_{1} 3q_{2}}{60b_{0}^{4}} t^{5} & … \\ b t & & b_{0} & & 0 t & & \frac{3}{4b_{0}} t^{2} & & 0 t^{3} & & _{\frac{39}{96b_{0}^{3}}} t^{4} & & 0 t^{5} & … \\ c t & & b_{0} & & 0 t & & \frac{3}{4b_{0}} t^{2} & & 0 t^{3} & & _{\frac{39}{96b_{0}^{3}}} t^{4} & & 0 t^{5} & … \\ f t & & 0 & & 3t & & 0 t^{2} & & _{\frac{6 q_{1} q_{2}}{2b_{0}^{2}}} t^{3} & & 0 t^{4} & & \frac{2q_{1}^{2} 7q_{1}q_{2} 2q_{2}^{2} 27q_{1} 27q_{2} 90}{20b_{0}^{4}} t^{5} & … \end{matrix}

(5.60)

The functions b and c coincide up to fifth order. We therefore suppose that b t c t for all values of t. Analogously to the previous case, there are two parameters q_{1} and q_{2} which can be chosen freely. We will prove that b c and that q_{1} and q_{2} are the only free parameters later on. Now, we consider the equations (5.39) under the assumption that b 0 f 0 0. It is easy to see that in this situation we necessarily have c 0 ^{2} a_{1} 0 ^{2} a_{2} 0 ^{2}. Let a_{0} : a_{1} 0 . Since there are two possibilities for the sign of a_{2} 0 and c 0 , there are four kinds of initial value problems which we have to consider. Because of the symmetry of (5.39) which we have mentioned in the previous case, we can assume that the sign of a_{1} 0 and c 0 is the same. Therefore, the only two subcases which we have to consider are a_{1} 0 a_{2} 0 and a_{1} 0 a_{2} 0 . The initial condition a_{1} 0 a_{2} 0 yields the following power series:

\begin{matrix}a_{1} t & & a_{0} & & 0 t & & \frac{1}{a_{0}} t^{2} & & 0 t^{3} & & \frac{q 21}{24a_{0}^{3}} t^{4} & & 0 t^{5} & … \\ a_{2} t & & a_{0} & & 0 t & & \frac{1}{a_{0}} t^{2} & & 0 t^{3} & & \frac{q 21}{24a_{0}^{3}} t^{4} & & 0 t^{5} & … \\ b t & & 0 & & t & & 0 t^{2} & & \frac{q}{6a_{0}^{2}} t^{3} & & 0 t^{4} & & \frac{8q^{2} 42q 9}{120a_{0}^{4}} t^{5} & … \\ c t & & a_{0} & & 0 t & & \frac{1}{2a_{0}} t^{2} & & 0 t^{3} & & \frac{q 6}{24a_{0}^{3}} t^{4} & & 0 t^{5} & … \\ f t & & 0 & & 6t & & 0 t^{2} & & \frac{2 q 3}{a_{0}^{2}} t^{3} & & 0 t^{4} & & \frac{11q^{2} 54q 123}{10a_{0}^{4}} t^{5} & … \end{matrix}

(5.61)

where q is a free parameter. Next, we consider the equations (5.39) under the assumption that b 0 f 0 0 and a_{1} 0 a_{2} 0 . By a short calculation, we obtain a system of quadratic equations for the first derivatives of the metric functions at t 0. b^{}0 and f^{}0 have to be in ℝ 0 for geometric reasons. Otherwise, the length of a great circle on the universal cover S^{3} of K U 1 _{1,1} would be O t^{2} for small t and the metric thus would not be smooth. Therefore, we can exclude some solutions of the quadratic system. Nevertheless, the two possible solutions a_{1}^{}0 , a_{2}^{}0 , b^{}0 , c^{}0 , f^{}0 0, 0, 1, 0, 6 and a_{1}^{}0 , a_{2}^{}0 , b^{}0 , c^{}0 , f^{}0 0, 0, 1, 0, 6 remain. It is easy to see that if a_{1} t , a_{2} t , b t , c t , f t is a solution of (5.39), a_{2} t , a_{1} t , b t , c t , f t is a solution of (5.39), too. Since this symmetry maps a solution with b^{}0 1 into a solution with b^{}0 1, we only need to consider the case where b^{}0 1. In this case, we obtain:

\begin{matrix}a_{1} t & & a_{0} & & 0 t & & \frac{1}{a_{0}} t^{2} & & 0 t^{3} & & \frac{3q 23}{24a_{0}^{3}} t^{4} & & 0 t^{5} & … \\ a_{2} t & & a_{0} & & 0 t & & \frac{q 2}{a_{0}} t^{2} & & 0 t^{3} & & \frac{12q^{2} 25q 7}{24a_{0}^{3}} t^{4} & & 0 t^{5} & … \\ b t & & 0 & & t & & 0 t^{2} & & \frac{1}{6a_{0}^{2}} t^{3} & & 0 t^{4} & & \frac{39q^{2} 114q 25}{240a_{0}^{4}} t^{5} & … \\ c t & & a_{0} & & 0 t & & \frac{q}{2a_{0}} t^{2} & & 0 t^{3} & & \frac{3q^{2} 13q 3}{24a_{0}^{3}} t^{4} & & 0 t^{5} & … \\ f t & & 0 & & 6t & & 0 t^{2} & & \frac{4}{a_{0}^{2}} t^{3} & & 0 t^{4} & & \frac{3q^{2} 18q 175}{20a_{0}^{4}} t^{5} & … \end{matrix}

(5.62)

As usual, q can be chosen freely. Our aim is to prove that for any of the three initial value problems and any choice of the parameters a_{0}, b_{0}, q, q_{1}, q_{2} there exists a unique smooth metric which solves the initial value problem. In order to do this, we need a condition which guarantees the smoothness of the metric. We take a look at the power series which we have found and see that the Taylor polynomials of low order which correspond to the horizontal part of the metric are even functions and those which correspond to the vertical part are odd. We therefore propose the following sufficient conditions for the smoothness:

1. Let M, g be a cohomogeneity-one orbifold whose principal orbit is an Aloff-Wallach space which is N^{1,1}. Furthermore, let the matrix representation of g with respect to the basis e_{1}, …, e_{7} from page 150 be of type (5.31). If

(a) a, b, c, and f are analytic,

(b) a 0 f 0 0,

(c) b 0 ^{2} c 0 ^{2},

(d) the metric on the collapsing space K U 1 _{k,l} has sectional curvature \frac{1}{t^{2}} O \frac{1}{t} for small t,

(e) a, f are odd, and

(f) b, c are even,

g can be smoothly extended to the singular orbit ℂℙ^{2}.

2. Let M, g be a cohomogeneity-one orbifold whose principal orbit is N^{1,1}. Furthermore, let the matrix representation of g with respect to the basis e_{1}, …, e_{7} from page 150 be a diagonal matrix of type (5.33). If

(a) a_{1}, a_{2}, b, c, and f are analytic,

(b) a_{1} 0 a_{2} 0 f 0 0,

(c) b 0 ^{2} c 0 ^{2},

(d) the metric on the collapsing space K U 1 _{1,1} S^{3} ℤ_{2} has sectional curvature \frac{1}{t^{2}} O \frac{1}{t} for small t, (e) a_{1}, a_{2}, f are odd, and

(f) b, c are even,

g can be smoothly extended to the singular orbit ℂℙ^{2}.

3. Let M, g be a cohomogeneity-one manifold whose principal orbit is N^{1,1}. Furthermore, let the matrix representation of g with respect to the basis e_{1}, …, e_{7} from page 150 be a diagonal matrix of type (5.33). If

(a) a_{1}, a_{2}, b, c, and f are analytic,

(b) b 0 f 0 0,

(c) a_{1}^{2} 0 a_{2}^{2} 0 c^{2} 0 ,

(d) the metric on the collapsing sphere K U 1 _{1,1} S U 2 U 1 U 1 _{1, 2} S^{3} has sectional curvature \frac{1}{t^{2}} O \frac{1}{t} for small t,

(e) b, f are odd, and

(f) a_{1}, a_{2}, c are even,

g can be extended smoothly to the singular orbit ℂℙ^{2}.

We will work with Theorem 3.2.18. If we are able to show that the maps from S^{m} 𝔭^{}to S^{2} 𝔭 (or S^{2} 𝔭^{}) which correspond to the even derivatives of the horizontal (or vertical) part of the metric are K-equivariant, our statement follows by the same arguments as in Section 5.2 and 5.3. We therefore will describe those maps in detail. It is possible to consider all of the three cases simultaneously. We start with the vertical part g^{v} of the metric. Let f_{i} _{i I} be the family of the metric functions which correspond to g^{v}. We want to prove that we can choose the f_{i} as arbitrary odd functions. Analogously to the previous sections, it suffices to prove that the even derivatives of g^{v} can be chosen freely. Let m 1. As we have explained in Section 3.2, there exists a certain map ϕ : S^{2m} 𝔭^{}S^{2} 𝔭^{}which is associated to the 2m^{th} derivative of g^{v}. ϕ satisfies:

ϕ  ^{m} dt dt  \frac{^{2m}}{t^{2m}}g^{v}

where t is the radial direction in the normal space 𝔭^{}. On the orthogonal complement of m dt dt , ϕ vanishes. dt dt is invariant with respect to the action of K. The orbit of the K-action on any v 𝔭^{}0 is K-equivariantly diffeomorphic to K U 1 _{k,l}. Since the f_{i} define a K-invariant metric on K U 1 _{k,l}, the right hand side of the above equation is K-invariant, too. We thus can conclude that ϕ is a K-equivariant map. Next, we prove the smoothness condition for the horizontal part of the metric. This can be done by a geometric argument. Since we have to prove the smoothness of the metric at the singular orbit only, we can restrict our considerations to a tubular neighborhood of the singular orbit. That neighborhood has the structure of a fiber bundle over ℂℙ^{2}. We define a bilinear form g_{h} on that bundle which coincides in the horizontal direction with g and vanishes in the vertical direction. Since SU 3 acts transitively on the set of the fibers of the bundle, we only need to consider the restriction of g_{h} to a single fiber. We describe a construction which also yields g_{h}: Let γ be a geodesic which intersects all orbits perpendicularly. Without loss of generality, we assume that γ 0 ℂℙ^{2}. We first define g_{h} on γ only. By the action of K, g_{h} is extended to a tensor field on a fiber of the tubular neighborhood. Let k K be an element which maps γ t to γ t . Since we assume that g_{h} γ is even, k. g_{h} γ t g_{h} γ t . Since we furthermore assume that g_{h} γ 0 is K-invariant, the extension of g_{h} by the action of K is well-defined. Since g_{h} γ is a smooth function and k^{2m} U 1 _{k,l} K, the derivative \frac{^{2m}}{t^{2m}} _{t 0} g_{h} γ is well-defined, too. We conclude that the map ϕ : S^{2m} 𝔭^{}S^{2} 𝔭 which corresponds to the 2m^{th} derivative of g_{h} is also well-defined. Its K-equivariance follows from the construction which we have made and we have proven our statement.

In order to check if our power series satisfy the smoothness conditions, we have to calculate the values which the f_{i}^{}0 have to take. Let q be the metric with constant sectional curvature 1 on K U 1 _{k,l}. As in Section 5.2 and 5.3, we need to have:

\frac{}{t} ; ; ; _{t 0} e_{i} _{g} e_{i} _{q}

for all e_{i} whose length vanishes at the singular orbit. Before we can check if the above equation is satisfied, we have to describe q explicitly. The metric q on SU 2 with constant sectional curvature 1 is given by q X, Y \frac{1}{2}tr XY for all X, Y 𝔰𝔲 2 . The map π : SU 2 K U 1 _{k,l} with π k : kU 1 _{k,l} where SU 2 is canonically embedded into K SU 3 is a covering map. q therefore has to satisfy:

dπ e_{i} _{q} e_{i} q .

With help of this formula, we see that

1. in the case where a 0 f 0 0, e_{1} _{q} e_{2} _{q} 1 and a^{}0 thus has to equal 1,

2. in the case where a_{1} 0 a_{2} 0 f 0 , e_{1} _{q} e_{2} _{q} 1 and a_{1}^{}0 and a_{2}^{}0 thus have to equal 1,

3. in the case where b 0 f 0 0, e_{3} _{q} e_{4} _{q} 1 and b^{}0 thus has to equal 1.

In all three cases, these conditions are indeed satisfied. We next consider the condition on f^{}0 . The matrix diag i, i 𝔰𝔲 2 which is of unit length is mapped by dπ in the first two cases to

{ \atop }\begin{matrix}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 0 \end{matrix} { \atop }𝔲 1 _{k,l} T_{eU 1 _{k,l}}N^{k,l}

and in the third case to

{ \atop }\begin{matrix}i & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & i \end{matrix} { \atop }𝔲 1 _{1,1} T_{eU 1 _{1,1}}N^{1,1} .

We project the above two matrices onto 𝔪 and see that the two tangent vectors can be identified with

\frac{k l}{2Δ} e_{7} 𝔪 and \frac{1}{6}e_{7} 𝔪 .

These vectors are of unit length with respect to q. It therefore follows that

1. in the case where a 0 f 0 0, e_{7} _{q} \frac{2Δ}{k l} and f^{}0 thus has to equal \frac{2Δ}{k l},

2. in the case where a_{1} 0 a_{2} 0 f 0 , e_{7} _{q} 3 and f^{}0 thus has to equal 3,

3. in the case where b 0 f 0 0, e_{7} _{q} 6 and f^{}0 thus has to equal 6.

Since f^{}0 indeed takes these values, the first summands of our power series satisfy all of the smoothness conditions. We will prove that for any choice of the initial metric and the free parameters q, q_{1}, and q_{2} there exists a unique formal power series solution of our equations which satisfies the smoothness conditions. The cohomogeneity-one metric g g_{t} dt^{2}, which has to be analytic, is determined by the power series

g_{t} _{m 0}^{}g_{m}t^{m} .

The coefficient g_{m} satisfies an equation of type

L_{m} g_{m} P_{m} g_{0}, …, g_{m 1} ,

where L_{m} is a linear map and P_{m} is a polynomial. As in the other cases which we have considered in this section, we will show our result by calculating the rank of L_{m}. The smoothness conditions we will prove with help of symmetry arguments. All of these steps can be carried out no matter if Assumption 3.2.19 is satisfied. In the cases where that assumption holds, the convergence of the power series follows directly from the results of Eschenburg and Wang. In the other cases, the convergence follows by the same argument as we have made earlier in this section. The reason again is that the cohomogeneity-one Einstein condition is an equation which operates on the space S^{2} 𝔭 S^{2} 𝔭^{}if the metric is diagonal.

We will carry out the program which we have outlined above for each of our cases separately and start with the case where N^{k,l} N^{1,1} and a 0 f 0 0. The system (5.36) can be rewritten as:

\begin{matrix}a^{}abc &  & b^{2} c^{2} a^{2} a \frac{k l}{2Δ} bcf \\ ab^{}bc &  & a^{2} c^{2} b^{2} b \frac{l}{2Δ}acf \\ abc^{}c &  & a^{2} b^{2} c^{2} c \frac{k}{2Δ}abf \\ a^{2}b^{2}c^{2}f^{} &  & \frac{k l}{2Δ} b^{2}c^{2}f^{2} \frac{l}{2Δ}a^{2}c^{2}f^{2} \frac{k}{2Δ}a^{2}b^{2}f^{2} \end{matrix}  (5.63)

We assume that m 2 and take the m^{th} derivatives of the first three equations and the m 1 ^{st} derivative of the fourth equation. Analogously to the cases where the singular orbit is SU 3 U 1 ^{2} or S^{5}, we obtain:

\begin{matrix}m 1 b_{0}^{2}a^{m} 0 & & 2b_{0}^{2}a^{m} 0 \frac{k l}{2Δ} b_{0}^{2}f^{m} 0 R_{1} \\ mb_{0}^{2}b^{m} 0 & & 2b_{0}c^{m} 0 2b_{0}b^{m} 0 b_{0} R_{2} \\ mb_{0}^{2}c^{m} 0 & & 2b_{0}b^{m} 0 2b_{0}c^{m} 0 b_{0} R_{3} \\ 2 m 1 \frac{2Δ}{k l} b_{0}^{4}a^{m} 0 m m 1 b_{0}^{4}f^{m} 0 & & 2 m 1 \frac{k l}{2Δ} \frac{2Δ}{k l} b_{0}^{4}f^{m} 0 R_{4} \end{matrix}

In the above system, R_{1}, …, R_{4} are polynomials which depend on the lower derivatives of a, b, c, and f only. We have already inserted b 0 c 0 b_{0}, a^{}0 1, and f^{}0 \frac{2Δ}{k l}. The reason for the factor m m 1 in the fourth equation is that the term a^{2}b^{2}c^{2}f^{m} 0 has the multiplicity m m 1 if we apply the product rule m 1 times to a^{2}b^{2}c^{2}f^{}. We obtain the following linear system for the m^{th} derivatives of the functions a, b, c, and f:

; ; ; \begin{matrix}m 1 b_{0}^{2} & 0 & 0 & \frac{k l}{2Δ} b_{0}^{2} \\ 0 & m 2 b_{0}^{2} & 2b_{0}^{2} & 0 \\ 0 & 2b_{0}^{2} & m 2 b_{0}^{2} & 0 \\ m 1 \frac{4Δ}{k l} b_{0}^{4} & 0 & 0 & m 2 m 1 b_{0}^{4} \end{matrix} ; ; ; ; ; ; \begin{matrix}a^{m} 0 \\ b^{m} 0 \\ c^{m} 0 \\ f^{m} 0 \end{matrix} ; ; ; ; ; ; \begin{matrix}R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \end{matrix} ; ; ;

By a short calculation, we see that the matrix on the left hand side has rank 4, except for m 3, where the rank is 3. Since we have already calculated the power series (5.59), we know that the solution space of the above linear system is one-dimensional if m 3. We therefore have proven that there exists a formal power series solution of (5.36) for any choice of b_{0} and q. Moreover, that solution is unique. q thus is the only free parameter even in the cases where 𝔭 and 𝔭^{}have an equivalent U 1 _{k,l}-submodule in common. If we replace a t , b t , c t , f t by a t , b t , c t , f t , we again obtain a solution of (5.36) with the same values of b 0 and f^{}0 . Therefore, we have a t a t , b t b t , c t c t , f t f t and the power series satisfies all of the smoothness conditions. Since the power series converges, we finally have proven the existence of a family of local cohomogeneity-one metrics with holonomy Spin 7 .

Next, we turn to the case where k l 1 and a_{1} 0 a_{2} 0 f 0 0. For our calculations, we rewrite the system (5.39) as:

\begin{matrix}a_{1}^{}a_{2}bcf &  & b^{2} c^{2} a_{1}^{2} a_{2}f 3 a_{1}^{2} a_{2}^{2} bc \frac{1}{3}bcf^{2} \\ a_{1}a_{2}^{}bcf &  & b^{2} c^{2} a_{2}^{2} a_{1}f 3 a_{2}^{2} a_{1}^{2} bc \frac{1}{3}bcf^{2} \\ a_{1}a_{2}b^{}bc &  & \frac{1}{2} a_{1}^{2} c^{2} b^{2} a_{2}b \frac{1}{2} a_{2}^{2} c^{2} b^{2} a_{1}b \frac{1}{6}a_{1}a_{2}cf \\ a_{1}a_{2}bc^{}c &  & \frac{1}{2} a_{1}^{2} b^{2} c^{2} a_{2}c \frac{1}{2} a_{2}^{2} b^{2} c^{2} a_{1}c \frac{1}{6}a_{1}a_{2}bf \\ a_{1}a_{2}b^{2}c^{2}f^{} &  & 3 a_{1} a_{2} ^{2}b^{2}c^{2} \frac{1}{3}b^{2}c^{2}f^{2} \frac{1}{6}a_{1}a_{2}c^{2}f^{2} \frac{1}{6}a_{1}a_{2}b^{2}f^{2} \end{matrix}  (5.64)

We assume that m 2 and consider the m 1 ^{st} derivatives of the above five equations at t 0. As in the previous case, we directly insert the values of b 0 , c 0 , a_{1}^{}0 , a_{2}^{}0 , and f^{}0 , which are determined by the power series (5.60), and obtain the following equations:

3m m 1 b_{0}^{2}a_{1}^{m} 0 3 m 1 b_{0}^{2}a_{2}^{m} 0 m 1 b_{0}^{2}f^{m} 0  2 m 1 b_{0}^{2} 3a_{2}^{m} 0 f^{m} 0
6 m 1 b_{0}^{2} a_{1}^{m} 0 a_{2}^{m} 0
2 m 1 b_{0}^{2}f^{m} 0 R_{1}
3 m 1 b_{0}^{2}a_{1}^{m} 0 3m m 1 b_{0}^{2}a_{2}^{m} 0 m 1 b_{0}^{2}f^{m} 0  2 m 1 b_{0}^{2} 3a_{1}^{m} 0 f^{m} 0
6 m 1 b_{0}^{2} a_{2}^{m} 0 a_{1}^{m} 0
2 m 1 b_{0}^{2}f^{m} 0 R_{2}
m m 1 b_{0}^{2}b^{m} 0  2 m 1 b_{0}^{2}c^{m} 0 b_{0}^{2}b^{m} 0
R_{3}
m m 1 b_{0}^{2}c^{m} 0  2 m 1 b_{0}^{2}b^{m} 0 b_{0}^{2}c^{m} 0
R_{4}
m m 1 b_{0}^{4}f^{m} 0 3 m 1 b_{0}^{4}a_{1}^{m} 0 3 m 1 b_{0}^{4}a_{2}^{m} 0  2 m 1 b_{0}^{4}f^{m} 0 R_{5}

In the above system, R_{1}, …, R_{5} are polynomials which depend on the lower derivatives of the coefficients of the metric only. The factors of type m m 1 appear for the same reason as in the previous case. Our equations are equivalent to the following linear system:

L_{m}  ;  ;  ;  ;  ;  ; \begin{matrix}a_{1}^{m} 0 \\ a_{2}^{m} 0 \\ b^{m} 0 \\ c^{m} 0 \\ f^{m} 0 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \end{matrix}  ;  ;  ;  ;  ; (5.65)

where

L_{m} ; ; \begin{matrix}3m 6 m 1 b_{0}^{2} & 3 m 1 b_{0}^{2} & 0 & 0 & m 1 b_{0}^{2} \\ 3 m 1 b_{0}^{2} & 3m 6 m 1 b_{0}^{2} & 0 & 0 & m 1 b_{0}^{2} \\ 0 & 0 & m 1 m 2 b_{0}^{2} & 2 m 1 b_{0}^{2} & 0 \\ 0 & 0 & 2 m 1 b_{0}^{2} & m 1 m 2 b_{0}^{2} & 0 \\ 3 m 1 b_{0}^{4} & 3 m 1 b_{0}^{4} & 0 & 0 & m 2 m 1 b_{0}^{4} \end{matrix} ; ;

The rank of L_{3} is 3 and for all other m 2 it is 5. Since we have been able to calculate the power series expansion (5.60) up to fifth order, the system (5.65) has a solution for m 3. Our initial value problem thus has a formal power series solution for any choice of b_{0}, q_{1}, and q_{2}. Moreover, we have proven that the two free parameters of third order are indeed the only ones. If we replace a_{1} t , a_{2} t , b t , c t , f t by a_{1} t , a_{2} t , b t , c t , f t , we obtain a new solution of (5.39) with the same values of b 0 , a_{1}^{}0 , and a_{2}^{}0 . We conclude that b and c are even and that a_{1}, a_{2}, and f are odd. All of the smoothness conditions thus are satisfied. The system (5.39) is also preserved if we interchange b and c. By the same arguments as above, we can conclude that any solution of our initial value problem satisfies b t c t . With help of the convergence result of Eschenburg and Wang [32], it follows that the power series converges to a metric whose holonomy is contained in Spin 7 .

Finally, we consider the case where k l 1 and b 0 f 0 0. There are two subcases, namely a_{2} 0 a_{1} 0 and a_{2} 0 a_{1} 0 . The subcase a_{2} 0 a_{1} 0 we investigate first.

We take the m 1 ^{st} derivatives of the first, second, and fifth equation of (5.64) and the m^{th} derivatives of the third and fourth equation. As usual, we assume that m 2 and obtain the following equations:

6m m 1 a_{0}^{2}a_{1}^{m} 0  6 m 1 a_{0} 2a_{0}c^{m} 0 2a_{0}a_{1}^{m} 0
3 m 1 a_{0} 2a_{0}a_{1}^{m} 0 2a_{0}a_{2}^{m} 0
R_{1}
6m m 1 a_{0}^{2}a_{2}^{m} 0  6 m 1 a_{0} 2a_{0}c^{m} 0 2a_{0}a_{2}^{m} 0
3 m 1 a_{0} 2a_{0}a_{2}^{m} 0 2a_{0}a_{1}^{m} 0
R_{2}
m 1 a_{0}^{3}b^{m} 0  2a_{0}^{3}b^{m} 0 \frac{1}{6}a_{0}^{3}f^{m} 0 R_{3}
ma_{0}^{3}c^{m} 0  a_{0}^{2} a_{0}a_{1}^{m} 0 a_{0}c^{m} 0
a_{0}^{2} a_{0}a_{2}^{m} 0 a_{0}c^{m} 0 R_{4}
12 m 1 a_{0}^{4}b^{m} 0 m m 1 a_{0}^{4}f^{m} 0  2 m 1 a_{0}^{4}f^{m} 0 R_{5}

where R_{1}, …, R_{5} are polynomials which depend on the lower derivatives of a_{1}, a_{2}, b, c, and f only. The above equations are equivalent to a linear system of type (5.65) where

L_{m} ; ; \begin{matrix}6 m 1 m 3 a_{0}^{2} & 6 m 1 a_{0}^{2} & 0 & 12 m 1 a_{0}^{2} & 0 \\ 6 m 1 a_{0}^{2} & 6 m 1 m 3 a_{0}^{2} & 0 & 12 m 1 a_{0}^{2} & 0 \\ 0 & 0 & m 1 a_{0}^{3} & 0 & \frac{1}{6} a_{0}^{3} \\ a_{0}^{3} & a_{0}^{3} & 0 & m 2 a_{0}^{3} & 0 \\ 0 & 0 & 12 m 1 a_{0}^{4} & 0 & m 2 m 1 a_{0}^{4} \end{matrix} ; ;

For m 3 the rank of L_{m} is 4. Since (5.61) is a power series expansion up to fifth order, the system (5.65) has in that case a solution. If m 2 or m 4, L_{m} is of full rank. Therefore, there exists for any choice of a_{0} and q a unique formal power series solution of our initial value problem. Since we can replace a_{1} t , a_{2} t , b t , c t , f t by a_{1} t , a_{2} t , b t , c t , f t without changing the system (5.39), we can prove by the same methods as in the previous cases that a_{1}, a_{2}, and c are even and that b and f are odd. The power series (5.61) converges and we thus have proven the existence of a family of local cohomogeneity-one metrics whose holonomy is contained in Spin 7 . From the fact that the system (5.39) is preserved if we interchange a_{1} and a_{2}, we can conclude that all of those metrics satisfy a_{1} t a_{2} t .

Next, we investigate the subcase a_{2} 0 a_{1} 0 . We again take the m 1 ^{st} derivatives of the first, second, and fifth equation of (5.64) and the m^{th} derivatives of the other two equations. As before, we assume without loss of generality that b^{}0 1. We obtain the following five equations:

6m m 1 a_{0}^{2}a_{1}^{m} 0  6 m 1 a_{0} 2a_{0}c^{m} 0 2a_{0}a_{1}^{m} 0
3 m 1 a_{0} 2a_{0}a_{1}^{m} 0 2a_{0}a_{2}^{m} 0
R_{1}
6m m 1 a_{0}^{2}a_{2}^{m} 0  6 m 1 a_{0} 2a_{0}c^{m} 0 2a_{0}a_{2}^{m} 0
3 m 1 a_{0} 2a_{0}a_{2}^{m} 0 2a_{0}a_{1}^{m} 0
R_{2}
m 1 a_{0}^{3}b^{m} 0  \frac{1}{6}a_{0}^{3}f^{m} 0 R_{3}
ma_{0}^{3}c^{m} 0  a_{0}^{2} a_{0}a_{1}^{m} 0 a_{0}c^{m} 0
a_{0}^{2} a_{0}a_{2}^{m} 0 a_{0}c^{m} 0 R_{4}
12 m 1 a_{0}^{4}b^{m} 0 m m 1 a_{0}^{4}f^{m} 0  24 m 1 a_{0}^{4}b^{m} 0
2 m 1 a_{0}^{4}f^{m} 0 R_{5}

where R_{1}, …, R_{5} are polynomials which depend on the lower derivatives of a_{1}, a_{2}, b, c, and f only. These equations are equivalent to a linear system of type (5.65) where

L_{m} ; ; \begin{matrix}6 m 1 m 3 a_{0}^{2} & 6 m 1 a_{0}^{2} & 0 & 12 m 1 a_{0}^{2} & 0 \\ 6 m 1 a_{0}^{2} & 6 m 1 m 1 a_{0}^{2} & 0 & 12 m 1 a_{0}^{2} & 0 \\ 0 & 0 & m 1 a_{0}^{3} & 0 & \frac{1}{6} a_{0}^{3} \\ a_{0}^{3} & a_{0}^{3} & 0 & ma_{0}^{3} & 0 \\ 0 & 0 & 12 m 1 a_{0}^{4} & 0 & m 1 m 2 a_{0}^{4} \end{matrix} ; ;

The rank of L_{m} is 5 for all m 3 and for m 2 it is 4. Since we have calculated the power series (5.62) up to fifth order, the system (5.65) has indeed a solution for m 2. We thus have proven that there exists a unique power series solution of our initial value problem for any choice of a_{0} and q. By the symmetry of (5.39) which we have used in the previous subcase, we can show that a_{1}, a_{2}, and c are even and that b and f are odd. Therefore, the power series satisfies all of the smoothness conditions. Since the series converges, we have proven the existence of a further family of local cohomogeneity-one metrics whose holonomy is contained in Spin 7 .

We will not calculate the dimensions of the spaces W_{m}^{h} and W_{2}^{v}. Nevertheless, we are able to make some statements on the number of cohomogeneity-one Einstein metrics whose principal orbit is an Aloff-Wallach space and whose singular orbit is ℂℙ^{2}. There are choices of k and l such that Assumption 3.2.19 is not satisfied. The action of the group U 1 _{k,l} splits 𝔪 into

𝕍_{k l} 𝕍_{2k l} 𝕍_{k 2l} ℝ ,

where 𝕍_{r} denotes the two-dimensional module on which U 1 _{k,l} acts with weight r. The action of U 1 _{k,l} on ℝ of course is trivial. We assume that 𝔨 𝔲 1 _{k,l} V_{1} V_{4}. In this situation, we have the following splittings:

𝔭^{}𝕍_{0} 𝕍_{k l} and 𝔭 𝕍_{2k l} 𝕍_{k 2l} .

The normal space 𝔭^{}is not a vector space but a quotient of a four-dimensional space by a group of type ℤ_{k l}. The above formula for the splitting of 𝔭^{}thus should be understood as follows: The action of U 1 _{k,l} on 𝔭^{}can be lifted to an action on a four-dimensional vector space which splits into 𝕍_{0} 𝕍_{k l} . Since the action on 𝔭^{}can be lifted, Theorem 3.2.24 also holds with the obvious replacements, although the underlying space is an orbifold instead of a manifold. Assumption 3.2.19 is violated if and only if one of the modules 𝕍_{2k l} or 𝕍_{k 2l} is trivial or equivalent to 𝕍_{k l} . By a short calculation, we see that this is equivalent to

k, l 0, 1 , 1, 2 , 1, 0 , 2, 1 .

The case, where k l 1 and b 0 f 0 0, is equivalent to 𝔨 𝔲 1 _{1, 2} V_{1} V_{4}. Therefore, our assumption is in that situation not satisfied. If k l 0 and a 0 f 0 0, Assumption 3.2.19 is not satisfied, too. We have justified why Theorem 3.2.24 also holds if 3.2.19 is not true but the metric and its Ricci-tensor still take their values in S^{2} 𝔭 S^{2} 𝔭^{}. Moreover, we show in Appendix C.3 and C.4 that this weaker assumption is satisfied if the metric is diagonal. For this reason, we notwithstanding apply the theorem to all of the cases which we have considered. By taking a look at the sufficient smoothness conditions which we have found, we find lower bounds on dim W_{m}^{h} and dim W_{2}^{v}. In all of our cases, the metric on the singular orbit is described by a single parameter. We therefore have dim W_{0}^{h} 1. The second derivatives of the functions which correspond to the horizontal part of the metric can be chosen arbitrarily without violating the smoothness conditions. The number of those functions thus yields a lower bound on dim W_{2}^{h}. Since the third derivatives of the functions which correspond to the vertical part of the metric can be chosen arbitrarily, too, we also have a lower bound on dim W_{2}^{v}. All in all, we can at least prescribe the following initial values:

1. If N^{k,l} N^{1,1} and a 0 f 0 0, we can choose

(a) b 0 c 0 ,

(b) b c ^{}0 ,

(c) a^{}0 , and f^{}0

arbitrarily.

2. If k l 1 and a_{1} 0 a_{2} 0 f 0 0, we can choose

(a) b 0 c 0 ,

(b) b c ^{}0 ,

(c) a_{1}^{}0 , a_{2}^{}0 , and f^{}0

arbitrarily.

3. If k l 1 and b 0 f 0 0, we can choose

(a) a_{1} 0 a_{2} 0 c 0 ,

(b) two elements of a_{1}^{}0 , a_{2}^{}0 , f^{}0 ,

(c) b^{}0 , and f^{}0

arbitrarily.

As usual, the Einstein constant can be fixed at an arbitrary value, too. The above considerations put the free parameters q, q_{1}, q_{2} of our metrics in the context of the research of Eschenburg and Wang [32].

In the following, we consider SU 2 as a subgroup of K. Moreover, we consider Hom_{SU 2} S^{m 2} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}as a subset of Hom_{SU 2} S^{m} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}. S^{2} 𝔭 and S^{2} 𝔭^{}both split with respect to the action of SU 2 into 3𝕍_{2}^{ℝ} 𝕍_{0}^{ℝ}. If m is odd, S^{m} 𝔭^{} decomposes into SU 2 -submodules with odd weights only. If m 2 is even, the complement of S^{m} 𝔭^{}S^{m 2} 𝔭^{}decomposes into SU 2 -submodules whose weights are greater than 2. With help of these facts, we can easily show that

Hom_{SU 2} S^{m 2} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}Hom_{SU 2} S^{m} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}

for all m 1. Let φ : S^{m 2} 𝔭^{}S^{2} 𝔭 S^{2} 𝔭^{}be a linear, K-equivariant map. φ has to be SU 2 -equivariant, too. From the above formula it follows that φ has to vanish on the complement of S^{m} 𝔭^{}in S^{m 2} 𝔭^{}. This argument proves that

Hom_{K} S^{m 2} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}Hom_{K} S^{m} 𝔭^{}, S^{2} 𝔭 S^{2} 𝔭^{}

for all m 1. Therefore, we have dim W_{m 2}^{h} dim W_{m 2}^{v} 0. We conclude with help of Theorem 3.2.24 that in the diagonal case where we can apply that theorem there are no free initial conditions of higher order than those which we have found in the previous paragraph. The focus of this thesis is on metrics whose holonomy is contained in Spin 7 . Therefore, we will not further discuss the issue of cohomogeneity-one Einstein metrics in this context.

Since we have not explicitly solved the systems (5.36) and (5.39), we cannot prove under which conditions the metrics from this section are complete. Moreover, the arguments which we have made in Section 5.2 and 5.3 do not exclude the existence of compact solutions in our situation. We will illustrate this by an example: Let the principal orbit of the cohomogeneity-one orbifold which we consider be a generic Aloff-Wallach space N^{k,l} with k l 0. Moreover, we assume that there are two singular orbits which are located at t 0 and t T for a T 0. If a 0 f 0 0, we have f^{}0 \frac{2Δ}{k l} 0. If in addition b T f T 0, it follows that f^{}T \frac{2Δ}{l} 0. In this situation, we cannot conclude that any of the functions a, b, c, or f has an additional zero in 0, T . Therefore, we can neither prove nor exclude the existence of compact examples by our methods. We finally sum up the results of this section:

Theorem 5.4.6. Let M, Ω be a Spin 7 -orbifold with a cohomogeneity-one action of SU 3 which preserves Ω. The metric associated to Ω we denote by g. In this situation, the following statements are true:

1. All principal orbits are SU 3 -equivariantly diffeomorphic to an Aloff-Wallach space N^{k,l}. We assume from now on that k, l is coprime.

2. (a) If N^{k,l} is generic, the matrix representation of g with respect to the basis e_{1}, …, e_{7} from page 150 is of type (5.31).

(b) If k 1 and l 0, the matrix representation of g with respect to the basis e_{1}, e_{2}, e_{5}, e_{6}, e_{3}, e_{4}, e_{7} is of type (5.32).

(c) If k 1 and l 1, the matrix representation of g with respect to the basis e_{1}, e_{2}, e_{7}, e_{3}, e_{4}, e_{5}, e_{6} is of type (5.33).

3. (a) We assume that N^{k,l} is generic or N^{1,0} and that g is diagonal with respect to the basis from page 150. Furthermore, we assume that there is a continuous map γ : 0, 1 Γ ^{4} T^{}M such that γ 0 Ω, γ 1 is a four-form of type (5.35), and γ t is for all t 0, 1 a SU 3 -invariant four-form with the same associated metric as Ω. In this situation, Ω already is a four-form of type (5.35) or Ω is not parallel.

(b) We assume that k 1, l 1, and that g is diagonal. If Ω is of type (5.43), there exists a parallel four-form of type (5.38) on M which has the same associated metric and orientation as Ω or Ω is not parallel.

4. (a) If N^{k,l} is generic or N^{1,0} and Ω is of type (5.35), Ω is parallel if and only if the

system (5.36) is satisfied. In that case, the holonomy of g is all of Spin 7 .

(b) If k l 1 and Ω is of type (5.38), Ω is parallel if and only if the system (5.39) is satisfied. In that case, the holonomy of g is contained in Spin 7 . If a_{1} a_{2} and a_{1}^{2} b^{2} c^{2}, the holonomy is contained in SU 4 .

If M has a singular orbit, which has to be the case if M, Ω is parallel and complete, it is SU 3 U 1 ^{2}, S^{5}, SU 3 SO 3 , or ℂℙ^{2}. If the singular orbit is S^{5} or SU 3 SO 3 , N^{k,l} has to be N^{1,0}. If the singular orbit is SU 3 U 1 ^{2}, there are no restrictions on k and l. If the singular orbit is ℂℙ^{2}, we can assume without loss of generality that the function a vanishes at the singular orbit (or if k l 1, a_{1} and a_{2} vanish at the singular orbit). In this situation, the only restriction on k and l is that k l 0. For any choice of the principal and the singular orbit, the orbifold M is a manifold, except in the case where the singular orbit is ℂℙ^{2} and k l 1. In that case, M is a D^{4} ℤ_{k l}-bundle over ℂℙ^{2}, where D^{4} is a disc in ℝ^{4} and ℤ_{k l} is generated by:

{ \atop }\begin{matrix}e^{\frac{2πi}{k l}} & 0 \\ 0 & e^{\frac{2πi}{k l}} \end{matrix} { \atop }.

From now on, we assume that the singular orbit is at t 0.

1. If the singular orbit is SU 3 U 1 ^{2} and N^{k,l} is generic or N^{1,0}, the four-form (5.35) cannot be parallel.

2. We assume that the singular orbit is SU 3 U 1 ^{2}, the principal orbit is N^{1,1}, and f 0 0. For any choice of the initial values a_{1} 0 , a_{2} 0 , b 0 , c 0 ℝ 0 with a_{1} 0 a_{2} 0 0, there exists a unique parallel four-form Ω of type (5.38) for all t 0, ϵ and an ϵ 0. Ω cannot be smoothly extended to the singular orbit. If we replace the principal orbit by N^{1,1} ℤ_{2} where ℤ_{2} is generated by diag i, i, 1 , there also exists a unique parallel four-form Ω for any choice of the initial values. In this new situation, it is possible to extend Ω smoothly to the singular orbit.

3. We assume that the singular orbit is S^{5} and the principal orbit is N^{1, 1} . The space of all SU 3 -invariant metrics on the singular orbit is two-dimensional. For any of those metrics and any choice of the signs of b 0 and c 0 , there exists a unique smooth and parallel four-form Ω of type (5.38) on a tubular neighborhood of the singular orbit.

4. We assume that the singular orbit is SU 3 SO 3 and the principal orbit is N^{1, 1} . N^{1, 1} is SU 3 -equivariantly diffeomorphic to SU 3 SO 2 where the group SO 2 is generated by e_{1}. The embedding of SO 3 into SU 3 shall be the standard one. We fix the basis e_{7}, e_{3}, e_{5}, e_{4}, e_{6}, e_{2}, e_{8} of the tangent space of SU 3 SO 2 . Let Ω be a SU 3 -invariant Spin 7 -structure whose associated metric is diagonal and which can be connected by a path of such structures to a Spin 7 -structure with a frame of type (5.55). In this situation, Ω already is a Spin 7 -structure with a frame of type (5.55). There exists no parallel four-form on a tubular neighborhood of the singular orbit which belongs to the above class of Spin 7 -structures. Moreover, there does not even exist such a parallel four-form on a neighborhood of a principal orbit.

5. We assume that the singular orbit is ℂℙ^{2} and that the principal orbit is a generic Aloff-Wallach space, N^{0, 1} , or N^{1,0}. Furthermore, let a 0 f 0 0. The space of all SU 3 -invariant metrics on the singular orbit is one-dimensional. For any choice of the invariant metric on ℂℙ^{2}, of the signs of b 0 and c 0 , and of f^{}0 , there exists a unique smooth and parallel four-form Ω of type (5.35) on a tubular neighborhood of the singular orbit.

6. We assume that the singular orbit is ℂℙ^{2} and that the principal orbit is N^{1,1}. Furthermore, let a_{1} 0 a_{2} 0 f 0 0. The space of all SU 3 -invariant metrics on the singular orbit is one-dimensional. For any choice of the invariant metric on ℂℙ^{2}, of the signs of b 0 and c 0 , of a_{1}^{}0 , and of a_{2}^{}0 , there exists a unique smooth and parallel four-form Ω of type (5.38) on a tubular neighborhood of the singular orbit. Moreover, we have b t c t for all values of t.

7. We assume that the singular orbit is ℂℙ^{2} and that the principal orbit is N^{1,1}. Furthermore, let b 0 f 0 0 and a_{1} 0 a_{2} 0 . The space of all SU 3 -invariant metrics on the singular orbit is one-dimensional. For any choice of the invariant metric on ℂℙ^{2}, of the signs of a_{1} 0 and c 0 , and of b^{}0 , there exists a unique smooth and parallel four-form Ω of type (5.38) on a tubular neighborhood of the singular orbit. Moreover, we have a_{1} t a_{2} t for all values of t.

8. We assume that the singular orbit is ℂℙ^{2} and that the principal orbit is N^{1,1}. Furthermore, let b 0 f 0 0, a_{1} 0 a_{2} 0 , and b^{}0 1. The space of all SU 3 -invariant metrics on the singular orbit is one-dimensional. For any choice of the invariant metric on ℂℙ^{2}, of the signs of a_{1} 0 and c 0 , and of c^{}0 , there exists a unique smooth and parallel four-form Ω of type (5.38) on a tubular neighborhood of the singular orbit.

Remark 5.4.7. 1. For any of the above results which holds for a principal orbit N^{k,l}, there is an analogous result which holds for the other Aloff-Wallach spaces which are N^{k,l}. We have already discussed in the course of this section that if the principal orbit is covered by N^{1,1}, the singular orbit is SU 3 U 1 ^{2}, and the metric is diagonal, it suffices to consider the initial condition f 0 0. If the isotropy group of the singular orbit is conjugate to SO 3 and the isotropy group of the principal orbit is as usual generated by diag i, i, 0 and not by e_{1}, the metric has to be non-diagonal. The cases where k, l 1, 2 , 2, 1 and a 0 f 0 0 are both equivalent to the initial value problem with k l 1 and b 0 f 0 0 which we have included in our theorem. In the last case we have mentioned, we necessarily have b^{}0 1. We did not need to consider the case b^{}0 1, since there is a symmetry of the system (5.39) which changes b t into b t . For the above reasons, we have investigated all possible initial conditions for the systems (5.36) and (5.39).

2. The metrics which are associated to any of the Spin 7 -structures from the above theorem are diagonal. At least if k l 1, there are further cohomogeneity-one Spin 7 - structures whose associated metric is diagonal. We suppose that those structures have a similar behavior.

With help of Theorem 3.2.24, we also have obtained results on cohomogeneity-one Einstein metrics with an Aloff-Wallach space as principal orbit:

Theorem 5.4.8. Let M be a cohomogeneity-one orbifold whose principal orbit is an Aloff-Wallach space N^{k,l}. We assume that M has a singular orbit, which has to be one of those from Theorem 5.4.6. Let I ℝ be an interval which is isometric to M SU 3 . Any diagonal SU 3 -invariant metric on M is described by the four functions a, b, c, f : I ℝ if N^{k,l} is generic or N^{1, 1} . If k l 1, the metric is described by the functions a_{1}, a_{2}, b, c, f : I ℝ. We denote the off-diagonal coefficients of the metric g by β_{i,j} : g e_{i}, e_{j} . Without loss of generality, we assume that the singular orbit is at 0 I. In this situation, the following statements are true:

1. Let N^{k,l} be generic and the singular orbit be SU 3 U 1 ^{2}. For any choice of a_{0}, b_{0}, c_{0}, f_{3}, λ ℝ, there exists a unique SU 3 -invariant Einstein metric on a tubular neighborhood of SU 3 U 1 ^{2} such that:

(a) a 0 a_{0}, b 0 b_{0}, c 0 c_{0},

(b) f^{}0 f_{3}, and

(c) the Einstein constant is λ.

2. Let k 1, l 0, and the singular orbit be SU 3 U 1 ^{2}. For any choice of a_{0}, b_{0}, c_{0}, { \atop β_{1,5,3}}, { \atop β_{1,6,3}}, f_{3}, λ ℝ, there exists a unique SU 3 -invariant Einstein metric on a tubular neighborhood of SU 3 U 1 ^{2} such that:

(a) a 0 a_{0}, b 0 b_{0}, c 0 c_{0},

(b) β_{1,5}^{}0 { \atop β_{1,5,3}}, β_{1,6}^{}0 { \atop β_{1,6,3}},

(c) f^{}0 f_{3}, and

(d) the Einstein constant is λ.

3. Let N^{k,l} be generic, the singular orbit be ℂℙ^{2}, and a 0 f 0 0. For any choice of b_{0}, b_{2}, a_{3}, f_{3}, λ ℝ, there exists a SU 3 -invariant Einstein metric on a tubular neighborhood of ℂℙ^{2} such that:

(a) b 0 c 0 b_{0},

(b) b c ^{}0 b_{2}, (c) a^{}0 a_{3}, f^{}0 f_{3}, and

(d) the Einstein constant is λ.

4. Let k l 1, the singular orbit be ℂℙ^{2}, and a_{1} 0 a_{2} 0 f 0 0. For any choice of b_{0}, b_{2}, a_{1,3}, a_{2,3}, f_{3}, λ ℝ, there exists a SU 3 -invariant Einstein metric on a tubular neighborhood of ℂℙ^{2} such that:

(a) b 0 c 0 b_{0},

(b) b c ^{}0 b_{2},

(c) a_{1}^{}0 a_{1,3}, a_{2}^{}0 a_{2,3}, f^{}0 f_{3}, and

(d) the Einstein constant is λ.

For the case, where the principal orbit is N^{1,1} ℤ_{2}, we have a similar result:

Theorem 5.4.9. Let M be a cohomogeneity-one manifold with principal orbit N^{1,1} ℤ_{2} where ℤ_{2} is generated by diag i, i, 1 . Since the principal orbit is covered by N^{1,1}, we can use the same notation as in Theorem 5.4.8. We assume that M has SU 3 U 1 ^{2} as a singular orbit, that the singular orbit is located at t 0, and that f 0 0. In this situation, the following statement is true:

For any choice of a_{0}, b_{0}, c_{0}, a_{1}, { \atop β_{1,2,1}}, f_{3}, λ ℝ, there exists a unique SU 3 -invariant Einstein metric on a tubular neighborhood of SU 3 U 1 ^{2} such that:

1. a_{1} 0 a_{2} 0 a_{0}, b 0 b_{0}, c 0 c_{0},

2. a_{1} a_{2} ^{}0 a_{1}, β_{1,2}^{}0 { \atop β_{1,2,1}},

3. f^{}0 f_{3}, and

4. the Einstein constant is λ.

There are certain cases where Assumption 3.2.19 is not satisfied, but Theorem 3.2.24 still holds true. In those cases, we obtain further results on cohomogeneity-one Einstein metrics with an Aloff-Wallach space as principal orbit:

Theorem 5.4.10. Let M be a cohomogeneity-one manifold whose principal orbit is an Aloff-Wallach space N^{k,l}. We assume that M has a singular orbit at t 0 and use the same notation as in Theorem 5.4.8. In this situation, the following statements are true:

1. Let k 1, l 1, and the singular orbit be S^{5}. For any choice of b_{0}, f_{0}, b_{1}, { \atop β_{3,5,1}}, { \atop β_{3,6,1}}, a_{3}, λ ℝ, there exists a unique SU 3 -invariant Einstein metric on a tubular neighborhood of S^{5} such that:

(a) b 0 c 0 b_{0}, f 0 f_{0},

(b) b c ^{}0 b_{1}, β_{3,5}^{}0 { \atop β_{3,5,1}}, β_{3,6}^{}0 { \atop β_{3,6,1}},

(c) a^{}0 a_{3}, and

(d) the Einstein constant is λ.

2. Let k, l 0, 1 , 1, 0 , the singular orbit be ℂℙ^{2}, and a 0 f 0 0. For any choice of b_{0}, b_{2}, a_{3}, f_{3}, λ ℝ, there exists a SU 3 -invariant Einstein metric on a tubular neighborhood of ℂℙ^{2} such that:

(a) b 0 c 0 b_{0},

(b) b c ^{}0 b_{2},

(c) a^{}0 a_{3}, f^{}0 f_{3}, and

(d) the Einstein constant is λ.

3. Let k l 1, the singular orbit be ℂℙ^{2}, and b 0 f 0 0. For any choice of a_{0}, a_{1,2}, a_{2,2}, b_{3}, f_{3}, λ ℝ, there exists a SU 3 -invariant Einstein metric on a tubular neighborhood of ℂℙ^{2} such that:

(a) a_{1} 0 a_{2} 0 c 0 a_{0},

(b) a_{1} a_{2} ^{}0 a_{1,2}, a_{1} a_{2} 2c ^{}0 a_{2,2},

(c) b^{}0 b_{3}, f^{}0 f_{3}, and

(d) the Einstein constant is λ.

There is one case left open for which we have not proven that it is possible to apply Theorem 3.2.24. We thus state the result of our calculations as a conjecture:

Conjecture 5.4.11. In the situation of Theorem 5.4.10, let k l 1, the singular orbit be SU 3 U 1 ^{2}, f 0 0, and β_{1,7} t β_{2,7} t 0 for all t. For any choice of a_{0}, b_{0}, c_{0}, { \atop β_{3,5,1}}, { \atop β_{3,6,1}}, a_{2}, { \atop β_{1,2,2}}, f_{3}, λ ℝ, there exists a unique SU 3 -invariant Einstein metric with β_{1,7} t β_{2,7} t 0 on a tubular neighborhood of SU 3 U 1 ^{2} such that:

1. a_{1} 0 a_{2} 0 a_{0}, b 0 b_{0}, c 0 c_{0},

2. β_{3,5}^{}0 { \atop β_{3,5,1}}, β_{3,6}^{}0 { \atop β_{3,6,1}},

3. a_{1} a_{2} ^{}0 a_{2}, β_{1,2}^{}0 { \atop β_{1,2,2}},

4. f^{}0 f_{3}, and

5. the Einstein constant is λ.

If we specialize the above conjecture to the case of diagonal metrics, we obtain the following result:

Theorem 5.4.12. In the situation of Theorem 5.4.10, let k l 1, the singular orbit be SU 3 U 1 ^{2}, f 0 0, and the metric be diagonal for all t. For any choice of a_{0}, b_{0}, c_{0}, a_{2}, f_{3}, λ ℝ, there exists a unique diagonal SU 3 -invariant Einstein metric on a tubular neighborhood of SU 3 U 1 ^{2} such that:

1. a_{1} 0 a_{2} 0 a_{0}, b 0 b_{0}, c 0 c_{0},

2. a_{1} a_{2} ^{}0 a_{2}, 3. f^{}0 f_{3}, and

4. the Einstein constant is λ.

Remark 5.4.13. 1. If we could show that Ric e_{1}, e_{7} Ric e_{2}, e_{7} 0 for any cohomogeneityone metric with principal orbit N^{1,1} and β_{1,7} t β_{2,7} t 0, Conjecture 5.4.11 would be proven. Alternatively, it may be possible to generalize the Theorem of Eschenburg and Wang [32] to arbitrary cohomogeneity-one Einstein manifolds which do not necessarily satisfy Assumption 3.2.19. The generalized theorem likely predicts free parameters which are associated to Hom_{K} S^{m} 𝔭^{}, 𝔭 𝔭^{}. If we could prove such a theorem, we could also prove the conjecture and even drop the restriction β_{1,7} t β_{2,7} t 0.

2. As in Theorem 5.4.6, any of the above results which holds for an Aloff-Wallach space N^{k,l} can be carried over to all Aloff-Wallach spaces which are related to N^{k,l} by . Let K be the isotropy group of the SU 3 -action on the singular orbit. In some cases, we can replace K by a conjugate K^{}of K such that we still have U 1 _{k,l} K^{}SU 3 . In those cases, we can change the whole metric by an isometry to a metric with singular orbit SU 3 K. For the same reasons as in Remark 5.4.7, our results cover up to a conjugation of K almost all cohomogeneity-one Einstein metrics with an Aloff-Wallach space as principal orbit. The only exception is the case where the singular is orbit SU 3 SO 3 , which we have considered briefly only.

At the end of this section, we compare our results with those of other authors: Remark 5.4.14. 1. The four-form (5.35) has been described in the literature before, for

example in Kanno, Yasui [48]. In [49], the same authors also present the form (5.38) which is defined on a product of N^{1,1} and an interval. With help of these forms, the authors derive two systems of ordinary differential equations for the reduction of the holonomy which are equivalent to our systems. There is a further method to find our sufficient condition for the holonomy reduction: The condition for the Ricci-flatness of a diagonal cohomogeneity-one metric can be described with help of a Lagrangian. By constructing a superpotential, we obtain the metrics which satisfy the equations (5.36) as a subclass of the Ricci-flat metrics. This ansatz has been made by Cvetič et al. [22] and by Kanno, Yasui [48]. The contribution of this thesis is the discussion on the existence of further homogeneous G_{2}-structures or non-diagonal metrics on the Aloff-Wallach spaces which also yield cohomogeneity-one metrics with special holonomy.

2. Some of our metrics can also be obtained by the construction of Bazaikin [5]. The starting point of this construction is a cone over a seven-dimensional compact 3-Sasakian manifold M. It should be noted that there are only few compact regular 3-Sasakian manifolds of dimension 7, namely S^{7}, ℝℙ^{7}, and N^{1,1} (see Boyer, Galicki [13] and Friedrich, Kath [36]). We are mainly interested in the last case, although the construction is the same for any M. The holonomy of the cone metric is contained in Sp 2 . It is possible to deform the structure on the cone in such a way that the holonomy group Hol becomes Spin 7 . Since the 3-Sasakian structure on N^{1,1} is homogeneous, the deformed cones over that space are cohomogeneity-one manifolds. The condition Hol Spin 7 can be written as a system of ordinary differential equations. These equations are equivalent to our system (5.39) under the additional restriction that b c. The conical singularity can be resolved into a six- or a four-dimensional space, which is determined by the 3-Sasakian structure. If M N^{1,1}, these spaces become SU 3 U 1 ^{2} and ℂℙ^{2}. For both cases, Bazaikin investigated the smoothness near the singular orbit. It is also proven that in the SU 3 U 1 ^{2}-case the principal orbit has to be replaced by N^{1,1} ℤ_{2} in order to make the metric smooth. The main result of Bazaikin is the following: Let the conical singularity be resolved into a six-dimensional space. For a special choice of the initial values, the differential equations have a solution which is asymptotically conical (AC), i.e. it approaches a cone metric for t . The holonomy of the AC-metric is SU 4 . Metrics of that kind have first been considered by Berard-Bergery [7] and by Page, Pope [58]. For a more general choice of the initial values, there is a family of non-homothetic metrics with holonomy Spin 7 , which depends on one parameter. These metrics are asymptotically locally conical (ALC), i.e. they approach a circle bundle over a cone for t . If M N^{1,1} ℤ_{2}, the ALC-metrics coincide with those of our metrics which have SU 3 U 1 ^{2} as a singular orbit and satisfy b t c t and b 0 a_{1} 0 0. The global behavior of the metrics with singular orbit ℂℙ^{2} and the free parameters of third order are not the subject of [5]. In a forthcoming paper [6], Bazaikin and Malkovich will deform a cone metric whose holonomy is SU 4 . By this deformation, they will obtain further examples of cohomogeneity-one metrics with holonomy Spin 7 . Since we have allowed b to be c, our examples with singular orbit SU 3 U 1 ^{2} complement the analysis of [5].

3. In the paper of Cvetič et al. [24], cohomogeneity-one metrics with singular orbit S^{5} and principal orbit N^{1, 1} are investigated. Those metrics correspond to solutions of (5.36) with k 1, l 1, a 0 0, and b 0 c 0 . For small values of ; ; \frac{f 0}{b 0} ; ; the solution

is ALC and for great values of ; ; \frac{f 0}{b 0} ; ; it is incomplete. In the limiting case, we obtain an AC metric. All these observations are made by numerical methods. Our discussion of the smoothness conditions and our proof that for any choice of b 0 and f 0 there actually are unique local metrics which solve (5.36) is new.

4. The possible singular orbit SU 3 SO 3 of a cohomogeneity-one manifold with principal orbit N^{1, 1} is investigated in this thesis for the first time. Therefore, our discussion of special metrics with this orbit structure is new, too.

5. The metrics with an Aloff-Wallach space N^{k,l} which is N^{1,1} as principal and ℂℙ^{2} as singular orbit are intensively studied in the literature. In this context, the papers of Cvetič et al. [24], of Gukov, Sparks [39], and of Kanno, Yasui [48] should be mentioned. In [24], the first summands of the power series expansion of the corresponding initial value problem are calculated and the free parameter q of third order is discovered. The construction of the authors can also be carried over to the case k l 1 and a_{1} t a_{2} t . Numerical simulations for different values of q are made. It turns out that there is a critical value q_{0} 0, such that the metric is AC if q q_{0} and ALC if q q_{0}. The authors give further heuristic arguments which affirm the numerical results. Cvetič et al. also construct for any choice of k, l for which S U 2 U 1 U 1 _{k,l} is not S^{2} S^{1} an explicit solution of the differential equations which corresponds to a metric with singular orbit ℂℙ^{2}. This metric is asymptotically locally conical. It is possible to characterize the metric by a choice q of the parameter q. q depends in a non-trivial way on k and l and is calculated for special orbits N^{k,l}. The authors also discuss the topology of the coset spaces S U 2 U 1 U 1 _{k,l} S^{3} ℤ_{k l} and the set of all values of k, l which have to be considered. The weak G_{2}-structures on N^{k,l}, which are important for the description of the AC metrics, are explicitly given, too.

Kanno and Yasui [48] worked on this issue at the same time as Cvetič et al. Therefore, the papers [48] and [24] partly overlap. In particular, the free parameter q, the explicit solution with singular orbit ℂℙ^{2}, and the structure of S U 2 U 1 U 1 _{k,l} are discussed in [48], too.

In the paper of Gukov and Sparks [39], the connections between metrics with holonomy Spin 7 and M-theory are investigated. One chapter of this work is devoted to cohomogeneity-one orbifolds with an Aloff-Wallach space as principal orbit. The type of the orbifold singularities is important from a physical perspective. For this reason, the topology of the spaces S U 2 U 1 U 1 _{k,l} is studied in detail. Moreover, a special case of the explicit metric with singular orbit ℂℙ^{2} is first discovered in [39]. In our notation, that metric satisfies k 0, l 1, and a 0 f 0 0.

Although much research has been done on metrics of the above kind, our interpretation of the free parameter q with help of the results of Eschenburg and Wang [32] is new. Furthermore, we have rigorously proven that q is the only free parameter and that the power series indeed converges against a smooth metric which is defined on a tubular neighborhood of ℂℙ^{2}.

6. In [49], Kanno and Yasui investigate cohomogeneity-one metrics with special holonomy and N^{1,1} as principal orbit. They also obtain the system (5.39), which we have used throughout this thesis, as a sufficient condition for the holonomy reduction. By a natural ansatz for the Kähler form, they show that in the case a_{2} a_{1} and a_{1}^{2} b^{2} c^{2} the holonomy reduces to a subgroup of SU 4 . This result we have proven independently. Kanno and Yasui moreover prove that in this special situation the system (5.39) has an explicit solution which depends solely on the two parameters b 0 and c 0 . All of the resulting metrics are asymptotically conical. For most choices of the two parameters the metric has holonomy SU 4 . If we further assume that b 0 0, we obtain a metric with singular orbit ℂℙ^{2}, which has holonomy Sp 2 . This metric coincides with the hyper-Kähler metric on T^{}ℂℙ^{2} which has been discovered by Calabi [18]. Other references for this metric are Dancer, Swann [28] and Cvetič et al. [22]. Kanno and Yasui [49] also study the possible AC- and ALC-solutions of (5.39) which are defined for a sufficiently large t. This analysis is done with help of a power series in \frac{1}{t}. The authors also calculate the first terms of the power series expansion of (5.39) for the initial conditions a_{1} 0 a_{2} 0 0, b 0 , c 0 0, and f 0 0. The solutions of (5.39) with these initial conditions correspond to metrics whose principal orbit is covered by N^{1,1} and whose singular orbit is SU 3 U 1 ^{2}. For the initial conditions b 0 f 0 0 and c 0 a_{1} 0 a_{2} 0 , the same calculations are made and the authors also observe the free parameter of third (second) order.

The case where a_{1} 0 a_{2} 0 f 0 0 and the two parameters q_{1} and q_{2} of third order seem to be discussed in this thesis for the first time. For all of the four cases with singular orbit ℂℙ^{2}, we have interpreted the free parameter(s) in the context of Eschenburg’s and Wang’s work [32] and have proven that there are no further free parameters. Moreover, we have shown an existence and smoothness result for these metrics. For the case SU 3 U 1 ^{2}, we have made a similar analysis. All of these results are original work of the author.

7. Cohomogeneity-one Einstein metrics with an Aloff-Wallach space as principal orbit are also studied by Dancer and Wang [29]. In their paper, they consider the cohomogeneityone Einstein condition as a Hamiltonian system. They assume that its solution has a Painlevé expansion and study the behavior of the cohomogeneity-one Einstein metric for great values of t. If the solution of the system exists for arbitrarily large t, the metric is AC or ALC. If the metric is non-compact and exists for t t_{0} only, at least one of the metric functions has a pole of type O t_{0} t ^{α} with α ℚ. The AC- and ALC-cases and the possible poles are classified by the authors. All of their results describe the metric for t (or t t_{0}) only. We consider the behavior of the Einstein metric near a singular orbit. Therefore it is, except for the global examples which we describe in this remark, unclear if our metrics evolve into those of Dancer and Wang or vice versa.

8. On the quaternionic projective space ℍℙ^{2}, there is a certain isometric action of U 3 which is of cohomogeneity one. Its principal orbit is the Aloff-Wallach space N^{1,0} SU 3 U 1 U 1 ^{2}, where the abelian factor of SU 3 U 1 acts by right-multipli-cation with elements of Norm_{SU 3}U 1 _{1,0}. The cohomogeneity-one action has two singular orbits, namely ℂℙ^{2} and S^{5}. An explicit description of this action can be found in a paper of Püttmann and Rigas [59]. The cohomogeneity-one Einstein metrics in this thesis which have an Aloff-Wallach space as principal orbit and are neither ℍℙ^{2} nor one of the examples with holonomy Spin 7 are new contributions of the author.

Parallel Spin 7 -manifolds with reducible principal orbits

6.1 General considerations

Again, let M, Ω be a parallel Spin 7 -manifold admitting an action of cohomogeneity one which preserves Ω. In this chapter, we will consider the case where the principal orbit is of type G H U 1 , where G U 1 is the group acting with cohomogeneity one and H G is the isotropy group. Unlike in Chapter 5, we will not try to classify all G U 1 -invariant parallel Spin 7 -structures on M. Instead, we will focus on the question if any interesting examples exist. In the first section of this chapter, we will consider the equations for the holonomy reduction without specifying G or H. In the second section, we will consider a special reducible principal orbit.

There is one class of examples which is easy to obtain. Let M^{}, ω be a parallel G_{2}-manifold with principal orbit G H. Then, the Riemannian product M : M^{}U 1 carries a canonical Spin 7 -structure Ω such that M, Ω is a parallel Spin 7 -manifold with principal orbit G H U 1 . The holonomy of this space is contained in G_{2}. Our main concern is if there exists any parallel cohomogeneity-one Spin 7 -structure on M which is not of this type. A possible example would be a warped product of M^{}and U 1 . In order to decide this question, we have to take a closer look at the different structures on M.

Since G H U 1 admits a G_{2}-structure, we can show by the same arguments as in Lemma 3.2.27 that G H admits a SU 3 -structure: Let u : T_{p,1}G H U 1 ℝ^{7} be an element of the G_{2}-structure which maps tangent vectors in the U 1 -direction into the space x_{1}, 0, …, 0 x_{1} ℝ ℝ^{7}. The SU 3 -structure on G H is given by the restrictions T_{p}G H ℝ^{6} 0, x_{2}, …, x_{7} x_{2}, …, x_{7} ℝ of those u. We will describe the relation between the G_{2}- and the SU 3 -structure on the level of tensor fields. First, we choose a frame e_{1}, …, e_{7} of the G_{2}-structure on G H U 1 . The three-form ω associated to the G_{2}-structure is as usual given by:

ω e^{123} e^{145} e^{167} e^{246} e^{257} e^{347} e^{356} .

We assume without loss of generality that the direction of the circle U 1 is e_{1}. Our claim is that e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7} is a frame of the SU 3 -structure on G H. We therefore replace from now on e_{7} by e_{7}. After this replacement, ω becomes:

ω e^{1} e^{23} e^{45} e^{67} e^{246} e^{257} e^{347} e^{356} .

On any manifold equipped with a SU 3 -structure, there is a two-form α which becomes symplectic if the SU 3 -structure is torsion-free. Furthermore, we have a complex volume form. We will denote its real part by θ. With this notation, we obtain:

ω e^{1} α θ .  (6.1)

Since we can identify α and the complex volume form by our basis with the standard symplectic and complex volume form on ℂ^{3}, e_{2}, …, e_{7} is indeed a frame of the SU 3 -structure. By a short calculation, we can compute the following Hodge duals:

G H U 1 θ  e^{1} J^{}θ
G H U 1 e^{1} α  α α

where J is the almost complex structure associated to the SU 3 -structure. Since we will consider geometric structures on several different spaces, we will always index the Hodge star and the exterior differential with the space we consider. For the understanding of the above formulas it is important to note that the replacement of e_{7} by e_{7} has changed the orientation. The Hodge star operator _{G H U 1} is defined by the former orientation induced by the G_{2}-structure. For the four-form associated to the G_{2}-structure we therefore have:

G H U 1 ω α α e^{1} J^{}θ .  (6.2)

We now consider another G_{2}-structure on a certain space. Let M^{o} be the union of all principal orbits of M. Furthermore, let I^{o} : M^{o} G U 1 . We define M^{o^{}}as G H I^{o}. Let t be the coordinate of I^{o}, which we parameterize by arclength. There is a G-invariant G_{2}-structure on M^{o^{}}with a frame which is given by \frac{}{t}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7} . Analogously to formulas (6.1) and (6.2), we obtain for the three-form ω and the four-form _{M^{o^{}}}ω on M^{o^{}}:

ω  dt α θ
M^{o^{}}ω  α α dt J^{}θ

By applying d_{M^{o^{}}}to these equations, we obtain the condition for the holonomy reduction to a subgroup of G_{2}:

d_{M^{o^{}}}dt α dt d_{G H}α dt \frac{}{t}α  d_{G H}θ dt \frac{}{t}θ  0

2 d_{G H}α α 2dt \frac{}{t}α α d_{M^{o^{}}}dt J^{}θ dt d_{G H} J^{}θ

dt dt \frac{}{t}J^{}θ  0

As in Convention 3.2.32, we denote the Lie derivative L_{\frac{}{t}} shortly by \frac{}{t}. The above system

can be simplified to:

d_{G H}α dt d_{G H}θ \frac{}{t}θ dt  0
2 d_{G H}α α 2 \frac{}{t}α α dt d_{G H} J^{}θ dt  0

We split the above equations into those terms which contain an dt and those which do not contain a dt:

d_{G H}θ  0
\frac{}{t}θ  d_{G H}α
d_{G H}α α  0
2 \frac{}{t}α α  d_{G H} J^{}θ

Our aim is to compare these equations with the equation d_{M^{o}}Ω 0, which is equivalent to:

\frac{}{t} _{G H U 1} ω  d_{G H U 1} ω
d_{G H U 1} _{G H U 1} ω  0

There is a canonical G-equivariant diffeomorphism M^{o^{}}U 1 M^{o}. We will therefore identify those spaces. By inserting the equations (6.1) and (6.2), we obtain:

2 \frac{}{t}α α \frac{}{t}e^{1} J^{}θ e^{1}  \frac{}{t} J^{}θ  (6.3)
d_{G H U 1} e^{1} α e^{1} d_{G H U 1} α d_{G H U 1} θ
2 d_{G H U 1} α α d_{G H U 1} e^{1} J^{}θ e^{1} d_{G H U 1} J^{}θ  0  (6.4)

There exists a homogeneous vector field on G H U 1 which is directed along the circle U 1 and coincides with e_{1} at the point we consider. In the above formulas, we identify e_{1} with this vector field. The bundle of all k-forms has the following splitting:

kT^{}G H U 1  span e^{1}  ^{k 1} T^{}G H  ^{k}T^{}G H .

We denote the projection onto the second summand by π_{2}. Analogously, we denote the projection onto the first summand by p_{1} and define π_{1} by

e^{1} π_{1} β : p_{1} β β ^{}T^{}G H U 1 .

Our next step is to decompose the equations (6.3) and (6.4) with respect to the above splitting. Let β be an arbitrary k-form on G H which we extend by the U 1 -action to G H U 1 . We have:

d_{G H U 1} β d_{G H}β e^{1} L_{e_{1}}β .

Furthermore, we have:

L_{e_{1}} β X_{1}, …, X_{k} L_{e_{1}}β X_{1}, …, X_{k} _{i 1}^{k} β X_{1}, …, e_{1}, X_{i} , …, X_{k} .

We choose the X_{i} as vector fields on G H which we extend by the action of U 1 to vector fields on all of G H U 1 . Since G H U 1 is a Cartesian product, the commutator e_{1}, X_{i} vanishes. Furthermore, β X_{1}, …, X_{k} is constant in the e_{1}-direction. These considerations show that L_{e_{1}}β 0 and therefore

d_{G H U 1} β d_{G H}β ,

or equivalently π_{1} d_{G H U 1} β 0. Let q be a normal metric on G H U 1 . In the next two formulas, { \atop e_{1}} denotes the vector which is with respect to q dual to e^{1}. We have:

d_{G H U 1} e^{1} { \atop e_{1}}, e_{j}^{}q { \atop e_{1}}, { \atop e_{1}}, e_{j}^{}q { \atop e_{1}}, { \atop e_{1}} , e_{j}^{}0

and therefore π_{1} d_{G H U 1} e^{1} 0. Now let j, k 1. We have:

d_{G H U 1} e^{1} e_{j}^{}, e_{k}^{}q { \atop e_{1}}, e_{j}, e_{k} ^{}0 .

All in all, we have shown that d_{G H U 1} e^{1} 0. It is easy to see that \frac{}{t}e^{1} e_{j} \frac{}{t}g e_{1}, e_{j} . Since we have made no assumptions on the Spin 7 -structure or the metric, we do not assume that π_{1} \frac{}{t}e^{1} 0 or π_{2} \frac{}{t}e^{1} 0. Now we are able to split the equations (6.3) and (6.4):

π_{1} \frac{}{t}e^{1} e^{1} J^{}θ e^{1}  \frac{}{t} J^{}θ  e^{1} d_{G H}α
2 \frac{}{t}α α π_{2} \frac{}{t}e^{1} J^{}θ  d_{G H}θ
e^{1} d_{G H} J^{}θ  0
2 d_{G H}α α  0

We can simplify this to:

\begin{matrix}π_{1} \frac{}{t}e^{1} J^{}θ \frac{}{t} J^{}θ &  & d_{G H}α \\ 2 \frac{}{t}α α π_{2} \frac{}{t}e^{1} J^{}θ &  & d_{G H}θ \\ d_{G H} J^{}θ &  & 0 \\ d_{G H}α α &  & 0 \end{matrix}  (6.5)

We now consider the case where M is a Riemannian product of a circle and a G_{2}-manifold. In this situation, we have \frac{}{t}e^{1} 0. Therefore, the above equations become:

\frac{}{t} J^{}θ  d_{G H}α
2 \frac{}{t}α α  d_{G H}θ
d_{G H} J^{}θ  0
d_{G H}α α  0

Next, we consider the SU 3 -structure which is determined by the metric g, the two-form α, and the complex volume-form whose real part is θ : J^{}θ. The subgroup of U 3 which leaves the complex volume-form invariant is in both cases the same. Therefore, the new SU 3 -structure coincides with the original SU 3 -structure. Since J^{}θ θ, the above equations become:

\frac{}{t} θ  d_{G H}α
2 \frac{}{t}α α  d_{G H} J^{}θ
d_{G H} θ  0
d_{G H}α α  0

These are exactly the conditions for the torsion-freeness of the G_{2}-structure on M^{o^{}}. We have proven the following theorem:

Theorem 6.1.1. Let G H be a six-dimensional homogeneous space admitting a G-invariant SU 3 -structure. We assume that there is an interval I ℝ and a parallel G-invariant G_{2}-structure on N : G H I. We define M : N U 1 where ” ” denotes the Riemannian product. In this situation, there exists a canonical parallel G U 1 -invariant Spin 7 - structure on M.

Conversely, let N S^{1} be a Riemannian product equipped with a parallel Spin 7 -structure which is preserved by a cohomogeneity-one action by a compact, connected Lie group G U 1 . We assume that the U 1 -factor acts canonically on S^{1}. Then there exists a G-invariant G_{2}- structure on N, which is parallel.

Remark 6.1.2. The above theorem and the equations (6.5) do not forbid the existence of parallel cohomogeneity-one Spin 7 -structures which are not Riemannian products with a circle. In that case, the G_{2}-structure on M^{o^{}}has to have non-vanishing intrinsic torsion. Our hope is that if we choose a G_{2}-structure on M^{o^{}}with a certain intrinsic torsion, we can extend it to a parallel Spin 7 -structure on M. Because of Theorem 6.1.1, this Spin 7 -structure will not be a Riemannian product. Unfortunately, we are not able to easily deduce a theorem of the following type:

”If the G_{2}-structure on M^{o^{}}G H I^{o} has the following intrinsic torsion, then there exists a parallel Spin 7 -structure on the Cartesian product M^{o} M^{o^{}}U 1 : ...”

The following considerations illustrate the difficulties we face: Let the G_{2}-structure ω on M^{o^{}} be nearly parallel. The equations dω λ ω and d ω 0 can be written as:

\begin{matrix}d_{G H}θ &  & λα α \\ d_{G H}α &  & \frac{}{t}θ λJ^{}θ \\ d_{G H}α α &  & 0 \\ 2 \frac{}{t}α α &  & d_{G H}J^{}θ \end{matrix}  (6.6)

Now, we consider the equation dΩ 0 on M. We assume that π_{2} \frac{}{t}e^{1} vanishes and that π_{1} \frac{}{t}e^{1} is a constant function which we denote by λ, too. Then, we have:

\begin{matrix}λθ \frac{}{t} θ &  & d_{G H}α \\ 2 \frac{}{t}α α &  & d_{G H}J^{}θ \\ d_{G H} θ &  & 0 \\ d_{G H}α α &  & 0 \end{matrix}  (6.7)

If λ 0, (6.6) and (6.7) are not equivalent. If we make other natural assumptions on \frac{}{t}e^{1}, we do not get a system of type (6.6), either. For other choices of the intrinsic torsion of ω, we face similar difficulties.

In order to decide if there are examples of parallel cohomogeneity-one Spin 7 -structures with a principal orbit of type G H U 1 , whose holonomy is not contained in G_{2}, we will consider a concrete example.

6.2. THE PRINCIPAL ORBIT SU 3 U 1 ^{2} U 1 225

6.2 The principal orbit SU 3 U 1 ^{2} U 1

For our considerations, we choose SU 3 U 1 ^{2} U 1 as principal orbit. In the paper of Cleyton and Swann [20], cohomogeneity-one G_{2}-structures with different principal orbits have been treated in great detail. Since the principal orbit SU 3 U 1 ^{2} is one of the most interesting ones, our choice of the principal orbit seems promising, too. We assume without loss of generality that U 1 ^{2} SU 3 is the maximal torus whose Lie algebra is given by:

2𝔲 1 : ; ; { \atop }\begin{matrix}ix & 0 & 0 \\ 0 & iy & 0 \\ 0 & 0 & iz \end{matrix} { \atop } ; ; ; ; ; x, y, z ℝ, x y z 0 ; ; .

SU 3 U 1 ^{2} U 1 and the Aloff-Wallach spaces are both of type SU 3 U 1 U 1 ^{2}. Since π_{1} SU 3 U 1 ^{2} U 1 ℤ and π_{1} N^{k,l} 0 , the principal orbit we consider is not an Aloff-Wallach space. The aim of this section is to study the parallel Spin 7 -structures of cohomogeneity one with SU 3 U 1 ^{2} U 1 as principal orbit and to compare them with the parallel cohomogeneity-one G_{2}-structures with principal orbit SU 3 U 1 ^{2}. In order to do this, we will briefly review the results of [20] on the SU 3 U 1 ^{2}-case. First, we choose an appropriate basis of 𝔰𝔲 3 . We define e_{1}, …, e_{6} 𝔰𝔲 3 as in Section 5.4 and define

e_{7} : { \atop }\begin{matrix}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 0 \end{matrix} { \atop }e_{8} : { \atop }\begin{matrix}i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & 2i \end{matrix} { \atop }.

As usual, we define a biinvariant background metric q on 𝔰𝔲 3 by q X, Y : tr XY and denote the q-orthogonal complement of 2𝔲 1 𝔰𝔲 3 by 𝔪. Since the basis e_{1}, …, e_{8} is orthogonal with respect to q, e_{1}, …, e_{6} spans 𝔪 and e_{7}, e_{8} spans 2𝔲 1 . The matrix representation of ad_{e_{7}} _{𝔪} and ad_{e_{8}} _{𝔪} with respect to the basis e_{1}, …, e_{6} is, as in Section 5.4, given by:

ad_{e_{7}} _{𝔪} ; ; ; ; ; ; ; \begin{aligned}0 & 2 \\ 2 & 0 \\ & & 0 & 1 \\ & & 1 & 0 \\ & & & & 0 & 1 \\ & & & & 1 & 0 \end{aligned} ; ; ; ; ; ; ; ad_{e_{8}} _{𝔪} ; ; ; ; ; ; ; \begin{aligned}0 & 0 \\ 0 & 0 \\ & & 0 & 3 \\ & & 3 & 0 \\ & & & & 0 & 3 \\ & & & & 3 & 0 \end{aligned} ; ; ; ; ; ; ; .

𝔪 decomposes therefore into the following 2𝔲 1 -modules:

V_{1} : span e_{1}, e_{2}
V_{2} : span e_{3}, e_{4}
V_{3} : span e_{5}, e_{6}

e_{7} and e_{8} act with different weights on the modules V_{i}. Therefore, V_{1}, V_{2}, and V_{3} are pairwise inequivalent. By the same arguments as in Chapter 5 we can show that the matrix representation of g with respect to the basis e_{1}, …, e_{6} has to be of the following type:

; ; ; ; ; ; ; \begin{aligned}a^{2} & 0 \\ 0 & a^{2} \\ & & b^{2} & 0 \\ & & 0 & b^{2} \\ & & & & c^{2} & 0 \\ & & & & 0 & c^{2} \end{aligned} ; ; ; ; ; ; ; with a, b, c ℝ 0 .

Conversely, any such matrix defines a SU 3 -invariant metric on SU 3 U 1 ^{2}. Next, we construct a frame of an invariant SU 3 -structure on SU 3 U 1 ^{2}. If we first interchange e_{1}, e_{2} and then e_{5}, e_{6} , and furthermore identify ℝ^{6} with ℂ^{3}, the matrix representation of ad_{e_{7}} _{𝔪} and ad_{e_{8}} _{𝔪} becomes:

ad_{e_{7}} _{𝔪} { \atop }\begin{matrix}2i & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & i \end{matrix} { \atop }and ad_{e_{8}} _{𝔪} { \atop }\begin{matrix}0 & 0 & 0 \\ 0 & 3i & 0 \\ 0 & 0 & 3i \end{matrix} { \atop }.

Since both of these matrices are contained in 𝔰𝔲 3 , a frame h_{1}, …, h_{6} of an invariant SU 3 -structure on SU 3 U 1 ^{2} whose associated metric is g is therefore given by:

\begin{matrix}h_{1} : \frac{1}{a}e_{2} & h_{2} : \frac{1}{a}e_{1} & h_{3} : \frac{1}{b}e_{3} \\ h_{4} : \frac{1}{b}e_{4} & h_{5} : \frac{1}{c}e_{6} & h_{6} : \frac{1}{c}e_{5} \end{matrix}  (6.8)

Our next question is if this SU 3 -structure is unique. More concretely, we want to determine the set of all invariant SU 3 -structures whose extension to a SO 6 -structure is the same. We can show by the same arguments as in the proof of Lemma 3.1.50 that this set can be generated by the action of

ϕ SO 6 ϕU 1 ^{2}ϕ^{1} SU 3

on the frame h_{1}, …, h_{6} . Any element of SU 3 is conjugate to an element of the maximal torus U 1 ^{2}. Since the action of SU 3 leaves the SU 3 -structure invariant, the above set can be replaced by

ϕ SO 6 ϕU 1 ^{2}ϕ^{1} U 1 ^{2}

or equivalently by Norm_{SO 6}U 1 ^{2}. Let

U 1 ^{3} : ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; cos θ_{1} sin θ_{1} sin θ_{1} cos θ_{1} cos θ_{2} sin θ_{2} sin θ_{2} cos θ_{2} cos θ_{3} sin θ_{3} sin θ_{3} cos θ_{3} ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; θ_{1}, θ_{2}, θ_{3} ℝ ; ; ; ; ; ; ; ; ; ; ; ; ; ;

be the standard maximal torus of SO 6 . U 1 ^{2} SU 3 SO 6 is the set of all elements of U 1 ^{3} satisfying θ_{1} θ_{2} θ_{3} 0. By a short calculation, we can see that the only 6 6-matrices ϕ with ϕ U 1 ^{2} U 1 ^{3}ϕ are of type

 ;  ;  ;  ;  ;  ;  ; \begin{aligned}x_{1} & y_{1} \\ y_{1} & x_{1} \\ & & x_{2} & y_{2} \\ & & y_{2} & x_{2} \\ & & & & x_{3} & y_{3} \\ & & & & y_{3} & x_{3} \end{aligned}  ;  ;  ;  ;  ;  ;  ;

All those ϕ also satisfy ϕ U 1 ^{2} U 1 ^{2}ϕ. Since we only search for matrices in SO 6 , we obtain Norm_{SO 6}U 1 ^{2} U 1 ^{3}. The action of U 1 ^{2} on h_{1}, …, h_{6} leaves the SU 3 - structure invariant. Therefore, the set of all SU 3 -structures is generated by a U 1 -action. We can choose the group acting on h_{1}, …, h_{6} as the set of all matrices in U 1 ^{3} with θ_{1} θ_{2} θ_{3} : θ. A frame of the most general invariant SU 3 -structure on SU 3 U 1 ^{2} therefore is:

h_{1} : \frac{1}{a} sin θ e_{1} \frac{1}{a} cos θ e_{2} h_{2} : \frac{1}{a} cos θ e_{1} \frac{1}{a} sin θ e_{2}
h_{3} : \frac{1}{b} cos θ e_{3} \frac{1}{b} sin θ e_{4} h_{4} : \frac{1}{b} sin θ e_{3} \frac{1}{b} cos θ e_{4}
h_{5} : \frac{1}{c} sin θ e_{5} \frac{1}{c} cos θ e_{6}  h_{6} : \frac{1}{c} cos θ e_{5} \frac{1}{c} sin θ e_{6}

(6.9)

with θ ℝ. Let M be a cohomogeneity-one manifold with principal orbit SU 3 U 1 ^{2}. As we have seen in the first section of this chapter, we can obtain a frame of an invariant G_{2}-structure on M by changing the sign of h_{6} and supplementing our basis with \frac{}{t}. We obtain:

f_{1} : \frac{}{t}  f_{2} : \frac{1}{a} sin θ e_{1} \frac{1}{a} cos θ e_{2}
f_{3} : \frac{1}{a} cos θ e_{1} \frac{1}{a} sin θ e_{2}  f_{4} : \frac{1}{b} cos θ e_{3} \frac{1}{b} sin θ e_{4}
f_{5} : \frac{1}{b} sin θ e_{3} \frac{1}{b} cos θ e_{4} f_{6} : \frac{1}{c} sin θ e_{5} \frac{1}{c} cos θ e_{6}
f_{7} : \frac{1}{c} cos θ e_{5} \frac{1}{c} sin θ e_{6}

(6.10)

The three-form which is associated to this G_{2}-structure we denote by ω_{θ}. We obtain:

\begin{aligned}ω_{θ} & abc cos 3θ e^{135} e^{146} e^{236} e^{245} abc sin 3θ e^{136} e^{145} e^{235} e^{246} \\ & a^{2} e^{12} dt b^{2} e^{34} dt c^{2} e^{56} dt . \end{aligned}  (6.11)

Furthermore, we have:

\begin{aligned}ω_{θ} & a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456} \\ & abc cos 3θ e^{136} e^{145} e^{235} e^{246} dt \\ & abc sin 3θ e^{135} e^{146} e^{236} e^{245} dt \end{aligned}  (6.12)

Next, we determine dω_{θ} and d ω_{θ}. Since dω_{θ} contains terms of type 4abc cos 3θ e^{ijkl}, cos 3θ has to vanish if we want the G_{2}-structure to be parallel. As we will see in Appendix D, it is possible to assume without loss of generality that θ \frac{π}{6} . Therefore, the equations for the reduction of the holonomy to a subgroup of G_{2} do not contain θ anymore. These equations are equivalent to:

\frac{a^{}}{a}  \frac{b^{2} c^{2} a^{2}}{abc}
\frac{b^{}}{b}  \frac{a^{2} c^{2} b^{2}}{abc}
\frac{c^{}}{c}  \frac{a^{2} b^{2} c^{2}}{abc}

The detailed calculations and arguments needed for the derivation of these equations can be found in Appendix D. We now show that the metrics we obtain as solutions of the above system have holonomy G_{2}. In order to do this, we assume that the holonomy is smaller. In that case, the holonomy is a subgroup of SU 3 and there exists a one-dimensional subspace of the tangent space on which the holonomy acts trivially. The holonomy bundle of SU 3 U 1 ^{2} is SU 3 -invariant, since the metric is invariant. Furthermore, the subbundle of the tangent bundle on which the holonomy acts trivially, is invariant, too. Its fibers have to be of positive dimension. We first consider the case where the fibers are one-dimensional. Since any submodule of 𝔪 is at least two-dimensional, the fiber of the bundle is span \frac{}{t} . In this situation, the manifold has to be a Riemannian product of SU 3 U 1 ^{2} and an interval. Therefore, we have a^{}b^{}c^{}0. By adding the above three equations up we obtain a b c 0, which is impossible. Next, we assume that the space on which the holonomy acts trivially is at least two-dimensional. That space has to contain V_{1}, V_{2}, or V_{3} as a subspace. We can conclude with help of the de Rham decomposition theorem that SU 3 U 1 ^{2} is a product, which has a two-dimensional torus as factor. Since π_{1} SU 3 U 1 ^{2} 0 , we can exclude this case, too.

Next, we construct parallel cohomogeneity-one Spin 7 -structures with principal orbit SU 3 U 1 ^{2} U 1 . In order to do this, we choose a basis of 𝔰𝔲 3 𝔲 1 . Let i : 𝔰𝔲 3 𝔰𝔲 3 𝔲 1 be the canonical embedding. The family

i e_{1} , …, i e_{6} , ; ; ; \begin{aligned}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ & & & i \end{aligned} ; ; ; , i e_{7} , i e_{8}

is a basis of 𝔰𝔲 3 𝔲 1 . For reasons of simplicity, we denote this new basis by e_{1}, …, e_{9} . span e_{1}, …, e_{7} can be identified with the tangent space of SU 3 U 1 ^{2} U 1 at a point p and e_{8}, e_{9} spans 2𝔲 1 𝔰𝔲 3 . The tangent space splits into the following irreducible 2𝔲 1 -modules:

V_{1} : span e_{1}, e_{2}
V_{2} : span e_{3}, e_{4}
V_{3} : span e_{5}, e_{6}
V_{4} : span e_{7}

Since V_{1}, …, V_{4} are pairwise inequivalent, any SU 3 U 1 -invariant metric g on the principal orbit has the following matrix representation with respect to e_{1}, …, e_{7} :

; ; ; ; ; ; ; ; ; ; \begin{aligned}a^{2} & 0 \\ 0 & a^{2} \\ & & b^{2} & 0 \\ & & 0 & b^{2} \\ & & & & c^{2} & 0 \\ & & & & 0 & c^{2} \\ & & & & & & f^{2} \end{aligned} ; ; ; ; ; ; ; ; ; ; with a, b, c, f ℝ 0 .

As in Section 6.1, we obtain a three- and a four-form associated to an invariant G_{2}-structure by replacing dt by fe^{7} in (6.11) and (6.12). Therefore, we have found a family of G_{2}-structures depending on a parameter θ. Since we do not want to introduce a new notation, we denote the three- and the four-form associated to these G_{2}-structures by the same symbols as before. We obtain:

ω_{θ} abc cos 3θ e^{135} e^{146} e^{236} e^{245} abc sin 3θ e^{136} e^{145} e^{235} e^{246}
a^{2}f e^{127} b^{2}f e^{347} c^{2}f e^{567}
ω_{θ} a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
abcf cos 3θ e^{1367} e^{1457} e^{2357} e^{2467}
abcf sin 3θ e^{1357} e^{1467} e^{2367} e^{2457}

with θ ℝ. We can see by the same arguments as for the principal orbit SU 3 U 1 ^{2} that these are all possible invariant G_{2}-structures. The evolution equation for the holonomy reduction is:

\frac{}{t} ω_{θ} dω_{θ} .

Furthermore, the G_{2}-structure on the principal orbit has to satisfy:

d ω_{θ} 0 .

Let e^{7}^{}be the one-form defined by e^{7}^{}e_{i}^{}: δ_{i}^{7}. Since e_{7} commutes with 𝔰𝔲 3 , we have de_{p}^{7}^{}0. Therefore, we obtain by a calculation, which is analogous to the one on page 268:

d ω_{θ} 4abcf sin 3θ e^{12347} e^{12567} e^{34567} .

Since we want d ω_{θ} to vanish, θ has to be in \frac{π}{3} ℤ. Next, we determine the system of ordinary differential equations for the holonomy reduction. The exterior derivative of the three-form ω_{θ} is:

\begin{aligned}dω_{θ} & 4abc cos 3θ e^{1234} e^{1256} e^{3456} \\ & a^{2}f b^{2}f c^{2}f e^{1367} e^{1457} e^{2357} e^{2467} . \end{aligned}

Furthermore, we have:

\frac{}{t} ω_{θ} 2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} 2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} 2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} cos 3θ 3θ^{}sin 3θ abcf e^{1367} e^{1457} e^{2357} e^{2467}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} sin 3θ 3θ^{}cos 3θ abcf e^{1357} e^{1467} e^{2367} e^{2457}

Therefore, the equations for the holonomy reduction are given by:

\frac{a^{}}{a} \frac{b^{}}{b}  2\frac{c}{ab} cos 3θ
\frac{a^{}}{a} \frac{c^{}}{c}  2\frac{b}{ac} cos 3θ
\frac{b^{}}{b} \frac{c^{}}{c}  2\frac{a}{bc} cos 3θ
\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} cos 3θ 3θ^{}sin 3θ  \frac{a^{2} b^{2} c^{2}}{abc}
\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} sin 3θ 3θ^{}cos 3θ  0

For analogous reasons as in the case where the principal orbit is SU 3 U 1 ^{2}, we can assume without loss of generality that θ 0. By adding the first three equations up, we obtain:

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{a^{2} b^{2} c^{2}}{abc} .

With the help of this relation, the above equations reduce to:

\frac{a^{}}{a}  \frac{b^{2} c^{2} a^{2}}{abc}
\frac{b^{}}{b}  \frac{a^{2} c^{2} b^{2}}{abc}
\frac{c^{}}{c}  \frac{a^{2} b^{2} c^{2}}{abc}
\frac{f^{}}{f}  0

The first three of these equations are exactly the equations we have obtained in the SU 3 U 1 ^{2}-case. Since f^{}0, the only examples we obtain for the principal orbit SU 3 U 1 ^{2} U 1 are Riemannian products of a circle and a parallel cohomogeneity-one G_{2}-manifold:

Theorem 6.2.1. Let M, Ω be a parallel Spin 7 -manifold with a SU 3 U 1 -action of cohomogeneity one which preserves Ω. We assume that the stabilizer group of the action on the principal orbit is U 1 ^{2} SU 3 . In this situation, M is the Riemannian product of a circle and a parallel G_{2}-manifold of cohomogeneity one.

Although we have obtained only negative results in this chapter, we still hope that there are new examples which are not of this simple type. One reason for our optimism is found in a paper of Gukov, Yau, and Zaslow [40]. In that paper, the authors construct parallel cohomogeneity-one G_{2}-manifolds whose principal orbit is SU 2 U 1 ^{3}. The metrics they obtain are not Riemannian products of a circle and a manifold with holonomy contained in SU 3 . Therefore, we hope that in the Spin 7 -case there also exist examples with the desired properties.

232 CHAPTER 6. THE REDUCIBLE PRINCIPAL ORBITS

Conclusion and outlook

We shortly recapitulate what we have done. One aim of this thesis was to find a partial classification result on parallel Spin 7 -manifolds of cohomogeneity one and many of our results indeed include some kind of classification.

In Theorem 4.2, we have listed all possible principal orbits of cohomogeneity-one Spin 7 - manifolds under some mild restrictions. Some of those orbits we have considered in detail and have found further classification results.

We first have taken a look at the Berger space B^{7} as principal orbit. In this case, the only parallel cohomogeneity-one Spin 7 -manifold is the cone over B^{7}. The reason for this is that B^{7} is an isotropy-irreducible homogeneous space.

In Section 5.2 and 5.3, we also have described all parallel cohomogeneity-one Spin 7 -manifolds (and -orbifolds) whose principal orbit is Q^{1,1,1} or M^{1,1,0} and which have a singular orbit. The following facts have simplified our arguments: First, all homogeneous metrics on those orbits are diagonal. This makes the space of all metrics which we had to consider sufficiently small. Second, the space of all G_{2}-structures on the principal orbit which have the same associated metric and orientation is diffeomorphic to a circle. We therefore did not have to make any simplifying assumptions on the cohomogeneity-one Spin 7 -structure. Third, the equations for the holonomy reduction can be solved explicitly, which was helpful, too. All of our examples have been considered in the literature before (see Berard-Bergery [8]; Stenzel [64]; Page, Pope [58]; Herzog, Klebanov [41]; Cvetič et al. [21], [25]). Nevertheless, our classification result has until now not been stated explicitly and we have introduced a new point of view on those metrics.

The case where the principal orbit is a product of a circle and another homogeneous space has been shortly investigated, too. The only principal orbit which we have explored in detail is SU 3 U 1 ^{2} U 1 . In this case, we also have been able to classify all homogeneous G_{2}- structures on the principal orbit. Furthermore, we have seen that in our situation the only cohomogeneity-one metrics whose holonomy is contained in Spin 7 are Riemannian products of a circle and a parallel cohomogeneity-one G_{2}-manifold. Since that kind of manifolds is classified by Cleyton and Swann [20], an analogous result for our case follows.

In the case where the principal orbit is an Aloff-Wallach space N^{k,l}, we had to make some simplifications. On the two exceptional Aloff-Wallach spaces N^{1,0} and N^{1,1}, there exist nondiagonal SU 3 -invariant metrics. We nevertheless have always assumed that the metric on the principal orbit is diagonal. For each type of Aloff-Wallach space, we have described the space of all homogeneous G_{2}-structures whose associated metric and orientation is an arbitrary but fixed one. We have been able to determine the type of the connected components of that space but not their number. Therefore, we have restricted our attention to G_{2}-structures of a special kind or equivalently to a subset of the connected components of that space. In the case k l 1, we moreover have considered a one-parameter family of G_{2}-structures in that space only. Under these restrictions, we have been able to classify all parallel cohomogeneityone Spin 7 -structures with a singular orbit. In our situation, there is no explicit solution of the equations for the holonomy reduction, except in special cases. Therefore, we could prove the existence of our metrics on a tubular neighborhood of the singular orbit only. Another difference to the other principal orbits is that sometimes our metrics depend on additional initial conditions of higher order. The number and order of those initial conditions we have described, too. We have reproduced the metrics which can be found in the literature (see Cvetič et al. [24]; Gukov, Sparks [39]; Kanno, Yasui [48], [49]; Bazaikin [5]), but also have found new examples.

Although we have proven many results on parallel cohomogeneity-one Spin 7 -manifolds with an Aloff-Wallach space N^{k,l} as principal orbit, there are still many questions left open. One of these questions is if there are any further Spin 7 -manifolds of this kind. In order to find further examples, we either have to consider non-diagonal metrics on N^{k,l} or if the metric is diagonal, at least other G_{2}-structures which do not satisfy our simplifying assumptions. On the singular orbit, the SU 3 -invariance forces the metric in most cases to be diagonal. The only exception is the case where the principal orbit is covered by N^{1,1} and the isotropy algebra of the singular orbit is generated by diag i, i, 2i and an element of span e_{1}, e_{2}, e_{7} which is not an e_{i}. Nevertheless, it may possible that the metric on the singular orbit becomes nondiagonal as soon as we move away from that orbit. Therefore, the existence of non-diagonal examples cannot be a priori excluded. Calculations which have been made by the author seem to indicate that the G_{2}-structure on the principal orbit is often not coclosed if the metric is non-diagonal or if the G_{2}-structure is different from the G_{2}-structure which we have chosen. If it is possible to make a more general ansatz for the coclosed G_{2}-structure on N^{k,l}, our differential equations contain more functions and are more difficult to handle. Nevertheless, it is possible to prove local existence results by the same methods which we have developed in this thesis.

Another interesting question is if our examples can be extended to complete or compact metrics. The holonomy of a compact parallel Spin 7 -manifold which is not covered by a Riemannian product is determined by its Â-genus. It therefore may be possible to find restrictions on the orbit structure of compact examples by calculating the Â-genus of cohomogeneity-one manifolds with two singular orbits and an Aloff-Wallach space as principal orbit. If we could make a statement on the long term behavior of the solutions of our differential equations, the compactness and completeness of our metrics could be verified. In the paper of Bazaikin [5], methods were introduced which may be helpful to answer these question in further cases, beside those which are treated in [5] or in his forthcoming paper [6]. All of the above questions can be the subject of future research.

In this thesis, we have investigated all irreducible principal orbits, except the Stiefel-manifold V^{5,2} SO 5 SO 3 and the sphere S^{7} Sp 2 Sp 1 . Cohomogeneity-one metrics with special holonomy whose principal orbit is covered by one of those spaces have been studied in the literature. On the cotangent bundle T^{}S^{4} of the four-dimensional sphere, there is a canonical SO 5 -action of cohomogeneity one. The principal orbit of this action is V^{5,2} and the zero section is a singular orbit with isotropy group SO 4 . On this space, a complete SO 5 - invariant metric with holonomy SU 4 is known. That metric is contained in the construction of Stenzel [64] and is explicitly described by Cvetič et al. [25]. The complete metric with holonomy Spin 7 which has been found by Bryant and Salamon [15] is a cohomogeneity-one metric with S^{7} as principal and S^{4} as singular orbit. Further metrics of a similar kind have been discovered in a series of papers by Cvetič et al. [23], [24], [26], [27]. Among them are cohomogeneity-one manifolds with principal orbit S^{7} ℤ_{4} and singular orbit ℂℙ^{3} as well as manifolds with principal orbit S^{7} and singular orbit S^{4}. The methods of Bazaikin [5] yield further examples of the first kind. All of those metrics are diagonal with respect to a natural choice of the basis of the tangent space. Furthermore, they are all non-compact. For both orbit structures, AC- and ALC-examples do exist. Since much research has been done on the principal orbits V^{5,2} and S^{7} (or S^{7} ℤ_{4} respectively), we have not investigated them in this thesis.

We will not further review the above results, but will shortly discuss if we can obtain any new examples by considering non-diagonal metrics. For this discussion, we denote the irreducible representations of 𝔰𝔲 2 in the same way as in the rest of this thesis. The tangent space of V^{5,2} splits into 𝕍_{2}^{ℝ} 𝕍_{2}^{ℝ} 𝕍_{0}^{ℝ} with respect to the isotropy group SO 3 . In the same way, the tangent space of the sphere splits into 𝕍_{1}^{ℂ} 3𝕍_{0}^{ℝ} with respect to Sp 1 . Since both splittings contain equivalent summands, there exist non-diagonal homogeneous metrics on both spaces. By a short calculation, we see that SO 3 SO 2 and Sp 1 Sp 1 are the identity components of the normalizers Norm_{SO 5}SO 3 and Norm_{Sp 2}Sp 1 . The second factor of SO 3 SO 2 acts on the second factor of 𝕍_{2}^{ℝ} ℝ^{2} by matrix multiplication. The second factor of Sp 1 Sp 1 acts as SO 3 on 3𝕍_{0}^{ℝ}. Any homogeneous metric on V^{5,2} or S^{7} can be made diagonal by the action of a suitable element of the normalizer. Let M, Ω be a parallel cohomogeneity-one Spin 7 -manifold with V^{5,2} or S^{7} as principal orbit. The associated metric on M we denote by g. We consider the restriction of the metric to a fixed principal orbit O. Let φ be a diffeomorphism of O which is induced by an element of the normalizer and makes the metric diagonal. The simultaneous action of φ on all principal orbits transforms g into a new metric g^{}, which is diagonal on O. If g^{}satisfies the equations for the holonomy reduction from the literature [24], [25], g^{}as a whole is diagonal, too. Let ω be the G_{2}-structure on the principal orbit O and let ω^{}: φ^{}ω. The only possibility to obtain a non-diagonal example which is not isometric to one of the known ones is if ω^{}satisfies one of the following two conditions:

1. The metric g^{}is always diagonal, but satisfies a system of differential equations which is different from the known ones. This can be the case only if ω^{}is an unknown coclosed G_{2}-structure on the principal orbit which yields a diagonal metric.

2. The restriction of g^{}to O is diagonal, but the equation \frac{}{t} ω^{}dω^{}changes g^{}into a non-diagonal metric. In this situation, ω^{}has to be an unknown coclosed G_{2}-structure, too.

Although it seems unlikely that such an ω^{}exists, only an analysis which is similar to that in Section 5.4 can fully answer the question if there are any non-obvious non-diagonal examples.

We finally remark that although there are still interesting questions open, the number of parallel cohomogeneity-one Spin 7 -manifolds is limited. The reason for this is that all possible principal orbits are contained in the table of Theorem 4.2 and that any parallel cohomogeneityone Spin 7 -structure is determined by the initial values at the singular orbit. Despite the fact that it may be possible to construct compact manifolds with exceptional holonomy by our methods, in the long run one has to turn to other techniques in order to construct such manifolds.

Appendix A

Calculations for the principal orbit Q^{1,1,1}

Let M, Ω be a parallel Spin 7 -manifold admitting a cohomogeneity-one action of SU 2 ^{3} which preserves Ω. From Theorem 4.2, it follows that the isotropy group of the action on the principal orbit is isomorphic to U 1 ^{2}. Furthermore, we have shown in Section 5.2 that the principal orbit is SU 2 ^{3}-equivariantly diffeomorphic to Q^{1,1,1}. The aim of this chapter is to express the equation dΩ 0 for the Ω which we have chosen in 5.2 as a set of equations for the metric functions.

Our Ω is defined by the frame (5.4) from page 108. We denote the linear map associated to this basis by ψ : 𝔪 span \frac{}{t} ℝ^{8} and obtain the following SU 2 ^{3}-invariant four-form:

Ω ψ^{}dx^{0123} dx^{0145} dx^{0167} dx^{0246} dx^{0257} dx^{0347} dx^{0356}
dx^{1247} dx^{1256} dx^{1346} dx^{1357} dx^{2345} dx^{2367} dx^{4567}
a^{2}f e^{712} dt b^{2}f e^{734} dt c^{2}f e^{765} dt
abc e^{136} dt abc e^{145} dt abc e^{235} dt abc e^{246} dt
abcf e^{7135} abcf e^{7146} abcf e^{7236} abcf e^{7245}
a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1265} b^{2}c^{2} e^{3465}
abcf e^{1357} abcf e^{1467} abcf e^{2367} abcf e^{2457}
a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
a^{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt
abc e^{136} dt abc e^{145} dt abc e^{235} dt abc e^{246} dt

By a straightforward but lengthy calculation we could show that this form is indeed invariant with respect to the action of SU 2 ^{3}. In order to obtain the equations for the holonomy reduction we have to determine dΩ. Let p Q^{1,1,1} be a point whose tangent space we identify with 𝔪. First, we calculate the exterior derivatives of the one-forms e^{i}^{}which satisfy e^{i}^{}e_{j}^{}δ_{j}^{i}. By Lemma 3.1.44 we obtain:

238 APPENDIX A. THE PRINCIPAL ORBIT Q^{1,1,1}

\begin{matrix}de_{p}^{1} & e_{p}^{27} \\ de_{p}^{2} & e_{p}^{17} \\ de_{p}^{3} & e_{p}^{47} \\ de_{p}^{4} & e_{p}^{37} \\ de_{p}^{5} & e_{p}^{67} \\ de_{p}^{6} & e_{p}^{57} \\ de_{p}^{7} & \frac{1}{3}e_{p}^{12} \frac{1}{3}e_{p}^{34} \frac{1}{3}e_{p}^{56} \end{matrix}

We are now able to compute dΩ:

dΩ \frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1357} dt abcf e^{27357} abcf e^{14757} abcf e^{13677}

\frac{1}{3}abcf e^{13512} e^{13534} e^{13556}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1467} dt abcf e^{27467} abcf e^{13767} abcf e^{14577}

\frac{1}{3}abcf e^{14612} e^{14634} e^{14656}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2367} dt abcf e^{17367} abcf e^{24767} abcf e^{23577}

\frac{1}{3}abcf e^{23612} e^{23634} e^{23656}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2457} dt abcf e^{17457} abcf e^{23757} abcf e^{24677}

\frac{1}{3}abcf e^{24512} e^{24534} e^{24556}

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} dt a^{2}b^{2} e^{27234} a^{2}b^{2} e^{11734} a^{2}b^{2} e^{12474} a^{2}b^{2} e^{12337}

2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} dt a^{2}c^{2} e^{27256} a^{2}c^{2} e^{11756} a^{2}c^{2} e^{12676} a^{2}c^{2} e^{12557}

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt b^{2}c^{2} e^{47456} b^{2}c^{2} e^{33756} b^{2}c^{2} e^{34676} b^{2}c^{2} e^{34557}

a^{2}f e^{2727} dt a^{2}f e^{1177} dt \frac{1}{3}a^{2}f e^{1212} dt e^{1234} dt e^{1256} dt

b^{2}f e^{4747} dt b^{2}f e^{3377} dt \frac{1}{3}b^{2}f e^{3412} dt e^{3434} dt e^{3456} dt

c^{2}f e^{6767} dt c^{2}f e^{5577} dt \frac{1}{3}c^{2}f e^{5612} dt e^{5634} dt e^{5656} dt

abc e^{2736} dt abc e^{1476} dt abc e^{1357} dt

abc e^{2745} dt abc e^{1375} dt abc e^{1467} dt

abc e^{1735} dt abc e^{2475} dt abc e^{2367} dt

abc e^{1746} dt abc e^{2376} dt abc e^{2457} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1357} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1467} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2367} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2457} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} dt

2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} dt

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

\frac{1}{3}a^{2}f e^{1234} dt e^{1256} dt

\frac{1}{3}b^{2}f e^{1234} dt e^{3456} dt

\frac{1}{3}c^{2}f e^{1256} dt e^{3456} dt

abc e^{2367} dt abc e^{1467} dt abc e^{1357} dt

abc e^{2457} dt abc e^{1357} dt abc e^{1467} dt

abc e^{1357} dt abc e^{2457} dt abc e^{2367} dt

abc e^{1467} dt abc e^{2367} dt abc e^{2457} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} \frac{1}{3}a^{2}f \frac{1}{3}b^{2}f e^{1234} dt

2\frac{a^{}}{a} 2\frac{c^{}}{c} \frac{1}{3}a^{2}f \frac{1}{3}c^{2}f e^{1256} dt

2\frac{b^{}}{b} 2\frac{c^{}}{c} \frac{1}{3}b^{2}f \frac{1}{3}c^{2}f e^{3456} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf 3abc e^{1357} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf 3abc e^{1467} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf 3abc e^{2367} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf 3abc e^{2457} dt

From this calculation, it follows that the necessary condition d ω 0 for the existence of a parallel Spin 7 -structure is automatically satisfied. Furthermore, we see that the condition for the holonomy reduction is equivalent to:

\begin{matrix}2\frac{a^{}}{a} 2\frac{b^{}}{b} & \frac{1}{3} \frac{f}{a^{2}} \frac{1}{3} \frac{f}{b^{2}} \\ 2\frac{a^{}}{a} 2\frac{c^{}}{c} & \frac{1}{3} \frac{f}{a^{2}} \frac{1}{3} \frac{f}{c^{2}} \\ 2\frac{b^{}}{b} 2\frac{c^{}}{c} & \frac{1}{3} \frac{f}{b^{2}} \frac{1}{3} \frac{f}{c^{2}} \\ \frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} & 3\frac{1}{f} \end{matrix}

We can simplify this system to:

\begin{matrix}a^{} & \frac{1}{6} \frac{f}{a} \\ b^{} & \frac{1}{6} \frac{f}{b} \\ c^{} & \frac{1}{6} \frac{f}{c} \\ f^{} & \frac{1}{6} \frac{f^{2}}{a^{2}} \frac{1}{6} \frac{f^{2}}{b^{2}} \frac{1}{6} \frac{f^{2}}{c^{2}} 3 \end{matrix}

These are the equations which we use throughout Section 5.2.

Appendix B

Calculations for the principal orbit M^{1,1,0}

Let M be a cohomogeneity-one manifold with principal orbit M^{1,1,0}. The frame (5.18) from page 132 defines a SU 3 SU 2 -invariant Spin 7 -structure on M. In this chapter, we will calculate the four-form Ω which is associated to this Spin 7 -structure. Furthermore, we express the condition dΩ 0 as a system of ordinary differential equations for the metric functions a, b, and c, which are defined by (5.17).

Let ψ : T_{p}M 𝔪 span \frac{}{t} ℝ^{8} be the linear map which corresponds to the frame (5.18). In our situation, Ω is the following four-form:

Ω ψ^{}dx^{0123} dx^{0145} dx^{0167} dx^{0246} dx^{0257} dx^{0347} dx^{0356}
dx^{1247} dx^{1256} dx^{1346} dx^{1357} dx^{2345} dx^{2367} dx^{4567}
b^{2}c dt e^{765} a^{2}c dt e^{712} a^{2}c dt e^{743} a^{2}b dt e^{614}
a^{2}b dt e^{623} a^{2}b dt e^{513} a^{2}b dt e^{524}
a^{2}bc e^{7613} a^{2}bc e^{7624} a^{2}bc e^{7514} a^{2}bc e^{7523}
a^{2}b^{2} e^{6512} a^{2}b^{2} e^{6543} a^{4} e^{1243}
a^{4} e^{1234} a^{2}b^{2} e^{1256} a^{2}b^{2} e^{3456}
a^{2}bc e^{1367} a^{2}bc e^{1457} a^{2}bc e^{2357} a^{2}bc e^{2467}
a^{2}b e^{135} dt a^{2}b e^{146} dt a^{2}b e^{236} dt a^{2}b e^{245} dt
a^{2}c e^{127} dt a^{2}c e^{347} dt b^{2}c e^{567} dt

By the considerations we have made on page 131, we see that Ω has to be SU 3 SU 2 - invariant. This fact could also be proven by explicitly considering the action of ad_{e_{8}}, ad_{e_{9}}, and ad_{e_{11}} on Ω. The Spin 7 -structure which we obtain by the SU 3 SU 2 -invariant extension of the frame is thus well-defined.

Before we turn our attention to dΩ, we have to know the exterior derivative at p M^{1,1,0} of the one-forms e^{i}^{}which are defined by e^{i}^{}e_{j}^{}δ_{j}^{i}:

242 APPENDIX B. THE PRINCIPAL ORBIT M^{1,1,0}

\begin{matrix}de_{p}^{1}^{} & 3e_{p}^{27}^{} \\ de_{p}^{2}^{} & 3e_{p}^{17}^{} \\ de_{p}^{3}^{} & 3e_{p}^{47}^{} \\ de_{p}^{4}^{} & 3e_{p}^{37}^{} \\ de_{p}^{5}^{} & 2e_{p}^{67}^{} \\ de_{p}^{6}^{} & 2e_{p}^{57}^{} \\ de_{p}^{7}^{} & \frac{3}{4}e_{p}^{12}^{}\frac{3}{4}e_{p}^{34}^{}\frac{1}{2}e_{p}^{56}^{} \end{matrix}

By the usual methods, we are now able to calculate dΩ:

dΩ 4a^{}a^{3} e^{1234} dt a^{4} 3e^{27234} 3e^{11734} 3e^{12474} 3e^{12337}

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1256} dt a^{2}b^{2} 3e^{27256} 3e^{11756} 2e^{12676} 2e^{12557}

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{3456} dt a^{2}b^{2} 3e^{47456} 3e^{33756} 2e^{34676} 2e^{34557}

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{1367} dt

a^{2}bc 3e^{27367} 3e^{14767} 2e^{13577} \frac{3}{4}e^{13612} \frac{3}{4}e^{13634} \frac{1}{2}e^{13656}

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{1457} dt

a^{2}bc 3e^{27457} 3e^{13757} 2e^{14677} \frac{3}{4}e^{14512} \frac{3}{4}e^{14534} \frac{1}{2}e^{14556}

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{2357} dt

a^{2}bc 3e^{17357} 3e^{24757} 2e^{23677} \frac{3}{4}e^{23512} \frac{3}{4}e^{23534} \frac{1}{2}e^{23556}

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{2467} dt

a^{2}bc 3e^{17467} 3e^{23767} 2e^{24577} \frac{3}{4}e^{24612} \frac{3}{4}e^{24634} \frac{1}{2}e^{24656}

a^{2}b 3e^{2735} dt 3e^{1475} dt 2e^{1367} dt

a^{2}b 3e^{2746} dt 3e^{1376} dt 2e^{1457} dt

a^{2}b 3e^{1736} dt 3e^{2476} dt 2e^{2357} dt

a^{2}b 3e^{1745} dt 3e^{2375} dt 2e^{2467} dt

a^{2}c 3e^{2727} dt 3e^{1177} dt \frac{3}{4}e^{1212} dt \frac{3}{4}e^{1234} dt \frac{1}{2}e^{1256} dt

a^{2}c 3e^{4747} dt 3e^{3377} dt \frac{3}{4}e^{3412} dt \frac{3}{4}e^{3434} dt \frac{1}{2}e^{3456} dt

b^{2}c 2e^{6767} dt 2e^{5577} dt \frac{3}{4}e^{5612} dt \frac{3}{4}e^{5634} dt \frac{1}{2}e^{5656} dt

4a^{}a^{3} e^{1234} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1256} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{3456} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{1367} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{1457} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{2357} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc e^{2467} dt

a^{2}b 3e^{2357} dt 3e^{1457} dt 2e^{1367} dt

a^{2}b 3e^{2467} dt 3e^{1367} dt 2e^{1457} dt

a^{2}b 3e^{1367} dt 3e^{2467} dt 2e^{2357} dt

a^{2}b 3e^{1457} dt 3e^{2357} dt 2e^{2467} dt

a^{2}c \frac{3}{4}e^{1234} dt \frac{1}{2}e^{1256} dt

a^{2}c \frac{3}{4}e^{1234} dt \frac{1}{2}e^{3456} dt

b^{2}c \frac{3}{4}e^{1256} dt \frac{3}{4}e^{3456} dt

4a^{}a^{3} \frac{3}{2}a^{2}c e^{1234} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} \frac{1}{2}a^{2}c \frac{3}{4}b^{2}c e^{1256} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc 8a^{2}b e^{1367} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc 8a^{2}b e^{1457} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc 8a^{2}b e^{2357} dt

2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} a^{2}bc 8a^{2}b e^{2467} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} \frac{1}{2}a^{2}c \frac{3}{4}b^{2}c e^{3456} dt

The equation dΩ 0 therefore is equivalent to:

\begin{matrix}4\frac{a^{}}{a} & \frac{3}{2} \frac{c}{a^{2}} \\ 2\frac{a^{}}{a} 2\frac{b^{}}{b} & \frac{1}{2} \frac{c}{b^{2}} \frac{3}{4} \frac{c}{a^{2}} \\ 2\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} & 8\frac{1}{c} \end{matrix}

We can simplify this system and obtain:

\begin{matrix}\frac{a^{}}{a} & \frac{3}{8} \frac{c}{a^{2}} \\ \frac{b^{}}{b} & \frac{1}{4} \frac{c}{b^{2}} \\ \frac{c^{}}{c} & 8\frac{1}{c} \frac{1}{4} \frac{c}{b^{2}} \frac{3}{4} \frac{c}{a^{2}} \end{matrix}

These are exactly the equations (5.20) with which we work in Section 5.3.

Appendix C

Calculations for the Aloff-Wallach spaces as principal orbits

C.1 The Spin 7 -structure for the Aloff-Wallach spaces except N^{1,1} as principal orbits and the condition for its torsionfreeness

Let M be a cohomogeneity-one manifold whose principal orbit is an Aloff-Wallach space. In this section, we assume that the principal orbit is not the exceptional Aloff-Wallach space N^{1,1}. Our first aim is to calculate the SU 3 -invariant four-form Ω which is associated to the frame (5.34) from page 155. Furthermore, we will deduce a system of ordinary differential equations for the metric functions a, b, c, and f which is equivalent to dΩ 0.

Let p M be a point whose tangent space we identify with 𝔪 span \frac{}{t} , and let ψ : T_{p}M ℝ^{8} be the linear map which maps the frame to the standard basis of ℝ^{8}. For Ω we obtain:

Ω ψ^{}dx^{0123} dx^{0145} dx^{0167} dx^{0246} dx^{0257} dx^{0347} dx^{0356}
dx^{1247} dx^{1256} dx^{1346} dx^{1357} dx^{2345} dx^{2367} dx^{4567}
a^{2}f e^{712} dt b^{2}f e^{743} dt c^{2}f e^{765} dt
abc e^{146} dt abc e^{135} dt abc e^{245} dt abc e^{236} dt
abcf e^{7145} abcf e^{7136} abcf e^{7246} abcf e^{7235}
a^{2}b^{2} e^{1243} a^{2}c^{2} e^{1265} b^{2}c^{2} e^{4365}
abcf e^{1367} abcf e^{1457} abcf e^{2357} abcf e^{2467}
a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
a^{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt
abc e^{135} dt abc e^{146} dt abc e^{236} dt abc e^{245} dt

In order to calculate dΩ, we need the exterior derivatives of the one-forms e^{i}^{}which are defined by e^{i}^{}e_{j}^{}δ_{j}^{i}. These derivatives are given by:

\begin{matrix}de_{p}^{1} & 3k 3l e_{p}^{27} e_{p}^{35} e_{p}^{46} \\ de_{p}^{2} & 3k 3l e_{p}^{17} e_{p}^{36} e_{p}^{45} \\ de_{p}^{3} & e_{p}^{15} e_{p}^{26} 3l e_{p}^{47} \\ de_{p}^{4} & e_{p}^{16} e_{p}^{25} 3l e_{p}^{37} \\ de_{p}^{5} & e_{p}^{13} e_{p}^{24} 3k e_{p}^{67} \\ de_{p}^{6} & e_{p}^{14} e_{p}^{23} 3k e_{p}^{57} \\ de_{p}^{7} & \frac{k l}{k^{2} lk l^{2}} e_{p}^{12} \frac{l}{k^{2} lk l^{2}} e_{p}^{34} \frac{k}{k^{2} lk l^{2}} e_{p}^{56} \end{matrix}

where p : eU 1 _{k,l} N^{k,l}. We are now able to express dΩ in terms of our basis of T_{p}M:

dΩ  \frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1367} dt

abcf 3k 3l e^{27} e^{35} e^{46} e^{367}

abcf e^{1} e^{15} e^{26} 3l e^{47} e^{67}

abcf e^{13} e^{14} e^{23} 3k e^{57} e^{7}

abcf e^{136} \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1457} dt

abcf 3k 3l e^{27} e^{35} e^{46} e^{457}

abcf e^{1} e^{16} e^{25} 3l e^{37} e^{57}

abcf e^{14} e^{13} e^{24} 3k e^{67} e^{7}

abcf e^{145} \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2357} dt

abcf 3k 3l e^{17} e^{36} e^{45} e^{357}

abcf e^{2} e^{15} e^{26} 3l e^{47} e^{57}

abcf e^{23} e^{13} e^{24} 3k e^{67} e^{7}

abcf e^{235} \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}

C.1. THE SPIN(7)-STRUCTURE FOR THE ORBITS EXCEPT N^{1,1} 247

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2467} dt

abcf 3k 3l e^{17} e^{36} e^{45} e^{467}

abcf e^{2} e^{16} e^{25} 3l e^{37} e^{67}

abcf e^{24} e^{14} e^{23} 3k e^{57} e^{7}

abcf e^{246} \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} dt

a^{2}b^{2} 3k 3l e^{27} e^{35} e^{46} e^{234}

a^{2}b^{2} e^{1} 3k 3l e^{17} e^{36} e^{45} e^{34}

a^{2}b^{2} e^{12} e^{15} e^{26} 3l e^{47} e^{4}

a^{2}b^{2} e^{123} e^{16} e^{25} 3l e^{37}

2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} dt

a^{2}c^{2} 3k 3l e^{27} e^{35} e^{46} e^{256}

a^{2}c^{2} e^{1} 3k 3l e^{17} e^{36} e^{45} e^{56}

a^{2}c^{2} e^{12} e^{13} e^{24} 3k e^{67} e^{6}

a^{2}c^{2} e^{125} e^{14} e^{23} 3k e^{57}

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

b^{2}c^{2} e^{15} e^{26} 3l e^{47} e^{456}

b^{2}c^{2} e^{3} e^{16} e^{25} 3l e^{37} e^{56}

b^{2}c^{2} e^{34} e^{13} e^{24} 3k e^{67} e^{6}

b^{2}c^{2} de^{345} e^{14} e^{23} 3k e^{57}

a^{2}f 3k 3l e^{27} e^{35} e^{46} e^{27} dt

a^{2}f e^{1} 3k 3l e^{17} e^{36} e^{45} e^{7} dt

a^{2}f e^{12}  \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}  dt

b^{2}f e^{15} e^{26} 3l e^{47} e^{47} dt

b^{2}f e^{3} e^{16} e^{25} 3l e^{37} e^{7} dt

b^{2}f e^{34}  \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}  dt

c^{2}f e^{13} e^{24} 3k e^{67} e^{67} dt

c^{2}f e^{5} e^{14} e^{23} 3k e^{57} e^{7} dt

c^{2}f e^{56}  \frac{l k}{k^{2} lk l^{2}} e^{12} \frac{l}{k^{2} lk l^{2}} e^{34} \frac{k}{k^{2} lk l^{2}} e^{56}  dt

248 APPENDIX C. THE ALOFF-WALLACH SPACES AS PRINCIPAL ORBIT abc 3k 3l e^{27} e^{35} e^{46} e^{35} dt abc e^{1} e^{15} e^{26} 3l e^{47} e^{5} dt abc e^{13} e^{13} e^{24} 3k e^{67} dt abc 3k 3l e^{27} e^{35} e^{46} e^{46} dt abc e^{1} e^{16} e^{25} 3l e^{37} e^{6} dt abc e^{14} e^{14} e^{23} 3k e^{57} dt abc 3k 3l e^{17} e^{36} e^{45} e^{36} dt abc e^{2} e^{15} e^{26} 3l e^{47} e^{6} dt abc e^{23} e^{14} e^{23} 3k e^{57} dt abc 3k 3l e^{17} e^{36} e^{45} e^{45} dt abc e^{2} e^{16} e^{25} 3l e^{37} e^{5} dt abc e^{24} e^{13} e^{24} 3k e^{67} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1367} dt

0 0 0 0

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{1457} dt

0 0 0 0

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2357} dt

0 0 0 0

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf e^{2467} dt

0 0 0 0

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} dt

0 0 0 0

2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} dt

0 0 0 0

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

0 0 0 0 a^{2}f e^{2357} dt e^{2467} dt a^{2}f e^{1367} dt e^{1457} dt

a^{2}f \frac{l}{k^{2} lk l^{2}} e^{1234} dt \frac{k}{k^{2} lk l^{2}} e^{1256} dt

b^{2}f e^{1457} dt e^{2467} dt

b^{2}f e^{1367} dt e^{2357} dt

b^{2}f \frac{l k}{k^{2} lk l^{2}} e^{1234} dt \frac{k}{k^{2} lk l^{2}} e^{3456} dt

c^{2}f e^{1367} dt e^{2467} dt

c^{2}f e^{1457} dt e^{2357} dt

c^{2}f \frac{l k}{k^{2} lk l^{2}} e^{1256} dt \frac{l}{k^{2} lk l^{2}} e^{3456} dt

abc 3k 3l e^{2357} dt e^{3456} dt

abc e^{1256} dt 3l e^{1457} dt

abc e^{1234} dt 3k e^{1367} dt

abc 3k 3l e^{2467} dt e^{3456} dt

abc e^{1256} dt 3l e^{1367} dt

abc e^{1234} dt 3k e^{1457} dt

abc 3k 3l e^{1367} dt e^{3456} dt

abc e^{1256} dt 3l e^{2467} dt

abc e^{1234} dt 3k e^{2357} dt

abc 3k 3l e^{1457} dt e^{3456} dt

abc e^{1256} dt 3l e^{2357} dt

abc e^{1234} dt 3k e^{2467} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f}  abcf a^{2}f b^{2}f c^{2}f 3l abc 3k abc 3k 3l abc  e^{1367} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} abcf a^{2}f b^{2}f c^{2}f 3k abc 3l abc 3k 3l abc e^{1457} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f}  abcf a^{2}f b^{2}f c^{2}f 3k 3l abc 3l abc 3k abc  e^{2357} dt
\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f}  abcf a^{2}f b^{2}f c^{2}f 3k 3l abc 3k abc 3l abc  e^{2467} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} \frac{l}{k^{2} lk l^{2}} a^{2}f \frac{l k}{k^{2} lk l^{2}} b^{2}f 4abc e^{1234} dt

2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} \frac{k}{k^{2} lk l^{2}} a^{2}f \frac{l k}{k^{2} lk l^{2}} c^{2}f 4abc e^{1256} dt

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} \frac{k}{k^{2} lk l^{2}} b^{2}f \frac{l}{k^{2} lk l^{2}} c^{2}f 4abc e^{3456} dt

The above calculations prove that dΩ 0 is equivalent to:

\begin{matrix}\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} & \frac{a^{2} b^{2} c^{2}}{abc} \\ 2 \frac{a^{}}{a} \frac{b^{}}{b} & \frac{l}{k^{2} lk l^{2}} \frac{f}{b^{2}} \frac{k l}{k^{2} lk l^{2}} \frac{f}{a^{2}} 4\frac{c}{ab} \\ 2 \frac{a^{}}{a} \frac{c^{}}{c} & \frac{k}{k^{2} lk l^{2}} \frac{f}{c^{2}} \frac{k l}{k^{2} lk l^{2}} \frac{f}{a^{2}} 4\frac{b}{ac} \\ 2 \frac{b^{}}{b} \frac{c^{}}{c} & \frac{k}{k^{2} lk l^{2}} \frac{f}{c^{2}} \frac{l}{k^{2} lk l^{2}} \frac{f}{b^{2}} 4\frac{a}{bc} \end{matrix}

The above system can be simplified to:

\begin{matrix}\frac{a^{}}{a} & \frac{a^{2} b^{2} c^{2}}{abc} \frac{k l}{2 k^{2} lk l^{2}} \frac{f}{a^{2}} \\ \frac{b^{}}{b} & \frac{b^{2} c^{2} a^{2}}{abc} \frac{l}{2 k^{2} lk l^{2}} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{c^{2} a^{2} b^{2}}{abc} \frac{k}{2 k^{2} lk l^{2}} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & \frac{k l}{2 k^{2} lk l^{2}} \frac{f}{a^{2}} \frac{l}{2 k^{2} lk l^{2}} \frac{f}{b^{2}} \frac{k}{2 k^{2} lk l^{2}} \frac{f}{c^{2}} \end{matrix}

These equations are equivalent to those in the literature. If we reverse for example the direction of t, rescale f by a factor of 2 k^{2} lk l^{2} , and rename the coefficients k, l, k l , we obtain the equations for the holonomy reduction which have been found by Kanno and Yasui in [48].

In order to keep our notation short, we denote the factor k^{2} lk l^{2} by Δ. Furthermore, we choose the same direction of t as Kanno and Yasui, since we want to reduce the number of terms with a minus. We finally obtain the equations which we use throughout Section 5.4:

\begin{matrix}\frac{a^{}}{a} & \frac{b^{2} c^{2} a^{2}}{abc} \frac{k l}{2Δ} \frac{f}{a^{2}} \\ \frac{b^{}}{b} & \frac{c^{2} a^{2} b^{2}}{abc} \frac{l}{2Δ} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{a^{2} b^{2} c^{2}}{abc} \frac{k}{2Δ} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & \frac{k l}{2Δ} \frac{f}{a^{2}} \frac{l}{2Δ} \frac{f}{b^{2}} \frac{k}{2Δ} \frac{f}{c^{2}} \end{matrix}

C.2. THE SPIN(7)-STRUCTURE FOR THE PRINCIPAL ORBIT N^{1,1} 251

C.2 The Spin 7 -structure for the principal orbit N^{1,1} and the condition for its torsion-freeness

In this section, we assume that M is a cohomogeneity-one manifold whose principal orbit is the exceptional Aloff-Wallach space N^{1,1}. We consider the frame (5.37), which is defined on page 156. This frame defines a Spin 7 -structure of cohomogeneity one on M, whose associated metric is a diagonal metric of type (5.33). The aim of this section is to calculate the SU 3 -invariant four-form Ω which is associated to this Spin 7 -structure and to find a system of ordinary differential equations for the metric functions a_{1}, a_{2}, b, c, and f which is equivalent to dΩ 0.

First, we calculate the four-form. As usual, let p M be a point whose tangent space we identify with 𝔪 span \frac{}{t} . Furthermore, let ψ : T_{p}M ℝ^{8} be the linear map which maps the frame to the standard basis of ℝ^{8}. We obtain for Ω:

Ω ψ^{}dx^{0123} dx^{0145} dx^{0167} dx^{0246} dx^{0257} dx^{0347} dx^{0356}
dx^{1247} dx^{1256} dx^{1346} dx^{1357} dx^{2345} dx^{2367} dx^{4567}
a_{1}a_{2}f e^{712} dt b^{2}f e^{743} dt c^{2}f e^{765} dt
a_{1}bc e^{146} dt a_{1}bc e^{135} dt a_{2}bc e^{245} dt a_{2}bc e^{236} dt
a_{1}bcf e^{7145} a_{1}bcf e^{7136} a_{2}bcf e^{7246} a_{2}bcf e^{7235}
a_{1}a_{2}b^{2} e^{1243} a_{1}a_{2}c^{2} e^{1265} b^{2}c^{2} e^{4365}
a_{1}bcf e^{1367} a_{1}bcf e^{1457} a_{2}bcf e^{2357} a_{2}bcf e^{2467}
a_{1}a_{2}b^{2} e^{1234} a_{1}a_{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
a_{1}a_{2}f e^{127} dt b^{2}f e^{347} dt c^{2}f e^{567} dt
a_{1}bc e^{135} dt a_{1}bc e^{146} dt a_{2}bc e^{236} dt a_{2}bc e^{245} dt

The exterior derivatives of the one-forms which are dual to the Killing vector fields e_{i}^{}can be obtained by Lemma 3.1.44 from page 54:

\begin{matrix}de_{p}^{1} & 6e_{p}^{27} e_{p}^{35} e_{p}^{46} \\ de_{p}^{2} & 6e_{p}^{17} e_{p}^{36} e_{p}^{45} \\ de_{p}^{3} & e_{p}^{15} e_{p}^{26} 3 e_{p}^{47} \\ de_{p}^{4} & e_{p}^{16} e_{p}^{25} 3 e_{p}^{37} \\ de_{p}^{5} & e_{p}^{13} e_{p}^{24} 3 e_{p}^{67} \\ de_{p}^{6} & e_{p}^{14} e_{p}^{23} 3 e_{p}^{57} \\ de_{p}^{7} & \frac{2}{3}e_{p}^{12} \frac{1}{3}e_{p}^{34} \frac{1}{3}e_{p}^{56} \end{matrix}

where p : eU 1 _{1,1} N^{1,1}. We are now able to express dΩ in terms of our basis of T_{p}M:

dΩ  \frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{1}bcf e^{1367} dt

a_{1}bcf 6e^{27} e^{35} e^{46} e^{367} e^{1} e^{15} e^{26} 3e^{47} e^{67}

e^{13} e^{14} e^{23} 3e^{57} e^{7} e^{136} \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}

\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{1}bcf e^{1457} dt

a_{1}bcf 6e^{27} e^{35} e^{46} e^{457} e^{1} e^{16} e^{25} 3e^{37} e^{57}

e^{14} e^{13} e^{24} 3e^{67} e^{7} e^{145} \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}

\frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{2}bcf e^{2357} dt

a_{2}bcf 6e^{17} e^{36} e^{45} e^{357} e^{2} e^{15} e^{26} 3e^{47} e^{57}

e^{23} e^{13} e^{24} 3e^{67} e^{7} e^{235} \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}

\frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{2}bcf e^{2467} dt

a_{2}bcf 6e^{17} e^{36} e^{45} e^{467} e^{2} e^{16} e^{25} 3e^{37} e^{67}

e^{24} e^{14} e^{23} 3e^{57} e^{7} e^{246} \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}

\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{b^{}}{b} a_{1}a_{2}b^{2} e^{1234} dt

a_{1}a_{2}b^{2} 6e^{27} e^{35} e^{46} e^{234} e^{1} 6e^{17} e^{36} e^{45} e^{34}

e^{12} e^{15} e^{26} 3e^{47} e^{4} e^{123} e^{16} e^{25} 3e^{37}

\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{c^{}}{c} a_{1}a_{2}c^{2} e^{1256} dt

a_{1}a_{2}c^{2} 6e^{27} e^{35} e^{46} e^{256} e^{1} 6e^{17} e^{36} e^{45} e^{56}

e^{12} e^{13} e^{24} 3e^{67} e^{6} e^{125} e^{14} e^{23} 3e^{57}

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

b^{2}c^{2} e^{15} e^{26} 3e^{47} e^{456} e^{3} e^{16} e^{25} 3e^{37} e^{56}

e^{34} e^{13} e^{24} 3e^{67} e^{6} e^{345} e^{14} e^{23} 3e^{57} a_{1}a_{2}f 6e^{27} e^{35} e^{46} e^{27} dt e^{1} 6e^{17} e^{36} e^{45} e^{7} dt

e^{12}  \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}  dt

b^{2}f e^{15} e^{26} 3e^{47} e^{47} dt e^{3} e^{16} e^{25} 3e^{37} e^{7} dt

e^{34}  \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}  dt

c^{2}f e^{13} e^{24} 3e^{67} e^{67} dt e^{5} e^{14} e^{23} 3e^{57} e^{7} dt

e^{56}  \frac{2}{3}e^{12} \frac{1}{3}e^{34} \frac{1}{3}e^{56}  dt

a_{1}bc 6e^{27} e^{35} e^{46} e^{35} dt e^{1} e^{15} e^{26} 3e^{47} e^{5} dt

e^{13} e^{13} e^{24} 3e^{67} dt

a_{1}bc 6e^{27} e^{35} e^{46} e^{46} dt e^{1} e^{16} e^{25} 3e^{37} e^{6} dt

e^{14} e^{14} e^{23} 3e^{57} dt

a_{2}bc 6e^{17} e^{36} e^{45} e^{36} dt e^{2} e^{15} e^{26} 3e^{47} e^{6} dt

e^{23} e^{14} e^{23} 3e^{57} dt

a_{2}bc 6e^{17} e^{36} e^{45} e^{45} dt e^{2} e^{16} e^{25} 3e^{37} e^{5} dt

e^{24} e^{13} e^{24} 3e^{67} dt

\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{1}bcf e^{1367} dt

\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{1}bcf e^{1457} dt

\frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{2}bcf e^{2357} dt

\frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{2}bcf e^{2467} dt

\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{b^{}}{b} a_{1}a_{2}b^{2} e^{1234} dt

\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{c^{}}{c} a_{1}a_{2}c^{2} e^{1256} dt

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

a_{1}a_{2}f e^{2357} dt a_{1}a_{2}f e^{2467} dt a_{1}a_{2}f e^{1367} dt a_{1}a_{2}f e^{1457} dt

\frac{1}{3}a_{1}a_{2}f e^{1234} dt \frac{1}{3}a_{1}a_{2}f e^{1256} dt

b^{2}f e^{1457} dt b^{2}f e^{2467} dt b^{2}f e^{1367} dt b^{2}f e^{2357} dt

\frac{2}{3}b^{2}f e^{1234} dt \frac{1}{3}b^{2}f e^{3456} dt

c^{2}f e^{1367} dt c^{2}f e^{2467} dt c^{2}f e^{1457} dt c^{2}f e^{2357} dt

\frac{2}{3}c^{2}f e^{1256} dt \frac{1}{3}c^{2}f e^{3456} dt

6a_{1}bc e^{2357} dt a_{1}bc e^{3456} dt a_{1}bc e^{1256} dt 3a_{1}bc e^{1457} dt

a_{1}bc e^{1234} dt 3a_{1}bc e^{1367} dt

6a_{1}bc e^{2467} dt a_{1}bc e^{3456} dt a_{1}bc e^{1256} dt 3a_{1}bc e^{1367} dt

a_{1}bc e^{1234} dt 3a_{1}bc e^{1457} dt

6a_{2}bc e^{1367} dt a_{2}bc e^{3456} dt a_{2}bc e^{1256} dt 3a_{2}bc e^{2467} dt

a_{2}bc e^{1234} dt 3a_{2}bc e^{2357} dt

6a_{2}bc e^{1457} dt a_{2}bc e^{3456} dt a_{2}bc e^{1256} dt 3a_{2}bc e^{2357} dt

a_{2}bc e^{1234} dt 3a_{2}bc e^{2467} dt

\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{b^{}}{b} a_{1}a_{2}b^{2} \frac{1}{3}a_{1}a_{2}f \frac{2}{3}b^{2}f 2a_{1}bc 2a_{2}bc e^{1234} dt

\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{c^{}}{c} a_{1}a_{2}c^{2} \frac{1}{3}a_{1}a_{2}f \frac{2}{3}c^{2}f 2a_{1}bc 2a_{2}bc e^{1256} dt

\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{1}bcf a_{1}a_{2}f b^{2}f c^{2}f 6a_{1}bc 6a_{2}bc e^{1367} dt

\frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{1}bcf a_{1}a_{2}f b^{2}f c^{2}f 6a_{1}bc 6a_{2}bc e^{1457} dt

\frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{2}bcf a_{1}a_{2}f b^{2}f c^{2}f 6a_{1}bc 6a_{2}bc e^{2357} dt

\frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} a_{2}bcf a_{1}a_{2}f b^{2}f c^{2}f 6a_{1}bc 6a_{2}bc e^{2467} dt

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} \frac{1}{3}b^{2}f \frac{1}{3}c^{2}f 2a_{1}bc 2a_{2}bc e^{3456} dt

The equation dΩ 0 therefore is equivalent to the following system of ordinary differential equations:

\begin{matrix}\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{b^{}}{b} & \frac{1}{3} \frac{f}{b^{2}} \frac{2}{3} \frac{f}{a_{1}a_{2}} 2\frac{c}{a_{2}b} 2\frac{c}{a_{1}b} \\ \frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2\frac{c^{}}{c} & \frac{1}{3} \frac{f}{c^{2}} \frac{2}{3} \frac{f}{a_{1}a_{2}} 2\frac{b}{a_{2}c} 2\frac{b}{a_{1}c} \\ \frac{a_{1}^{}}{a_{1}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} & \frac{a_{2}}{bc} \frac{b}{a_{1}c} \frac{c}{a_{1}b} 6\frac{1}{f} 6\frac{a_{2}}{a_{1}f} \\ \frac{a_{2}^{}}{a_{2}} \frac{b^{}}{b} \frac{c^{}}{c} \frac{f^{}}{f} & \frac{a_{1}}{bc} \frac{b}{a_{2}c} \frac{c}{a_{2}b} 6\frac{a_{1}}{a_{2}f} 6\frac{1}{f} \\ 2\frac{b^{}}{b} 2\frac{c^{}}{c} & \frac{1}{3} \frac{f}{c^{2}} \frac{1}{3} \frac{f}{b^{2}} 2\frac{a_{1}}{bc} 2\frac{a_{2}}{bc} \end{matrix}

This system can be simplified to:

\begin{matrix}\frac{a_{1}^{}}{a_{1}} & \frac{a_{1}}{bc} 3\frac{a_{1}}{a_{2}f} 3\frac{a_{2}}{a_{1}f} \frac{b}{a_{1}c} \frac{c}{a_{1}b} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{a_{2}^{}}{a_{2}} & \frac{a_{2}}{bc} 3\frac{a_{1}}{a_{2}f} 3\frac{a_{2}}{a_{1}f} \frac{b}{a_{2}c} \frac{c}{a_{2}b} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{b^{}}{b} & \frac{1}{2} \frac{a_{1}}{bc} \frac{1}{2} \frac{a_{2}}{bc} \frac{1}{2} \frac{b}{a_{1}c} \frac{1}{2} \frac{b}{a_{2}c} \frac{1}{2} \frac{c}{a_{1}b} \frac{1}{2} \frac{c}{a_{2}b} \frac{1}{6} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{1}{2} \frac{a_{1}}{bc} \frac{1}{2} \frac{a_{2}}{bc} \frac{1}{2} \frac{b}{a_{1}c} \frac{1}{2} \frac{b}{a_{2}c} \frac{1}{2} \frac{c}{a_{1}b} \frac{1}{2} \frac{c}{a_{2}b} \frac{1}{6} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & 3\frac{a_{1}}{a_{2}f} 3\frac{a_{2}}{a_{1}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \frac{1}{6} \frac{f}{b^{2}} \frac{1}{6} \frac{f}{c^{2}} 6\frac{1}{f} \end{matrix}

For cosmetic reasons, we rearrange the summands on the right hand side:

\begin{matrix}\frac{a_{1}^{}}{a_{1}} & \frac{a_{1}^{2} b^{2} c^{2}}{a_{1}bc} 3\frac{a_{2}^{2} a_{1}^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{a_{2}^{}}{a_{2}} & \frac{a_{2}^{2} b^{2} c^{2}}{a_{2}bc} 3\frac{a_{1}^{2} a_{2}^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{b^{}}{b} & \frac{1}{2} \frac{b^{2} a_{1}^{2} c^{2}}{a_{1}bc} \frac{1}{2} \frac{b^{2} a_{2}^{2} c^{2}}{a_{2}bc} \frac{1}{6} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{1}{2} \frac{c^{2} a_{1}^{2} b^{2}}{a_{1}bc} \frac{1}{2} \frac{c^{2} a_{2}^{2} b^{2}}{a_{2}bc} \frac{1}{6} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & 3\frac{a_{1} a_{2} ^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \frac{1}{6} \frac{f}{b^{2}} \frac{1}{6} \frac{f}{c^{2}} \end{matrix}

The above system is equivalent to those equations which can be found in the literature. By changing t into t and f into 6f, we obtain the equations which have been considered by Kanno and Yasui in [49].

As in the previous section, we also reverse the direction of t. We finally obtain the system which we will use throughout Section 5.4:

\begin{matrix}\frac{a_{1}^{}}{a_{1}} & \frac{b^{2} c^{2} a_{1}^{2}}{a_{1}bc} 3\frac{a_{1}^{2} a_{2}^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{a_{2}^{}}{a_{2}} & \frac{b^{2} c^{2} a_{2}^{2}}{a_{2}bc} 3\frac{a_{2}^{2} a_{1}^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \\ \frac{b^{}}{b} & \frac{1}{2} \frac{a_{1}^{2} c^{2} b^{2}}{a_{1}bc} \frac{1}{2} \frac{a_{2}^{2} c^{2} b^{2}}{a_{2}bc} \frac{1}{6} \frac{f}{b^{2}} \\ \frac{c^{}}{c} & \frac{1}{2} \frac{a_{1}^{2} b^{2} c^{2}}{a_{1}bc} \frac{1}{2} \frac{a_{2}^{2} b^{2} c^{2}}{a_{2}bc} \frac{1}{6} \frac{f}{c^{2}} \\ \frac{f^{}}{f} & 3\frac{a_{1} a_{2} ^{2}}{a_{1}a_{2}f} \frac{1}{3} \frac{f}{a_{1}a_{2}} \frac{1}{6} \frac{f}{b^{2}} \frac{1}{6} \frac{f}{c^{2}} \end{matrix}

C.3. THE EINSTEIN CONDITION FOR THE ORBITS EXCEPT N^{1,1} 257

C.3 The Einstein condition for the generic Aloff-Wallach spaces and N^{1, 1} as principal orbits

Let M, g be a Riemannian cohomogeneity-one manifold whose principal orbit is an Aloff-Wallach space N^{k,l}. In this section, we assume that N^{k,l} is generic or N^{1, 1} . The restriction of the metric to a principal orbit we will denote by g, too. In the N^{1, 1} -case, we require that g is diagonal with respect to the basis e_{i} _{1 i 7} of 𝔪. If N^{k,l} is generic, this condition is automatically satisfied. In both cases, we are able to describe g by a matrix of type (5.31). The aim of this section is to express the Einstein condition as a system of ordinary differential equations for the metric functions a, b, c, and f. In order to do this, we have to calculate the Ricci-curvature of N^{k,l}. We will work with the Corollary 3.1.37 from page 51. Before we can make use of the formula from that Corollary, we first have to extend g S^{2} 𝔪 to an inner product on all of 𝔰𝔲 3 . This will be done by setting g e_{8}, e_{8} : 1. We choose the following g-orthonormal basis X_{i} _{1 i 8} of 𝔰𝔲 3 :

\begin{matrix}X_{1} : \frac{1}{a}e_{1} & X_{2} : \frac{1}{a}e_{2} & X_{3} : \frac{1}{b}e_{3} & X_{4} : \frac{1}{b}e_{4} \\ X_{5} : \frac{1}{c}e_{5} & X_{6} : \frac{1}{c}e_{6} & X_{7} : \frac{1}{f} e_{7} & X_{8} : e_{8} \end{matrix}

The explicit formulas for the de^{i}_{p}^{}contain most of the structure constants of the Lie algebra 𝔰𝔲 3 as coefficients. Therefore, the c_{ij}^{k} : g X_{i}, X_{j} , X_{k} can be obtained from the formulas on page 246 and the commutator relations for e_{8}. We have:

c_{27}^{1} 3k 3l \frac{1}{f}  c_{28}^{1} k l  c_{35}^{1}  \frac{a}{bc}  c_{46}^{1}  \frac{a}{bc}
c_{17}^{2} 3k 3l \frac{1}{f} c_{18}^{2} k l  c_{36}^{2} \frac{a}{bc}  c_{45}^{2}  \frac{a}{bc}
c_{15}^{3} \frac{b}{ac}  c_{26}^{3}  \frac{b}{ac}  c_{47}^{3} 3l\frac{1}{f}  c_{48}^{3} 2k l
c_{16}^{4} \frac{b}{ac}  c_{25}^{4} \frac{b}{ac}  c_{37}^{4} 3l\frac{1}{f}  c_{38}^{4} 2k l
c_{13}^{5}  \frac{c}{ab}  c_{24}^{5}  \frac{c}{ab}  c_{67}^{5} 3k \frac{1}{f}  c_{68}^{5} 2l k
c_{14}^{6}  \frac{c}{ab}  c_{23}^{6} \frac{c}{ab}  c_{57}^{6} 3k \frac{1}{f}  c_{58}^{6} 2l k
c_{12}^{7} \frac{k l}{k^{2} lk l^{2}} \frac{f}{a^{2}}  c_{34}^{7} \frac{l}{k^{2} lk l^{2}} \frac{f}{b^{2}} c_{56}^{7}  \frac{k}{k^{2} lk l^{2}} \frac{f}{c^{2}}
c_{12}^{8} \frac{k l}{k^{2} lk l^{2}} \frac{1}{a^{2}}  c_{34}^{8} \frac{2k l}{k^{2} lk l^{2}} \frac{1}{b^{2}} c_{56}^{8} \frac{2l k}{k^{2} lk l^{2}} \frac{1}{c^{2}}

The c_{ij}^{k} which are not contained in the above table and are not determined by the skewsymmetry vanish. We calculate the coefficients of the Killing form κ. The biinvariant metric q on 𝔰𝔲 3 which is defined by q X, Y : tr XY has to be a multiple of κ. We apply formula (3.2) and obtain:

κ X_{1}, X_{1}

2 c_{12}^{7}c_{17}^{2} c_{12}^{8}c_{18}^{2} c_{13}^{5}c_{15}^{3} c_{14}^{6}c_{16}^{4}

2 \frac{k l}{k^{2} lk l^{2}} \frac{f}{a^{2}} 3k 3l \frac{1}{f} \frac{k l}{k^{2} lk l^{2}} \frac{1}{a^{2}} k l  \frac{c}{ab} \frac{b}{ac}  \frac{c}{ab} \frac{b}{ac}

\frac{12}{a^{2}}

Since q e_{1}, e_{1} 2, we have κ 6q. Therefore:

\begin{matrix}κ X_{2}, X_{2} & \frac{12}{a^{2}} \\ κ X_{3}, X_{3} & κ X_{4}, X_{4}  \frac{12}{b^{2}} \\ κ X_{5}, X_{5} & κ X_{6}, X_{6}  \frac{12}{c^{2}} \\ κ X_{7}, X_{7} & \frac{36 k^{2} lk l^{2}}{f^{2}} \end{matrix}

The off-diagonal coefficients of κ vanish, since X_{i} _{1 i 8} is a q-orthogonal basis. We are now able to apply Corollary 3.1.37 and calculate the Ricci-curvature Ric of N^{k,l}. Since Ric S^{2} 𝔪 has to be U 1 _{k,l}-invariant, we can easily prove by Schur’s lemma that Ric is diagonal if N^{k,l} is generic. It is possible to prove that if N^{k,l} N^{1, 1} and g is diagonal, Ric is diagonal, too. We assume without loss of generality that k 1, l 1. Ric S^{2} 𝔪 S^{2} T_{eU 1 _{1, 1}} N^{1, 1} is invariant under all isometries which fix the point eU 1 _{1, 1} N^{1, 1} . The following maps are well-defined and fix eU 1 _{1, 1} :

gU 1 _{1, 1} hgh^{1} U 1 _{1, 1} where h Norm_{SU 3}U 1 _{1, 1} U 1 ^{2}

If the metric on N^{1, 1} is diagonal, all of these maps are isometries. Ric therefore has to be invariant under the action of the normalizer. By decomposing S^{2} 𝔪 into irreducible U 1 ^{2}- modules, we can deduce that Ric has to be diagonal, too. Therefore, the Ricci-tensor is in any case determined by the values of the four coefficients below:

Ric X_{1}, X_{1}  \frac{1}{2} c_{17}^{2} ^{2} c_{15}^{3} ^{2} c_{16}^{4} ^{2} c_{13}^{5} ^{2} c_{14}^{6} ^{2} c_{12}^{7} ^{2}

\frac{1}{4} c_{27}^{1} ^{2} c_{35}^{1} ^{2} c_{46}^{1} ^{2} c_{53}^{1} ^{2} c_{64}^{1} ^{2} c_{72}^{1} ^{2} \frac{1}{2}κ X_{1}, X_{1}

\frac{1}{2} 3k 3l ^{2} \frac{1}{f^{2}} \frac{b^{2}}{a^{2}c^{2}} \frac{b^{2}}{a^{2}c^{2}} \frac{c^{2}}{a^{2}b^{2}} \frac{c^{2}}{a^{2}b^{2}} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}}

\frac{1}{4} 3k 3l ^{2} \frac{1}{f^{2}} \frac{a^{2}}{b^{2}c^{2}} \frac{a^{2}}{b^{2}c^{2}} \frac{a^{2}}{b^{2}c^{2}} \frac{a^{2}}{b^{2}c^{2}} 3k 3l ^{2} \frac{1}{f^{2}} \frac{6}{a^{2}}

\frac{6}{a^{2}} \frac{1}{2} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}} \frac{a^{4} b^{4} c^{4}}{a^{2}b^{2}c^{2}}

Ric X_{3}, X_{3}  \frac{1}{2} c_{35}^{1} ^{2} c_{36}^{2} ^{2} c_{37}^{4} ^{2} c_{31}^{5} ^{2} c_{32}^{6} ^{2} c_{34}^{7} ^{2}

\frac{1}{4} c_{15}^{3} ^{2} c_{26}^{3} ^{2} c_{47}^{3} ^{2} c_{51}^{3} ^{2} c_{62}^{3} ^{2} c_{74}^{3} ^{2} \frac{1}{2}κ X_{3}, X_{3}

\frac{1}{2} \frac{a^{2}}{b^{2}c^{2}} \frac{a^{2}}{b^{2}c^{2}} 3l ^{2} \frac{1}{f^{2}} \frac{c^{2}}{a^{2}b^{2}} \frac{c^{2}}{a^{2}b^{2}} \frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}}

\frac{1}{4} \frac{b^{2}}{a^{2}c^{2}} \frac{b^{2}}{a^{2}c^{2}} 3l ^{2} \frac{1}{f^{2}} \frac{b^{2}}{a^{2}c^{2}} \frac{b^{2}}{a^{2}c^{2}} 3l ^{2} \frac{1}{f^{2}} \frac{6}{b^{2}}

\frac{6}{b^{2}} \frac{1}{2} \frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}} \frac{b^{4} a^{4} c^{4}}{a^{2}b^{2}c^{2}}

Ric X_{5}, X_{5}  \frac{1}{2} c_{53}^{1} ^{2} c_{54}^{2} ^{2} c_{51}^{3} ^{2} c_{52}^{4} ^{2} c_{57}^{6} ^{2} c_{56}^{7} ^{2}

\frac{1}{4} c_{13}^{5} ^{2} c_{24}^{5} ^{2} c_{31}^{5} ^{2} c_{42}^{5} ^{2} c_{67}^{5} ^{2} c_{76}^{5} ^{2} \frac{1}{2}κ X_{5}, X_{5}

\frac{1}{2} \frac{a^{2}}{b^{2}c^{2}} \frac{a^{2}}{b^{2}c^{2}} \frac{b^{2}}{a^{2}c^{2}} \frac{b^{2}}{a^{2}c^{2}} 3k ^{2} \frac{1}{f^{2}} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}}

\frac{1}{4} \frac{c^{2}}{a^{2}b^{2}} \frac{c^{2}}{a^{2}b^{2}} \frac{c^{2}}{a^{2}b^{2}} \frac{c^{2}}{a^{2}b^{2}} 3k ^{2} \frac{1}{f^{2}} 3k ^{2} \frac{1}{f^{2}} \frac{6}{c^{2}}

\frac{6}{c^{2}} \frac{1}{2} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}} \frac{c^{4} a^{4} b^{4}}{a^{2}b^{2}c^{2}}

Ric X_{7}, X_{7}  \frac{1}{2} c_{72}^{1} ^{2} c_{71}^{2} ^{2} c_{74}^{3} ^{2} c_{73}^{4} ^{2} c_{76}^{5} ^{2} c_{75}^{6} ^{2}

\frac{1}{4} c_{12}^{7} ^{2} c_{21}^{7} ^{2} c_{34}^{7} ^{2} c_{43}^{7} ^{2} c_{56}^{7} ^{2} c_{65}^{7} ^{2} \frac{1}{2}κ X_{7}, X_{7}

\frac{1}{2} 3k 3l ^{2} \frac{1}{f^{2}} 3k 3l ^{2} \frac{1}{f^{2}} 3l ^{2} \frac{1}{f^{2}} 3l ^{2} \frac{1}{f^{2}} 3k ^{2} \frac{1}{f^{2}} 3k ^{2} \frac{1}{f^{2}}

\frac{1}{4} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}} \frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}}

\frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}} \frac{18 k^{2} lk l^{2}}{f^{2}}

\frac{1}{2} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}} \frac{1}{2} \frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}} \frac{1}{2} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}}

With help of the formulas (3.13), we see that the metric g is Einstein with Einstein constant λ if and only if the following equations hold:

\frac{a^{}}{a} \frac{a^{2}}{a^{2}} \frac{a^{}}{a} 2\frac{a^{}}{a} 2\frac{b^{}}{b} 2\frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{a^{2}} \frac{1}{2} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}} \frac{a^{4} b^{4} c^{4}}{a^{2}b^{2}c^{2}}  λ
\frac{b^{}}{b} \frac{b^{2}}{b^{2}} \frac{b^{}}{b} 2\frac{a^{}}{a} 2\frac{b^{}}{b} 2\frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{b^{2}} \frac{1}{2} \frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}} \frac{b^{4} a^{4} c^{4}}{a^{2}b^{2}c^{2}}  λ
\frac{c^{}}{c} \frac{c^{2}}{c^{2}} \frac{c^{}}{c} 2\frac{a^{}}{a} 2\frac{b^{}}{b} 2\frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{c^{2}} \frac{1}{2} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}} \frac{c^{4} a^{4} b^{4}}{a^{2}b^{2}c^{2}}  λ

\frac{f^{}}{f} \frac{f^{2}}{f^{2}} \frac{f^{}}{f} 2\frac{a^{}}{a} 2\frac{b^{}}{b} 2\frac{c^{}}{c} \frac{f^{}}{f} \frac{1}{2} \frac{k l ^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{a^{4}} \frac{1}{2} \frac{l^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{b^{4}}

\frac{1}{2} \frac{k^{2}}{k^{2} lk l^{2} ^{2}} \frac{f^{2}}{c^{4}}  λ
2\frac{a^{}}{a} 2\frac{b^{}}{b} 2\frac{c^{}}{c} \frac{f^{}}{f}  λ

As we have remarked in 3.2.23, the above system should not be considered as over-determined, since it has unique short-time solutions for any choice of the metric and its first derivative on a principal orbit. The above equations are invariant under the change of the sign of t. Therefore, we may assume that t takes the same direction as in Section C.1. By a straightforward but lengthy calculation, it is possible to prove that the above system is automatically satisfied with λ 0 if a, b, c, and f satisfy the equations for the holonomy reduction from Section C.1. This is not surprising, since any metric whose holonomy is contained in Spin 7 has to be Ricci-flat. Nevertheless, that calculation is an additional test that our equations are correct.

C.4. THE EINSTEIN CONDITION FOR N^{1,1} 261

C.4 The Einstein condition for the exceptional Aloff-Wallach space N^{1,1} as principal orbit

Again, let M, g be a Riemannian cohomogeneity-one manifold whose principal orbit is an Aloff-Wallach space. In this section, we assume that the principal orbit is the exceptional Aloff-Wallach space N^{1,1} and that the restriction of the metric to any principal orbit is a diagonal metric of type (5.33). The aim of this section is to express the Einstein condition on g as a system of ordinary differential equations for the metric functions a_{1}, a_{2}, b, c, and f. As in the previous section, we will first calculate the Ricci-curvature of N^{1,1} with help of Corollary 3.1.37. By defining g e_{8}, e_{8} : 1, g is as before extended to an inner product on all of 𝔰𝔲 3 . We choose the following g-orthonormal basis X_{i} _{1 i 8} of 𝔰𝔲 3 :

\begin{matrix}X_{1} : \frac{1}{a_{1}} e_{1} & X_{2} : \frac{1}{a_{2}} e_{2} & X_{3} : \frac{1}{b}e_{3} & X_{4} : \frac{1}{b}e_{4} \\ X_{5} : \frac{1}{c}e_{5} & X_{6} : \frac{1}{c}e_{6} & X_{7} : \frac{1}{f} e_{7} & X_{8} : e_{8} \end{matrix}

If i, j, k 1, 2 , the coefficients c_{ij}^{k}, which describe the commutator relations of 𝔰𝔲 3 , remain the same as in the previous section. If exactly one of the indices is 1 (or 2), we have to replace the metric function a by a_{1} (or a_{2}). The remaining coefficients are:

\begin{matrix}c_{27}^{1} 6\frac{a_{1}}{a_{2}f} & c_{17}^{2} 6\frac{a_{2}}{a_{1}f} & c_{12}^{7} \frac{2}{3} \frac{f}{a_{1}a_{2}} \\ c_{28}^{1} 0 & c_{18}^{2} 0 & c_{12}^{8} 0 \end{matrix}

For the Killing-form κ, we obtain:

\begin{matrix}κ X_{1}, X_{1}  \frac{12}{a_{1}^{2}} & κ X_{2}, X_{2}  \frac{12}{a_{2}^{2}} & κ X_{3}, X_{3}  \frac{12}{b^{2}} & κ X_{4}, X_{4}  \frac{12}{b^{2}} \\ κ X_{5}, X_{5}  \frac{12}{c^{2}} & κ X_{6}, X_{6}  \frac{12}{c^{2}} & κ X_{7}, X_{7}  \frac{108}{f^{2}} \end{matrix}

The off-diagonal coefficients of κ vanish for the same reason as in the previous section. Since the Ricci-tensor has to be U 1 _{1,1}-invariant, we have Ric X_{3}, X_{3} Ric X_{4}, X_{4} and Ric X_{5}, X_{5} Ric X_{6}, X_{6} . Unlike in the generic case, span X_{1}, X_{2} is not an irreducible U 1 _{1,1}-module. Therefore, Ric X_{1}, X_{1} does not necessarily equal Ric X_{2}, X_{2} . By some calculations, which are similar to those in the previous section, we obtain:

Ric X_{1}, X_{1}  \frac{1}{2} 36\frac{a_{2}^{2}}{a_{1}^{2}f^{2}} \frac{b^{2}}{a_{1}^{2}c^{2}} \frac{b^{2}}{a_{1}^{2}c^{2}} \frac{c^{2}}{a_{1}^{2}b^{2}} \frac{c^{2}}{a_{1}^{2}b^{2}} \frac{4}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}}

\frac{1}{4} 36\frac{a_{1}^{2}}{a_{2}^{2}f^{2}} \frac{a_{1}^{2}}{b^{2}c^{2}} \frac{a_{1}^{2}}{b^{2}c^{2}} \frac{a_{1}^{2}}{b^{2}c^{2}} \frac{a_{1}^{2}}{b^{2}c^{2}} 36\frac{a_{1}^{2}}{a_{2}^{2}f^{2}} \frac{6}{a_{1}^{2}}

\frac{6}{a_{1}^{2}} \frac{2}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} 18\frac{a_{1}^{4} a_{2}^{4}}{a_{1}^{2}a_{2}^{2}f^{2}} \frac{a_{1}^{4} b^{4} c^{4}}{a_{1}^{2}b^{2}c^{2}}

Ric X_{2}, X_{2}  \frac{1}{2} 36\frac{a_{1}^{2}}{a_{2}^{2}f^{2}} \frac{b^{2}}{a_{2}^{2}c^{2}} \frac{b^{2}}{a_{2}^{2}c^{2}} \frac{c^{2}}{a_{2}^{2}b^{2}} \frac{c^{2}}{a_{2}^{2}b^{2}} \frac{4}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}}

\frac{1}{4} 36\frac{a_{2}^{2}}{a_{1}^{2}f^{2}} \frac{a_{2}^{2}}{b^{2}c^{2}} \frac{a_{2}^{2}}{b^{2}c^{2}} \frac{a_{2}^{2}}{b^{2}c^{2}} \frac{a_{2}^{2}}{b^{2}c^{2}} 36\frac{a_{2}^{2}}{a_{1}^{2}f^{2}} \frac{6}{a_{2}^{2}}

\frac{6}{a_{2}^{2}} \frac{2}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} 18\frac{a_{2}^{4} a_{1}^{4}}{a_{1}^{2}a_{2}^{2}f^{2}} \frac{a_{2}^{4} b^{4} c^{4}}{a_{2}^{2}b^{2}c^{2}}

Ric X_{3}, X_{3}  \frac{1}{2} \frac{a_{1}^{2}}{b^{2}c^{2}} \frac{a_{2}^{2}}{b^{2}c^{2}} 9\frac{1}{f^{2}} \frac{c^{2}}{a_{1}^{2}b^{2}} \frac{c^{2}}{a_{2}^{2}b^{2}} \frac{1}{9} \frac{f^{2}}{b^{4}}

\frac{1}{4} \frac{b^{2}}{a_{1}^{2}c^{2}} \frac{b^{2}}{a_{2}^{2}c^{2}} 9\frac{1}{f^{2}} \frac{b^{2}}{a_{1}^{2}c^{2}} \frac{b^{2}}{a_{2}^{2}c^{2}} 9\frac{1}{f^{2}} \frac{6}{b^{2}}

\frac{6}{b^{2}} \frac{1}{18} \frac{f^{2}}{b^{4}} \frac{b^{4} a_{1}^{4} c^{4}}{2a_{1}^{2}b^{2}c^{2}} \frac{b^{4} a_{2}^{4} c^{4}}{2a_{2}^{2}b^{2}c^{2}}

Ric X_{5}, X_{5}  \frac{1}{2} \frac{a_{1}^{2}}{b^{2}c^{2}} \frac{a_{2}^{2}}{b^{2}c^{2}} \frac{b^{2}}{a_{1}^{2}c^{2}} \frac{b^{2}}{a_{2}^{2}c^{2}} 9\frac{1}{f^{2}} \frac{1}{9} \frac{f^{2}}{c^{4}}

\frac{1}{4} \frac{c^{2}}{a_{1}^{2}b^{2}} \frac{c^{2}}{a_{2}^{2}b^{2}} \frac{c^{2}}{a_{1}^{2}b^{2}} \frac{c^{2}}{a_{2}^{2}b^{2}} 9\frac{1}{f^{2}} 9\frac{1}{f^{2}} \frac{6}{c^{2}}

\frac{6}{c^{2}} \frac{1}{18} \frac{f^{2}}{c^{4}} \frac{c^{4} a_{1}^{4} b^{4}}{2a_{1}^{2}b^{2}c^{2}} \frac{c^{4} a_{2}^{4} b^{4}}{2a_{2}^{2}b^{2}c^{2}}

Ric X_{7}, X_{7}  \frac{1}{2} 36\frac{a_{1}^{2}}{a_{2}^{2}f^{2}} 36\frac{a_{2}^{2}}{a_{1}^{2}f^{2}} 9\frac{1}{f^{2}} 9\frac{1}{f^{2}} 9\frac{1}{f^{2}} 9\frac{1}{f^{2}}

\frac{1}{4} \frac{4}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} \frac{4}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} \frac{1}{9} \frac{f^{2}}{b^{4}} \frac{1}{9} \frac{f^{2}}{b^{4}} \frac{1}{9} \frac{f^{2}}{c^{4}} \frac{1}{9} \frac{f^{2}}{c^{4}} \frac{54}{f^{2}}

\frac{36}{f^{2}} 18\frac{a_{1}^{2}}{a_{2}^{2}f^{2}} 18\frac{a_{2}^{2}}{a_{1}^{2}f^{2}} \frac{2}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} \frac{1}{18} \frac{f^{2}}{b^{4}} \frac{1}{18} \frac{f^{2}}{c^{4}}

Unfortunately, we cannot prove by the same arguments as in Section C.3 that the Riccitensor is diagonal. The reason for this is that those arguments can be applied only if g is invariant under the standard maximal torus of SU 3 . In that situation, we necessarily have a_{1}^{2} a_{2}^{2}. Since we want to choose a_{1} and a_{2} freely, we have to explicitly calculate the possible non-diagonal coefficients of the Ricci-tensor. Since the basis X_{i} _{1 i 8} is q-orthogonal, the summand \frac{1}{2}κ in the formula for the Ricci-curvature vanishes. We obtain:

Ric X_{1}, X_{2}  \frac{1}{2} c_{17}^{2}c_{27}^{2} c_{15}^{3}c_{25}^{3} c_{16}^{4}c_{26}^{4} c_{13}^{5}c_{23}^{5} c_{14}^{6}c_{24}^{6} c_{12}^{7}c_{22}^{7}

\frac{1}{4} c_{27}^{1}c_{27}^{2} c_{35}^{1}c_{35}^{2} c_{46}^{1}c_{46}^{2} c_{53}^{1}c_{53}^{2} c_{64}^{1}c_{64}^{2} c_{72}^{1}c_{72}^{2}

Ric X_{1}, X_{7}  \frac{1}{2} c_{17}^{2}c_{77}^{2} c_{15}^{3}c_{75}^{3} c_{16}^{4}c_{76}^{4} c_{13}^{5}c_{73}^{5} c_{14}^{6}c_{74}^{6} c_{12}^{7}c_{72}^{7}

\frac{1}{4} c_{27}^{1}c_{27}^{7} c_{35}^{1}c_{35}^{7} c_{46}^{1}c_{46}^{7} c_{53}^{1}c_{53}^{7} c_{64}^{1}c_{64}^{7} c_{72}^{1}c_{72}^{7}

0

Ric X_{2}, X_{7}  \frac{1}{2} c_{27}^{1}c_{77}^{1} c_{26}^{3}c_{76}^{3} c_{25}^{4}c_{75}^{4} c_{24}^{5}c_{74}^{5} c_{23}^{6}c_{73}^{6} c_{21}^{7}c_{71}^{7}

\frac{1}{4} c_{17}^{2}c_{17}^{7} c_{36}^{2}c_{36}^{7} c_{45}^{2}c_{45}^{7} c_{54}^{2}c_{54}^{7} c_{63}^{2}c_{63}^{7} c_{71}^{2}c_{71}^{7}

0

Ric X_{3}, X_{5}  \frac{1}{2} c_{35}^{1}c_{55}^{1} c_{36}^{2}c_{56}^{2} c_{37}^{4}c_{57}^{4} c_{31}^{5}c_{51}^{5} c_{32}^{6}c_{52}^{6} c_{34}^{7}c_{54}^{7}

\frac{1}{4} c_{15}^{3}c_{15}^{5} c_{26}^{3}c_{26}^{5} c_{47}^{3}c_{47}^{5} c_{51}^{3}c_{51}^{5} c_{62}^{3}c_{62}^{5} c_{74}^{3}c_{74}^{5}

0

Ric X_{3}, X_{6}  \frac{1}{2} c_{35}^{1}c_{65}^{1} c_{36}^{2}c_{66}^{2} c_{37}^{4}c_{67}^{4} c_{31}^{5}c_{61}^{5} c_{32}^{6}c_{62}^{6} c_{34}^{7}c_{64}^{7}

\frac{1}{4} c_{15}^{3}c_{15}^{6} c_{26}^{3}c_{26}^{6} c_{47}^{3}c_{47}^{6} c_{51}^{3}c_{51}^{6} c_{62}^{3}c_{62}^{6} c_{74}^{3}c_{74}^{6}

0

The above calculations prove that the Ricci-tensor is indeed diagonal. By inserting the coefficients of the Ricci-tensor into the formulas (3.13), we obtain the following system of ordinary differential equations which is equivalent to the Einstein condition:

\frac{a_{1}^{}}{a_{1}} \frac{a_{1}^{2}}{a_{1}^{2}} \frac{a_{1}^{}}{a_{1}} \frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2 \frac{b^{}}{b} 2 \frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{a_{1}^{2}} \frac{2}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} 18 \frac{a_{1}^{4} a_{2}^{4}}{a_{1}^{2}a_{2}^{2}f^{2}} \frac{a_{1}^{4} b^{4} c^{4}}{a_{1}^{2}b^{2}c^{2}}  λ
\frac{a_{2}^{}}{a_{2}} \frac{a_{2}^{2}}{a_{2}^{2}} \frac{a_{2}^{}}{a_{2}} \frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2 \frac{b^{}}{b} 2 \frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{a_{2}^{2}} \frac{2}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} 18 \frac{a_{2}^{4} a_{1}^{4}}{a_{1}^{2}a_{2}^{2}f^{2}} \frac{a_{2}^{4} b^{4} c^{4}}{a_{2}^{2}b^{2}c^{2}}  λ
\frac{b^{}}{b} \frac{b^{2}}{b^{2}} \frac{b^{}}{b} \frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2 \frac{b^{}}{b} 2 \frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{b^{2}} \frac{1}{18} \frac{f^{2}}{b^{4}} \frac{b^{4} a_{1}^{4} c^{4}}{2a_{1}^{2}b^{2}c^{2}}  \frac{b^{4} a_{2}^{4} c^{4}}{2a_{2}^{2}b^{2}c^{2}}  λ
\frac{c^{}}{c} \frac{c^{2}}{c^{2}} \frac{c^{}}{c} \frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2 \frac{b^{}}{b} 2 \frac{c^{}}{c} \frac{f^{}}{f}  \frac{6}{c^{2}} \frac{1}{18} \frac{f^{2}}{c^{4}} \frac{c^{4} a_{1}^{4} b^{4}}{2a_{1}^{2}b^{2}c^{2}}  \frac{c^{4} a_{2}^{4} b^{4}}{2a_{2}^{2}b^{2}c^{2}}  λ
\frac{f^{}}{f} \frac{f^{2}}{f^{2}} \frac{f^{}}{f} \frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2 \frac{b^{}}{b} 2 \frac{c^{}}{c} \frac{f^{}}{f}  \frac{36}{f^{2}} 18 \frac{a_{1}^{2}}{a_{2}^{2}f^{2}} 18 \frac{a_{2}^{2}}{a_{1}^{2}f^{2}} \frac{2}{9} \frac{f^{2}}{a_{1}^{2}a_{2}^{2}} \frac{1}{18} \frac{f^{2}}{b^{4}} \frac{1}{18} \frac{f^{2}}{c^{4}}  λ
\frac{a_{1}^{}}{a_{1}} \frac{a_{2}^{}}{a_{2}} 2 \frac{b^{}}{b} 2 \frac{c^{}}{c} \frac{f^{}}{f}  λ

As in the previous section, λ denotes the Einstein constant. The remarks which we have made on the system in Section C.3 are also true for the above equations: It is possible to change the sign of t without changing the equations. Therefore, we can assume that this sign is the same as in Section C.2. Furthermore, the above system is not overdetermined and we can prove that it is automatically satisfied with λ 0 if a_{1}, a_{2}, b, c, and f satisfy the first-order system which we have obtained in C.2.

264 APPENDIX C. THE ALOFF-WALLACH SPACES AS PRINCIPAL ORBIT

Appendix D

Calculations for the principal orbit SU 3 U 1 ^{2}

In this chapter, we will carry out the calculations which we have omitted in Section 6.2. In particular, we will deduce the equations for the reduction of the holonomy to a subgroup of G_{2}.

Let M, g be a Riemannian manifold equipped with an isometric SU 3 -action of cohomogeneity one with principal orbits of type SU 3 U 1 ^{2}. We have proven that any SU 3 - invariant G_{2}-structure on M has a frame which is described by (6.10). This frame depends on a parameter θ. As in Section 6.2, we denote the three-form which is associated to this frame by ω_{θ}. First, we compute ω_{0}. We obtain:

ω_{0} f^{123} f^{145} f^{167} f^{246} f^{257} f^{347} f^{356}
a^{2} e^{12} dt b^{2} e^{34} dt c^{2} e^{56} dt abc e^{236} abc e^{245} abc e^{135} abc e^{146}
abc e^{135} abc e^{146} abc e^{236} abc e^{245} a^{2} e^{12} dt b^{2} e^{34} dt c^{2} e^{56} dt

Let R_{θ} be the matrix

cos θ sin θ
sin θ cos θ
cos θ sin θ
sin θ cos θ
cos θ sin θ
sin θ cos θ

acting on the basis (6.8). It is easily possible to show the following relations:

266 APPENDIX D. THE PRINCIPAL ORBIT SU 3 U 1 ^{2}

R_{θ}^{}e^{12}  e^{12}
R_{θ}^{}e^{34}  e^{34}
R_{θ}^{}e^{56}  e^{56}
R_{θ}^{}e^{135} e^{146} e^{236} e^{245}  cos 3θ e^{135} e^{146} e^{236} e^{245}
sin 3θ e^{136} e^{145} e^{235} e^{246}
R_{θ}^{}e^{136} e^{145} e^{235} e^{246}  sin 3θ e^{135} e^{146} e^{236} e^{245}
cos 3θ e^{136} e^{145} e^{235} e^{246}

(D.1)

where the asterisk denotes the canonical action of GL 6 on ^{}T^{}M. Therefore, we obtain for the three-form ω_{θ}:

\begin{aligned}ω_{θ} & abc cos 3θ e^{135} e^{146} e^{236} e^{245} abc sin 3θ e^{136} e^{145} e^{235} e^{246} \\ & a^{2} e^{12} dt b^{2} e^{34} dt c^{2} e^{56} dt . \end{aligned}

By a short calculation, we can determine the four-form ω_{0}:

ω_{0} f^{1247} f^{1256} f^{1346} f^{1357} f^{2345} f^{2367} f^{4567}
dt abc e^{235} dt abc e^{246} dt abc e^{136} dt abc e^{145}
a^{2}b^{2} e^{2134} a^{2}c^{2} e^{2165} b^{2}c^{2} e^{3465}
abc e^{235} dt abc e^{246} dt abc e^{136} dt abc e^{145} dt
a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
abc e^{136} dt abc e^{145} dt abc e^{235} dt abc e^{246} dt

With help of the formulas (D.1), we obtain for ω_{θ}:

ω_{θ} a^{2}b^{2} e^{1234} a^{2}c^{2} e^{1256} b^{2}c^{2} e^{3456}
abc cos 3θ e^{136} e^{145} e^{235} e^{246} dt
abc sin 3θ e^{135} e^{146} e^{236} e^{245} dt

Analogously to Appendix C, we obtain for the exterior derivatives of the one-forms e^{i}^{}which are dual to the Killing vector fields e_{i}^{}:

\begin{matrix}de_{p}^{1} & e_{p}^{35} e_{p}^{46} \\ de_{p}^{2} & e_{p}^{36} e_{p}^{45} \\ de_{p}^{3} & e_{p}^{15} e_{p}^{26} \\ de_{p}^{4} & e_{p}^{16} e_{p}^{25} \\ de_{p}^{5} & e_{p}^{13} e_{p}^{24} \\ de_{p}^{6} & e_{p}^{14} e_{p}^{23} \end{matrix}

In the above equations, p SU 3 U 1 ^{2} denotes the point whose tangent space we identify with 𝔪. We are now able to determine exterior derivative of ω_{θ}:

dω_{θ}  \frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} cos 3θ 3θ^{}sin 3θ abc e^{135} e^{146} e^{236} e^{245} dt

abc cos 3θ e^{3535} e^{4635} e^{1155} e^{1265} e^{1313} e^{1324}

abc cos 3θ e^{3546} e^{4646} e^{1166} e^{1256} e^{1414} e^{1423}

abc cos 3θ e^{3636} e^{4536} e^{2156} e^{2266} e^{2314} e^{2323}

abc cos 3θ e^{3645} e^{4545} e^{2165} e^{2255} e^{2413} e^{2424}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} sin 3θ 3θ^{}cos 3θ abc e^{136} e^{145} e^{235} e^{246} dt

abc sin 3θ e^{3536} e^{4636} e^{1156} e^{1266} e^{1314} e^{1323}

abc sin 3θ e^{3545} e^{4645} e^{1165} e^{1255} e^{1413} e^{1424}

abc sin 3θ e^{3635} e^{4535} e^{2155} e^{2265} e^{2313} e^{2324}

abc sin 3θ e^{3646} e^{4546} e^{2166} e^{2256} e^{2414} e^{2423}

a^{2} e^{35} e^{46} e^{2} dt a^{2} e^{1} e^{36} e^{45} dt

b^{2} e^{15} e^{26} e^{4} dt b^{2} e^{3} e^{16} e^{25} dt

c^{2} e^{13} e^{24} e^{6} dt c^{2} e^{5} e^{14} e^{23} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} cos 3θ 3θ^{}sin 3θ abc e^{135} e^{146} e^{236} e^{245} dt

abc cos 3θ e^{3456} e^{1256} e^{1234} abc cos 3θ e^{3456} e^{1256} e^{1234}

abc cos 3θ e^{3456} e^{1256} e^{1234} abc cos 3θ e^{3456} e^{1256} e^{1234}

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} sin 3θ 3θ^{}cos 3θ abc e^{136} e^{145} e^{235} e^{246} dt

a^{2} e^{235} dt e^{246} dt a^{2} e^{136} dt e^{145} dt

b^{2} e^{145} dt e^{246} dt b^{2} e^{136} dt e^{235} dt

c^{2} e^{136} dt e^{246} dt c^{2} e^{145} dt e^{235} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} cos 3θ 3θ^{}sin 3θ abc e^{135} e^{146} e^{236} e^{245} dt

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} sin 3θ 3θ^{}cos 3θ abc e^{136} e^{145} e^{235} e^{246} dt

a^{2} b^{2} c^{2} e^{136} e^{145} e^{235} e^{246} dt

4abc cos 3θ e^{1234} e^{1256} e^{3456}

The equation dω_{θ} 0 therefore is equivalent to:

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} abc cos 3θ 3abc θ^{}sin 3θ  0
\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} abc sin 3θ 3abc θ^{}cos 3θ  a^{2} b^{2} c^{2}
4abc cos 3θ  0

Since a, b, c 0, cos 3θ has to be zero and the above system becomes:

\frac{a^{}}{a} \frac{b^{}}{b} \frac{c^{}}{c} \frac{a^{2} b^{2} c^{2}}{abc} .

We assume that θ \frac{π}{6} . In this situation, the sign on the left hand side is positive. By changing the orientation, we could replace \frac{}{t} by \frac{}{t} in the frame. That change would make the sign negative again. Therefore, our assumption is without loss of generality justified. Next, we compute d ω_{θ}:

d ω_{θ} 2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} dt

a^{2}b^{2} e^{35234} e^{46234} e^{13634} e^{14534} e^{12154} e^{12264} e^{12316} e^{12325}

2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} dt

a^{2}c^{2} e^{35256} e^{46256} e^{13656} e^{14556} e^{12136} e^{12246} e^{12514} e^{12523}

2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

b^{2}c^{2} e^{15456} e^{26456} e^{31656} e^{32556} e^{34136} e^{34246} e^{34514} e^{34523}

abc cos 3θ e^{35} e^{46} e^{36} e^{1} e^{15} e^{26} e^{6} e^{13} e^{14} e^{23} dt

abc cos 3θ e^{35} e^{46} e^{45} e^{1} e^{16} e^{25} e^{5} e^{14} e^{13} e^{24} dt

abc cos 3θ e^{36} e^{45} e^{35} e^{2} e^{15} e^{26} e^{5} e^{23} e^{13} e^{24} dt

abc cos 3θ e^{36} e^{45} e^{46} e^{2} e^{16} e^{25} e^{6} e^{24} e^{14} e^{23} dt

abc sin 3θ e^{35} e^{46} e^{35} e^{1} e^{15} e^{26} e^{5} e^{13} e^{13} e^{24} dt

abc sin 3θ e^{35} e^{46} e^{46} e^{1} e^{16} e^{25} e^{6} e^{14} e^{14} e^{23} dt

abc sin 3θ e^{36} e^{45} e^{36} e^{2} e^{15} e^{26} e^{6} e^{23} e^{14} e^{23} dt

abc sin 3θ e^{36} e^{45} e^{45} e^{2} e^{16} e^{25} e^{5} e^{24} e^{13} e^{24} dt

2\frac{a^{}}{a} 2\frac{b^{}}{b} a^{2}b^{2} e^{1234} dt 2\frac{a^{}}{a} 2\frac{c^{}}{c} a^{2}c^{2} e^{1256} dt 2\frac{b^{}}{b} 2\frac{c^{}}{c} b^{2}c^{2} e^{3456} dt

4abc sin 3θ e^{1234} e^{1256} e^{3456} dt Since we assume that θ \frac{π}{6} , the equation d ω_{θ} 0 is equivalent to:

\frac{a^{}}{a} \frac{b^{}}{b}  2\frac{c}{ab}
\frac{a^{}}{a} \frac{c^{}}{c}  2\frac{b}{ac}
\frac{b^{}}{b} \frac{c^{}}{c}  2\frac{a}{bc}

If we add these equations up, we obtain the equation for dω_{θ} 0. Therefore, the above system describes the condition for the holonomy reduction to a subgroup of G_{2}. This systems finally simplifies to:

\begin{matrix}\frac{a^{}}{a} & \frac{b^{2} c^{2} a^{2}}{abc} \\ \frac{b^{}}{b} & \frac{a^{2} c^{2} b^{2}}{abc} \\ \frac{c^{}}{c} & \frac{a^{2} b^{2} c^{2}}{abc} \end{matrix}  (D.2)

The system (D.2) coincides with the system which has been found by Cleyton and Swann 270 APPENDIX D. THE PRINCIPAL ORBIT SU 3 U 1 ^{2}

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