Lehrstuhl VII: Differentialgeometrie
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Item The symplectic Dirac and Dolbeault operators and the Lichnerowicz Laplacian(2022) Iskandarov, Artur; Schwachhoefer, Lorenz Johannes; Gutt, SimoneThe Laplacian operator $\mathcal{P}$ arising as a commutator of two symplectic Dirac operators introduced in [1] in the context of Schrödinger picture and rediscovered in [2] as a commutator of symplectic Dirac-Dolbeault operators in the Fock picture admits a natural geometric interpretation, which is described in this thesis. One part of the thesis deals with the central extension of the metaplectic group $Mp$ and its maximal compact subgroup $MU^c$. We discuss the existence and classification of $Mp-$ and $\Mp^c-$structures on almost symplectic manifolds and describe the correspondence with $Spin-$ resp. $Spin^c-$structures. After that we introduce Laplacians $\Delta^{1,0}$ and $\Delta^{0,1}$ on the complex symmetric algebra arising from the Lichnerowicz Laplacian $\Delta$ after the complexification of the tangent bundle and complex bilinear extension of the canonical Hermitian connection, when considering almost Kähler manifolds. Having explained the reason, why the symplectic spinor bundle can be regarded as the tensor bundle $L\otimes S^{*,0}(M)$, for some Hermitian line bundle $L$ over $M$, we adjust appropriately the symplectic Clifford multiplication and give then an interpretation of the symplectic Dirac-Dolbeault operators, regarding a symplectic manifold as almost Kähler equipped with the canonical Hermitian connection, and prove the main theorem, which can be stated as follows. Theorem: If the line bundle $L$ is trivial, the operator $\mathcal{P}$ coincides with $-\Delta^{1,0}$.Item G2-orbifolds with ADE-singularities(2017-04) Reidegeld, Frank; Schwachhöfer, LorenzMannigfaltigkeiten mit Holonomie G2 sind sowohl in der reinen Mathematik als auch der mathematischen Physik ein Gegenstand aktiver Forschung. Das Ziel dieser Arbeit ist es, zusätzlich zu den bekannten glatten Beispielen Orbifaltigkeiten mit einer G2-Struktur und Singularitäten von einem speziellen Typ zu finden. Hierzu modifizieren wir die bekannten Konstruktionsmethoden für G2-Mannigfaltigkeiten. Neben G2-Orbifaltigkeiten mit einer Vielzahl von unterschiedlichen Singularitäten finden wir auch eine glatte G2-Mannigfaltigkeit, deren Betti-Zahlen keinem bisher bekannten Beispiel entsprechen.Item Construction of nonnegatively curved invariant metrics on homogeneous disc bundles(2015) Kayaçelebi, Artanç; Schwachhöfer, Lorenz J.; Siburg, Karl FriedrichIn this thesis we analyze under which conditions posed on the boundary metric we can construct nonnegatively curved invariant metrics on homogeneous disc bundles. The metrics we consider are constructed with a method which goes back to Cheeger. In course of analyzing the above stated problem it is shown that an arbitrary invariant metric on a sphere with positive sectional curvature can be extended to a positively curved metric on the ball having the sphere as its boundary, in such a way that the metric is a warped product metric near the boundary. Moreover we analyze in detail under which conditions an invariant metric on the product of an interval and a homogeneous space admits a reparametrization such that the reparametrized metric has nonnegative resp. positive sectional curvature.Item Spin(7)-manifolds of cohomogeneity one(2008-10-27T11:56:50Z) Reidegeld, Frank; Schwachhöfer, Lorenz; Böhm, Christoph