Necessary and sufficient conditions for the existence of torsion-free covariant derivatives with prescribed curvature tensor
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Date
2022
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Abstract
Diese Dissertation behandelt das Problem, torsionsfreie kovariante Ableitungen mit vorgeschriebener Krümmung zu konstruieren. Die Krümmung einer solchen kovarianten Ableitung erfüllt notwendigerweise die Bianchi-Identitäten. Das Hauptresultat der Arbeit besagt, dass diese Bedingungen bereits hinreichend sind: jede analytische Krümmungsabbildung, die die Bianchi-Identitäten erfüllt, kann lokal als Krümmung einer eindeutigen torsionsfreien kovarianten Ableitung realisiert werden.
Als Anwendungen dieses Resultats in der Holonomietheorie können wir die lokalen Existenzresultate für Riemannsche Metriken mit spezieller Holonomie von Calabi, Yau, Bryant etc. vereinheitlichen und wesentlich vereinfachen
This dissertation deals with the problem of constructing torsion-free covariant derivatives with prescribed curvature. The curvature of a such covariant derivative necessarily satisfies the Bianchi identities. The main result of the present work asserts that these identities are enough to achieve this: any analytic curvature map which satisfies the Bianchi identities, can locally be realized as the curvature of a unique torsion-free covariant derivative. As applications of this result in Holonomy Theory, we can unify in a simple way all of the local existence results of Calabi, Yau, and Bryant, among others, for Riemannian metrics with special holonomy.
This dissertation deals with the problem of constructing torsion-free covariant derivatives with prescribed curvature. The curvature of a such covariant derivative necessarily satisfies the Bianchi identities. The main result of the present work asserts that these identities are enough to achieve this: any analytic curvature map which satisfies the Bianchi identities, can locally be realized as the curvature of a unique torsion-free covariant derivative. As applications of this result in Holonomy Theory, we can unify in a simple way all of the local existence results of Calabi, Yau, and Bryant, among others, for Riemannian metrics with special holonomy.
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Manifold, Curvature tensor, Torsion-free connection