Abstract
It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.
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The author is grateful to Dmitri V. Alekseevsky and Rod Gover for useful discussions.
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The author acknowledges the institutional support of University of Hradec Králové and the grant GA14–02476S of the Czech Science Foundation.
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Galaev, A.S. Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds. Ann Glob Anal Geom 51, 245–265 (2017). https://doi.org/10.1007/s10455-016-9533-1
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DOI: https://doi.org/10.1007/s10455-016-9533-1
Keywords
- Pseudo-Riemannian manifold
- Pseudo-Kählerian manifold
- Curvature tensor
- Covariant derivative of the curvature tensor
- Second Bianchi identity