Publication details
Covariant derivative of the curvature tensor of pseudo-Kahlerian manifolds
| Authors | |
|---|---|
| Year of publication | 2017 |
| Type | Article in Periodical |
| Magazine / Source | Annals of Global Analysis and Geometry |
| MU Faculty or unit | |
| Citation | |
| Web | full text |
| Doi | http://dx.doi.org/10.1007/s10455-016-9533-1 |
| Field | General mathematics |
| Keywords | Pseudo-Riemannian manifold; Pseudo-Kahlerian manifold; Curvature tensor; Covariant derivative of the curvature tensor; Second Bianchi identity |
| Description | 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-Kahlerian 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-Kahlerian 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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