Covariant derivative of the curvature tensor of pseudo-K\"ahlerian manifolds

TL;DR

This paper refines the decomposition of the covariant derivative of the curvature tensor for pseudo-K"ahlerian manifolds, introducing new classes of spaces and analyzing holonomy group representations.

Contribution

It extends known decompositions to pseudo-K"ahlerian manifolds and defines new classes of spaces generalizing symmetric and Einstein spaces.

Findings

Decomposition of the covariant derivative of the curvature tensor with respect to the pseudo-unitary group.

Introduction of natural classes of pseudo-K"ahlerian spaces generalizing symmetric and Einstein spaces.

Analysis of the holonomy group modules for non-locally symmetric pseudo-Riemannian manifolds.

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-K\"ahlerian 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-K\"ahlerian 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.