On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions

TL;DR

This paper surveys recent results on semi-Riemannian manifolds and submanifolds, especially hypersurfaces, satisfying generalized Einstein metric conditions expressed via the difference tensor involving curvature tensors.

Contribution

It introduces a family of generalized Einstein metric conditions involving the difference tensor and summarizes recent advances in their study.

Findings

Characterization of manifolds satisfying these conditions

Results on submanifolds and hypersurfaces with such properties

Connections between curvature tensors and Einstein-like conditions

Abstract

The difference tensor R.C-C.R of a semi-Riemannian manifold (M,g), dim M > 3, formed by its Riemannian-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0,6)-Tachibana tensors Q(A,T), where A is a symmetric (0,2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.