On semi-Riemannian manifolds satisfying some conformally invariant curvature condition

Abstract

The difference tensor R • C − C • R of a semi-Riemannian manifold (M, g), dim M ≥ 4, formed by its Riemann-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. 1

A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

“…The Riemannian ones were introduced by Defever and Deszcz in 1991 [13] (see also [15] and Chaki et al [7]). In [16] Deszcz proved that a quasi-Einstein Riemannian manifold with null Weyl tensor and few other conditions, is a warped product (+1) × q 2 M * , where M * is a (n − 1)-dimensional Riemannian manifold of constant curvature. Pseudo-Riemannian quasi-Einstein spaces arose in the study of exact solutions of Einstein's equations.…”

Curvature properties of Gödel metric

“…We mention that the Vaidya spacetime also satisfies (17) (see [35, Example 5.2]). We refer to [69] for recent results on manifolds satisfying (12) and (17). The conditions: (8), (10), (12), (14) and (17) or other conditions of this kind are called conditions of pseudosymmetry type.…”

On Chen Ideal Submanifolds Satisfying Some Conditions of Pseudo-symmetry Type

“…Very recently semi-Riemannian manifolds satisfying (23) were investigated in [24]. It is known that at every point of a hypersurface N in a space forms N n+1 (c), n ≥ 4, the tensors R · R − Q(S, R) and Q(g, C) are linearly dependent.…”