On generalized Robertson--Walker spacetimes satisfying some curvature condition

• Published in: Turkish journal of mathematics
• Main Authors: ARSLAN, Kadri, DESZCZ, Ryszard, EZENTAS, Ridvan, HOTLOS, Marian, MURATHAN, Cengizhan
• Format: Journal Article
• Language: English
• Published: TUBITAK 01.01.2014
• Subjects: Warped product, generalized Robertson > Walker spacetime, Einstein manifold, quasi-Einstein manifold, essentially conformally symmetric manifold, Tachibana tensor, generalized Einstein metric condition, pseudosymmetry type curvature condition, Ricci-pseudosymmetric hypersurface
• ISSN: 1303-6149; 1300-0098; 1303-6149
• DOI: 10.3906/mat-1304-3

We give necessary and sufficient conditions for warped product manifolds (M,g), of dimension \geqslant 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R . C - C . R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q(S,R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S - a g) \leqslant 1, for some a \in R, or non-quasi-Einstein.