On generalized Robertson-Walker spacetimes satisfying some curvature condition


ARSLAN K. , Deszcz R., EZENTAŞ R. , Hotlos M., MURATHAN C.

TURKISH JOURNAL OF MATHEMATICS, vol.38, no.2, pp.353-373, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 2
  • Publication Date: 2014
  • Doi Number: 10.3906/mat-1304-3
  • Title of Journal : TURKISH JOURNAL OF MATHEMATICS
  • Page Numbers: pp.353-373
  • Keywords: 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, HYPERSURFACES, GEOMETRY

Abstract

We give necessary and sufficient conditions for warped product manifolds (M, g), of dimension >= 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 - alpha g) <= 1, for some alpha is an element of R, or non-quasi-Einstein.