GRAY's DECOMPOSITION AND WARPED PRODUCT OF GENERALIZED RICCI RECURRENT SPACETIMES

Abstract

Generalized Ricci recurrent spacetimes (GR)_n are investigated in Gray's seven subspaces. It is proved that a (GR)_n spacetime in all subspaces but one is an Einstein spacetime. The subspace $ℐ$ cannot contain a (GR)_n spacetime. Further, the subspaces $ℐ\oplus A$ and $ℐ\oplus B$ reduce to $A$ and $B$, respectively. Next, we prove that a (GR)_n spacetime is Ricci semi-symmetric if and only if either the spacetime is Einstein or the vector field A^l is closed. Further, it is shown that the Ricci tensor of (GR)_n is Riemann compatible if A^l is closed. Finally, sufficient conditions are given on a (GR)_n warped product manifold to guarantee that the factor manifolds are Einstein. Moreover, it is shown that a generalized Ricci recurrent GRW spacetime is an Einstein spacetime.