Gray s decomposition on doubly warped product manifolds and applications

Abstract

A. Gray presented an interesting $O\left(n\right)$ invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold $M_i,i=1,2$ of a doubly warped product manifold $M{=}_{f_2}{M}_1{\times }_{f_1}{M}_2$ lie in the same Einstein-like class of $M$? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is proved. As an application, Einstein-like doubly warped product space-times of type $\mathcal{A},$ $\mathcal{B}$ or $\mathcal{P}$ are considered.