Gray’s decomposition on doubly warped product manifolds and applications

A. Gray presented an interesting O(n) 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 = (f2) M-1 x (f1) 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 A, B or P are considered.