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Gray’s decomposition on doubly warped product manifolds and applications

El-Sayied Hoda K. (Mathematics Department, Faculty of Science, Tanata University, Tanta, Egypt), hkelsayied1989@yahoo.com
Mantica Carlo A. (I.I.S. Lagrange, Via L. Modignani, Milan, Italy), carloalberto.mantica@libero.it
Shenawy Sameh (Basic Science Department, Modern Academy for Engineering and Technology, Maadi, Egypt), drshenawy@mail.com
Syied Noha (Basic Science Department, Modern Academy for Engineering and Technology, Maadi, Egypt), drnsyied@mail.com

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 Mi,i = 1,2 of a doubly warped product manifold M =f2 M1 x f1 M2 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.

Keywords: Codazzi Ricci tensor, doubly warped manifolds, Killing Ricci tensor, doubly warped space-times, Einstein-like manifolds