Gray’s decomposition on doubly warped product manifolds and applications

Abstract

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.

Almost Ricci–Bourguignon Solitons on Doubly Warped Product Manifolds

“…where the functions f i : E i → (0, ∞), i = 1, 2 are the warping functions of the doubly warped product [34][35][36][37][38]. The maps π i : E 1 × E 2 → E i are the natural projections of E onto E i whereas * denotes the pull-back operator on the tensors.…”

On the Geometry in the Large of Einstein-like Manifolds

“…Therefore, the Riemannian manifold is called an an Einstein-like manifold of type A (respectively, B or S) if ∇Ric ∈ C ∞ A (respectively, ∇Ric ∈ C ∞ B or ∇Ric ∈ C ∞ S). In the first case Ric is called the Killing-Ricci tensor (see [3]), while in the second case Ric is called the Codazzi-Ricci tensor (see [4]) and in the third case Ric is called the Sinyukov-Ricci tensor (see [5][6][7]).…”