# Gray s decomposition on doubly warped product manifolds and applications

## TL;DR

This paper investigates conditions under which factor manifolds of doubly warped product manifolds inherit Gray's decomposition properties, leading to new insights into Einstein-like structures and applications in space-time models.

## Contribution

It provides necessary and sufficient conditions on warping functions for factor manifolds to belong to the same Einstein-like class, extending Gray's decomposition to doubly warped products.

## Key findings

- Inheritance of Einstein-like properties under specific warping conditions
- Characterization of Einstein-like doubly warped space-times of types A, B, and P
- Extension of Gray's decomposition to complex manifold structures

## 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.