  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.
