Einstein-like doubly warped product manifolds

In this paper, it is proved that the factor manifolds $M_{i},i=1,2$ of a doubly warped product manifold $M=_{f_{2}}M_{1}\times _{f_{1}}M_{2}$ acquire the Einstein-like class type $\mathcal{A},$ $\mathcal{B}$, $\mathcal{P}$, $% \mathcal{I\oplus B}$, $\mathcal{I\oplus A}$, or $\mathcal{A\oplus B}$ of $M$ by imposing a sufficient condition on the warping functions in each case. As an application, Einstein-like doubly warped product spacetimes of type $% \mathcal{A},$ $\mathcal{B}$ or $\mathcal{P}$ are considered.