A family of multiply warped product semi-Riemannian Einstein metrics

In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an \begin{document}$ n $\end{document} -dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an \begin{document}$ (n-1) $\end{document} -dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.