On the structure of Einstein warped product semi-Riemannian manifolds

TL;DR

This paper investigates the structure of Einstein warped product semi-Riemannian manifolds, proving classification results under certain conditions and providing new examples of complete Einstein warped products.

Contribution

It classifies Einstein warped products with compact Ricci-flat fibers and conformal bases, and constructs new examples of complete Einstein warped product manifolds.

Findings

Warped products with compact Ricci-flat fibers are Riemannian products.

Classification of Einstein warped products with conformal bases invariant under translation.

New examples of complete Einstein warped product Riemannian manifolds.

Abstract

In this paper we consider a class of Einstein warped product semi-Riemannian manifolds $\widehat{M} = M^{n}\times_{f}N^{m}$ with $n\geq 3$ and $m\geq 2$. For $\widehat{M}$ with compact base and Ricci-flat fiber, we prove that $\widehat{M}$ is simply a Riemannian product space. Then, when the base $M$ is conformal to a pseudo-Euclidean space which is invariant under the action of a $(n-1)$-dimensional translation group, we classify all such spaces. Furthermore, we get new examples of complete Einstein warped products Riemannian manifolds.