Warped product Einstein metrics over spaces with constant scalar curvature

TL;DR

This paper investigates warped product Einstein metrics over spaces with constant scalar curvature, establishing rigidity results in three dimensions and providing examples and conditions for non-rigid cases in higher dimensions.

Contribution

It proves that three-dimensional bases with certain conditions are always rigid and introduces examples of non-rigid cases in higher dimensions.

Findings

Three-dimensional base spaces are always rigid under given conditions.

Examples of non-rigid warped product Einstein metrics in four dimensions.

Curvature conditions that characterize rigidity in higher dimensions.

Abstract

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.