Compact Einstein warped product manifolds

Abstract

The warped product \(B\times _f F\) of two Riemannian manifolds \((B,g_B)\) and \((F,g_F)\) with warping function \(f\) is the product manifold \(B\times F\) equipped with the warped product metric \(g_B + f^2 g_F\), where \(f\) is a positive function on \(B\). It is well-known that the notion of warped products plays some important roles in differential geometry as well as in general relativity. In this paper, we will survey recent results on the existence of compact Einstein warped product Riemannian manifolds.