Nonconstant warping functions on Einstein warped product manifolds with $2-$dimensional base

Abstract

In this paper, we study nonconstant warping functions on an Einstein warped product manifold $M=B \times _{f^2} F$ with a warped product metric $g=g_B +f(t)^2 g_F$. And we consider a $2-$dimensional base manifold $B$~ with a metric $g_B = dt^2 +(f'(t))^2 du^2 .$ As a result, we prove the following: if $M$ is an Einstein warped product manifold with a $2-$dimensional base, then there exist generally nonconstant warping functions $f(t).$