On the (2+2)-Einstein Warped Product Manifolds with $f$-curvature-Base

Abstract

We study the $(2+2)$-Einstein warped product manifolds, where the scalar curvature of the Base is a multiple of the warping function, and we called this condition (inside a warped product manifold) $f$-curvature-Base ($R_{f_B}$).The aim of this paper is to check if there are Base-manifolds with non-flat metrics that satisfy this condition, and this check was done in cases where $M$ and Fiber-manifold are not both non-Ricci-flat. As a results of the our cases we find that the "$f$-curvature-Base" is equivalent to requesting a flat metric on the Base-manifold.