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

TL;DR

This paper investigates (2+2)-Einstein warped product manifolds with a specific scalar curvature condition, concluding that such conditions imply the base manifold must be flat, thus characterizing these manifolds.

Contribution

It establishes that the $f$-curvature-Base condition in (2+2)-Einstein warped products necessitates a flat base manifold, providing a new characterization of these geometries.

Findings

The $f$-curvature-Base condition implies flatness of the base manifold.

Non-flat base manifolds do not satisfy the $f$-curvature-Base condition.

The study narrows the class of (2+2)-Einstein warped products satisfying this curvature condition.

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.