Curvature constrained on base for $(2+m)$-Einstein warped product manifolds

TL;DR

This paper investigates the influence of fiber dimension on curvature constraints in $(2+m)$-Einstein warped product manifolds, extending previous results and finding that fiber dimension does not alter the flatness condition of the base.

Contribution

It extends prior work on $f$-curvature-Base to higher fiber dimensions, showing the flatness condition remains unchanged for different fiber dimensions.

Findings

Fiber dimension does not affect the flatness condition of the base.

The results extend to $(2,m)$-PNDP manifolds with $R_{f_B}$.

The curvature constraints are independent of fiber dimension.

Abstract

For the studied cases in [10], the author showed that having the {\textit {$f$-curvature-Base}} ($R_{f_B}$) is equal to requiring a flat metric on the base-manifold. In [11] the authors introduced a new kind of Einstein warped product manifold, composed by positive-dimensional manifold and negative-dimensional manifold, the so called \textit{PNDP-manifolds} The aim of this paper is to extend the work done in [10] to $m$-dimensional fiber showing if the value of $m$ can influence the result, i.e., finding base-manifolds with non-flat metric for $dimF \neq 2$, and doing some considerations of the $(2, m)$-PNDP manifolds with $R_{f_B}$. As a result, we find out that the dimension of fiber-manifold does not change the result of [10].