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

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].