On the $R_{f_B}$ condition for the (2+m)-Einstein warped product manifolds and some almost-complex structure cases

Alexander Pigazzini, Cenap Ozel, Saeid Jafari

Published 2019-12-30Version 1

For the studied cases in [14], the author showed that having the $f$-curvature-Base ($R_{f_B}$) is equal to requiring a flat metric on the base-manifold. In [15] the authors used the condition $R_{f_B}$ on (2+m)-Einstein warped product manifold, in the case $\lambda=\mu=0$, to built a new kind of manifolds, composed by positive-dimensional manifold and negative-dimensional manifold, the so called POLJ-manifolds. This have opened up the possibility of seeking the existence of others $(n, m)$-POLJ manifolds. The aim of this paper is to extend the work done in [14] to $m$-dimensional fiber, and show if the value of $m$ can influence the result, i.e., finding base-manifolds with non-flat metric for $dimF \neq 2$ and, at the same time, check if in these cases can exists other kinds of (2, m)-POLJ manifolds. In the second part of the paper, we put in relation a particular type of almost-Hermitian manifolds with POLJ-manifolds and we proceeded to give the definitions of: almost Hermitian-POLJ manifold (aH-POLJs) and pseudo almost Hermitian-POLJ manifolds (paH-POLJs). As a result, we find out that the dimension of fiber-manifold does not change the result of [14]. Moreover, we show that the (2,m)-almost hyperbolic Hermitian-POLJ manifolds with flat fiber and quasi-constant curvature, and (2, m)-quasi-Einstein almost hyperbolic Hermitian-POLJ manifolds with flat fiber and quasi-constant curvature can not exist. Finally we add a Special Remark about the possible use of the $(n,-n)$-POLJs in superconductor graphene theory.