Para-Kenmotsu metric as a $\eta$-Ricci soliton

Abstract

The purpose of the paper is to study of Para-Kenmotsu metric as a $\eta$-Ricci soliton. The paper is organized as follows: * If an $\eta$-Einstein para-Kenmotsu metric represents an $\eta$-Ricci soliton with flow vector field $V$, then it is Einstein with constant scalar curvature $r = -2n(2n+1)$. * If a para-Kenmotsu metric $g$ represents an $\eta$-Ricci soliton with the flow vector field $V$ being an infinitesimal paracontact transformation, then $V$ is strict and the manifold is an Einstein manifold with constant scalar curvature $r = -2n(2n+1)$. * If a para-Kenmotsu metric $g$ represents an $\eta$-Ricci soliton with non-zero flow vector field $V$ being collinear with $\xi$, then the manifold is an Einstein manifold with constant scalar curvature $r = -2n(2n+1)$. Finally, we cited few examples to illustrate the results obtained.