Korean J. Math. Vol. 29 No. 2 (2021) pp.445-453
DOI: https://doi.org/10.11568/kjm.2021.29.2.445

# 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.

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