$\eta$-Ricci solitons on para-Kenmotsu manifolds

Abstract

In the context of paracontact geometry, $\eta$-Ricci solitons are considered on manifolds satisfying certain curvature conditions: $(\xi ,\cdot {)}_R\cdot S=0$, $(\xi ,\cdot {)}_S\cdot R=0$, $(\xi ,\cdot {)}_{W_2}\cdot S=0$ and $(\xi ,\cdot {)}_S\cdot W_2=0$. We prove that on a para-Kenmotsu manifold $(M,\phi ,\xi ,\eta ,g)$, the existence of an $\eta$-Ricci soliton implies that $(M,g)$ is quasi-Einstein and if the Ricci curvature satisfies $(\xi ,\cdot {)}_R\cdot S=0$, then $(M,g)$ is Einstein. Conversely, we give a sufficient condition for the existence of an $\eta$-Ricci soliton on a para-Kenmotsu manifold.

arXiv comment16 pages