$\eta$-Ricci Solitons on $\beta$-Kenmotsu Manifolds

Abstract

$\eta $-Ricci solitons on $\beta $-Kenmotsu manifold satisfying certain curvature conditions $R(\xi,X).S=0$, $S(\xi,X).R=0$, $W_{2}\left(\xi ,X\right).S=0$ and $S\left(\xi,X\right).W_{2}=0$. We proved that in $\beta $-Kenmotsu manifold $(M,\phi,\xi,\eta,g)$. Then the existence of an $\eta $-Ricci solitons implies that M is Einstein manifold and if the Ricci curvature tensor satisfies, $S(\xi,X).R=0$, then Ricci solitons M is steady. If the condition $\mu =0$, then $\lambda =0$, which shows that $\lambda $ is steady.