$*$-Conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds

Abstract

The goal of our present paper is to deliberate $*$-conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds. Here we have shown that a Kenmotsu metric as a $*$-conformal $\eta$-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have evolved the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $*$-conformal $\eta$-Ricci soliton. Next, we have contrived $*$-conformal $\eta$-Ricci soliton admitting $(\kappa,\mu)^\prime$-almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$. Finally we have constructed some examples to illustrate the existence of $*$-conformal $\eta$-Ricci soliton, gradient almost $*$-conformal $\eta$-Ricci soliton on Kenmotsu manifold and $(\kappa,\mu)^\prime$-almost Kenmotsu manifolds.