Characterization of almost $*$-conformal $\eta$-Ricci soliton on para-Kenmotsu manifolds

Abstract

The goal of this research paper is to deliberate $*$-conformal $\eta$-Ricci soliton and gradient almost $*$-conformal $\eta$-Ricci soliton within the framework of para-Kenmotsu manifolds as a characterization of Einstein metrics. Here, we explore that a para-Kenmotsu metric as a $*$-conformal $\eta$-Ricci soliton is Einstein metric if the soliton vector field is contact and the vector field is strictly infinitesimal contact transformation. Next, we turn up the nature of the soliton and discover the scalar curvature when the manifold admitting $\ast$-conformal $\eta$-Ricci soliton on para-Kenmotsu manifold. After that, we have shown the characterization of the vector field when the manifold satisfies $\ast$-conformal $\eta$-Ricci soliton. Further, we have developed the delineation of the para-Kenmotsu manifold or the nature of the potential vector field when the manifold admits gradient almost $*$-conformal $\eta$-Ricci soliton. Then, we have studied gradient $\ast$-conformal $\eta$-Ricci soliton to yield the nature of Riemannian curvature tensor and enactment of potential vector field on para-Kenmotsu manifold. Lastly, we give an example of conformal $*$-$\eta$-Ricci soliton, gradient almost conformal $*$-$\eta$-Ricci soliton on para-Kenmotsu manifold to prove our findings.