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

TL;DR

This paper investigates $*$-conformal $ abla$-Ricci solitons within Kenmotsu manifolds, establishing conditions under which these manifolds are Einstein, Ricci flat, or locally isometric to specific product spaces, with illustrative examples.

Contribution

It characterizes $*$-conformal $ abla$-Ricci solitons on Kenmotsu manifolds, including conditions for Einstein metrics and Ricci flatness, and provides explicit examples of such structures.

Findings

Kenmotsu metric is Einstein if the soliton vector field is contact.

Manifold is Ricci flat and locally isometric to $ ext{H}^{n+1}(-4) imes ext{R}^n$ under certain conditions.

Constructed examples of $*$-conformal $ abla$-Ricci solitons and related structures.

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.