# $*$-Ricci solitons and gradient almost $*$-Ricci solitons on Kenmotsu manifolds

## TL;DR

This paper investigates $*$-Ricci and gradient almost $*$-Ricci solitons on Kenmotsu manifolds, establishing conditions for soliton constants, curvature, and Einstein properties, with explicit examples provided.

## Contribution

It proves new results about the nature of $*$-Ricci solitons on Kenmotsu manifolds, including conditions for Einstein metrics and the behavior of potential vector fields.

## Key findings

- Soliton constant $\\lambda$ is zero for $*$-Ricci solitons on Kenmotsu manifolds.
- 3-dimensional Kenmotsu manifolds with $*$-Ricci solitons have constant sectional curvature -1.
- If potential vector field is collinear with characteristic vector field, the manifold is Einstein.

## Abstract

In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $\lambda$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $\xi$, then $M$ is Einstein and soliton vector field is equal to $\xi$. Finally, we prove that if $g$ is a gradient almost $*$-Ricci soliton, then either $M$ is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of $M$. We verify our result by constructing examples for both $*$-Ricci soliton and gradient almost $*$-Ricci soliton.