Kenmotsu metric as conformal $\eta$-Ricci soliton

TL;DR

This paper characterizes Kenmotsu manifolds admitting conformal η-Ricci solitons, showing conditions under which they are Einstein and exploring their behavior under deformations, with an example provided.

Contribution

It introduces the study of conformal η-Ricci solitons on Kenmotsu manifolds, establishing their properties, invariance under deformations, and providing explicit examples.

Findings

An η-Einstein Kenmotsu manifold with conformal η-Ricci soliton is Einstein.

Relation between potential vector field and Reeb vector field in gradient cases.

Invariance of conformal η-Ricci solitons under certain deformations.

Abstract

The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $\eta$-Ricci soliton. Here, we have investigated the nature of the conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $\eta$-Einstein Kenmotsu manifold admitting conformal $\eta$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $\eta$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $\eta$-Ricci soliton on Kenmotu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal $\eta$-Ricci soliton on Kenmotsu manifold.