Analysis of a special type of soliton on Kenmotsu manifolds

TL;DR

This paper investigates special soliton structures on Kenmotsu manifolds, proving conditions under which these manifolds are Einstein or Ricci flat, and providing an explicit example of such a soliton.

Contribution

It establishes new conditions linking almost $*$-Ricci-Bourguignon solitons to Einstein and Ricci flat properties on Kenmotsu manifolds, including an explicit example.

Findings

Kenmotsu manifolds with almost $*$-R-B-S are $ ext{eta}$-Einstein.

Manifolds with certain nullity distributions and almost $*$-R-B-S are Ricci flat.

Gradient almost $*$-R-B-S with scalar curvature preservation are Einstein.

Abstract

In this paper, we aim to investigate the properties of an almost $*$-Ricci-Bourguignon soliton (almost $*-$R-B-S for short) on a Kenmotsu manifold (K-M). We start by proving that if a Kenmotsu manifold (K-M) obeys an almost $*-$R-B-S, then the manifold is $\eta$-Einstein. Furthermore, we establish that if a $(\kappa, -2)'$-nullity distribution, where $\kappa<-1$, has an almost $*$-Ricci-Bourguignon soliton (almost $*-$R-B-S), then the manifold is Ricci flat. Moreover, we establish that if a K-M has almost $*$-Ricci-Bourguignon soliton gradient and the vector field $\xi$ preserves the scalar curvature $r$, then the manifold is an Einstein manifold with a constant scalar curvature given by $r=-n(2n-1)$. Finaly, we have given en example of a almost $*-$R-B-S gradient on the Kenmotsu manifold.