Analysis of a special type of soliton on Kenmotsu manifolds

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.