Certain results on almost $\ast$-Schouten solitons and gradient almost $\ast$-Schouten solitons

Abstract

The main object in this paper is to deliberate the characterizations of an almost $*$-Schouten soliton and gradient almost $*$-Schouten soliton on Kenmotsu manifold. In particular, we have proved that if a Kenmotsu manifold admits an almost $*$-Schouten soliton with non-zero potential vector field $V$ collinear to the Reeb vector field $\zeta$, then the manifold becomes an $\eta$-Einstein manifold. Furthermore, it is shown that if a $(\kappa, \mu)'$-almost Kenmotsu manifold admitting an almost $*$-Schouten soliton with $\kappa < 1$, then it is locally isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$. Lastly, we show that if a metric $g$ of Kenmotsu manifold endows with a gradient almost $*$-Schouten soliton and $\zeta$ leaves the scalar curvature $r$ invariant, then the manifold is an Einstein manifold with constant scalar curvature $r=-2n(2n+1)$. Lastly, we provide an example of 5-dimensional Kenmotsu manifolds admitting an almost $*$-Schouten soliton.