Geometry of almost contact metrics as an almost ∗-η-Ricci–Bourguignon solitons

Abstract

In this paper, we give some characterizations by considering almost ∗-[Formula: see text]-Ricci–Bourguignon soliton as a Kenmotsu metric. It is shown that if a Kenmotsu metric endows a ∗-[Formula: see text]-Ricci–Bourguignon soliton, then the curvature tensor [Formula: see text] with the soliton vector field [Formula: see text] is given by the expression [Formula: see text] Next, we show that if an almost Kenmotsu manifold such that [Formula: see text] belongs to [Formula: see text]-nullity distribution where [Formula: see text] acknowledges a ∗-[Formula: see text]-Ricci–Bourguignon soliton satisfying [Formula: see text], then the manifold is Ricci-flat and is locally isometric to [Formula: see text]. Moreover if the metric admits a gradient almost ∗-[Formula: see text]-Ricci–Bourguignon soliton and [Formula: see text] leaves the scalar curvature [Formula: see text] invariant on a Kenmotsu manifold, then the manifold is an [Formula: see text]-Einstein. Also, if a Kenmotsu metric represents an almost ∗-[Formula: see text]-Ricci–Bourguignon soliton with potential vector field [Formula: see text] is pointwise collinear with [Formula: see text], then the manifold is an [Formula: see text]-Einstein.