Certain results on gradient almost η - Ricci-Bourguignon Soliton

Abstract

The present research article deals with the study of almost η -Ricci- Bourguignon soliton and gradient almost η-Ricci-Bourguignon soliton on almost Kenmotsu manifolds. It is shown that if the metric of a Kenmotsu manifold M²ⁿ⁺¹ admits a gradient almost η-Ricci-Bourguignon soliton, then it is η-Einstein. More-over, if the manifold is complete and ξ leaves the scalar curvature invariant, then it is locally isometric to Hyperbolic space H²ⁿ⁺¹(-1). Next, we demonstrate that if a (κ; μ) almost Kenmotsu manifold admits an almost η-Ricci-Bourguignon soliton, then the manifold is η -Einstein. Besides, we explore the condition for non-normal almost Kenmotsu manifolds satisfying gradient almost η-Ricci-Bourguignon soliton. In addition, we have also investigated an almost η-Ricci-Bourguignon soliton on (κ; μ) ' -almost Kenmotsu manifold.