On K-contact metric manifolds satisfying an almost gradient Ricci–Bourguignon soliton

Abstract

The main purpose of the paper is to study an almost gradient Ricci–Bourguignon soliton (RB soliton) within the framework of [Formula: see text]-contact manifolds and [Formula: see text]-contact manifolds. First, we prove that if complete [Formula: see text]-contact manifold endows a gradient RB soliton, then the manifold is compact Sasakian and isometric to unit sphere [Formula: see text]. Next, we show that if a complete contact metric satisfies an almost RB soliton with a non-zero potential vector field is collinear with the Reeb vector field [Formula: see text] and the Reeb vector field [Formula: see text] acting as an eigenvector of the Ricci operator, then it is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field [Formula: see text]. Lastly, we prove that if the metric of a non-Sasakian [Formula: see text]-contact manifold is an almost gradient RB soliton, then it is flat in dimension 3 and in higher dimensions it is locally isometric to [Formula: see text].