Some results on almost η-Ricci-Bourguignon solitons

Abstract

We determine a formula to compute the defining function of a gradient almost Ricci-Bourguignon soliton by means of the potential vector field, providing in this sense two examples. We also give a rigidity result in this case. Moreover, we deduce some curvature properties of solitons with torse-forming potential vector field. Further we generalize some of the results to almost η-Ricci-Bourguignon solitons and prove that a gradient almost Ricci-Bourguignon soliton on a doubly warped product manifold induces gradient almost η-Ricci-Bourguignon solitons on its factor manifolds.