Some characterizations on Gradient Almost η-Ricci-Bourguignon Solitons

Abstract The aim of this paper is to characterize some equations of structures for gradient almost η-Ricci Bourguignon solitons which generalize the equivalent for gradient η-Ricci-Bourguignon solitons. We prove that a gradient almost η-Ricci-Bourguignon soliton is gradient almost 1/ωu-traceless Ricci soliton with a well defined potential function f. Moreover we investigate that an Einstein manifold of constant scalar curvature is isometric to a space form with a well defined potential function f. Finally, we derived an integral formula for gradient compact case. We show that a compact nontrivial almost 1/ωu-traceless Ricci soliton with constant scalar curvature or conformal potential vector field, is isometric to a standard unit sphere, hyperbolic space and Euclidean space.