Abstract

The aim of this paper is to prove some results about almost Yamabe soliton and almost Ricci-Bourguignon soliton with special soliton vector field. In fact, we prove that every compact non-trivial almost Ricci-Bourguignon soliton with constant scalar curvature is isometric to a Euclidean sphere. Then we show that every compact almost Ricci-Bourguignon soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Finally, we prove that a complete almost Ricci-Bourguignon soliton $(M,g, V, \lambda, \rho)$ has $V$ as the contact vector field of a contact manifold $M$ with metric $g$ and its Reeb vector field is geodesic, then it becomes a Ricci-Bourguignon soliton and $g$ has constant scalar curvature. In particular, if $V$ is strict, then $g$ is a compact Sasakian Einstein.

Keywords: Almost Ricci-Bourguignon soliton; contact manifold; Einstein Sasakian.

MSC: 53C25, 53E20, 53C21

DOI: 10.57016/MV-053OSOH9

Pages:  1$-$8