Some results on Ricci-Bourguignon solitons and almost solitons

Abstract

We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon $almost$ solitons and prove some results about them which generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci-Bourguignon solitons and compact gradient Ricci-Bourguignon almost solitons. Finally, using the integral formula we show that a compact gradient Ricci-Bourguignon almost soliton is isometric to an Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

v2-added two examples of Ricci-Bourguignon almost solitons and some minor cosmetic changes, including a change in the title. 10 pages. Final version, to appear in Canadian Mathematical Bulletin