Some results on Ricci-Bourguignon solitons and almost solitons

TL;DR

This paper extends the theory of Ricci-Bourguignon solitons by proving new results, introducing almost solitons, and deriving integral formulas, with applications to characterizing compact solutions as spheres under certain conditions.

Contribution

It generalizes known results for Ricci solitons to Ricci-Bourguignon solitons and introduces Ricci-Bourguignon almost solitons, providing new integral formulas and geometric characterizations.

Findings

Generalized results for Ricci-Bourguignon solitons

Introduced Ricci-Bourguignon almost solitons and proved related results

Showed compact gradient Ricci-Bourguignon almost solitons are spheres under specific conditions

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.