Some New Characterizations of Trivial Ricci–Bourguignon Solitons

Abstract

A Ricci–Bourguignon soliton is a self-similar solution to the Ricci–Bourguignon flow equation, and a Ricci–Bourguignon soliton is called trivial if its potential field is zero or killing. Each trivial Ricci–Bourguignon soliton is an Einstein manifold. The main purpose of this paper is to discover geometric conditions on compact Ricci–Bourguignon solitons for which the solitons are trivial. In Section 3, we establish three new characterizations for a compact connected Ricci–Bourguignon soliton to be trivial. In Section 4, we discover three conditions which assure that a compact gradient Ricci–Bourguignon soliton is trivial. Some applications of our results are also presented.