Some New Characterizations of Trivial Ricci-Bourguignon Solitons

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.