Abstract: Firstly, by using  the divergence theorem and  geometric analysis methods, we  studied the rigidity problem of compact η-Ricci-Bourguignon solitons, and obtained two key integral formulas for  the potential vector field of the soliton and the dual vector field of η. Secondly, under different integral conditions, we obtain rigidity results of the soliton, which  proves that the soliton is either an η-Einstein manifold or an Einstein manifold.

Key words: η-Ricci-Bourguignon soliton, Einstein manifold, Killing vector field, dual vector field