Abstract: By using the divergence theorem and some important inequalities on Riemannian manifolds, combined with  the method of geometric analysis, we studied rigidity problems of compact gradient Ricci-Yamabe solitons, and obtained rigidity result of the nontrivial compact gradient Ricci-Yamabe solitons being equidistant from Euclidean sphere under appropriate conditions. In addition, under the assumption of positive scalar curvature, we proved that n(4≤n≤6) dimensional compact gradient shrinking Ricci-Yamabe solitons that satisfied L^n/2 integral pinched condition must be Einstein manifolds.

Key words: gradient Ricci-Yamabe soliton, rigidity, integral pinched condition, scalar curvature