On the rigidity of harmonic-Ricci solitons

Abstract: In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not necessarily gradient solitons while, in the complete non-compact case, we restrict our attention to steady and shrinking gradient solitons. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry a Riemannian manifold equipped with a smooth map $\varphi$, called $\varphi$-curvatures, which are a natural generalization of the standard curvature tensors in the setting of harmonic-Ricci solitons.