On the rigidity of harmonic-Ricci solitons

TL;DR

This paper introduces the concept of rigidity in harmonic-Ricci solitons, characterizes it through modified curvature tensors, and extends known results for Ricci solitons to a broader geometric setting.

Contribution

It generalizes rigidity results for Ricci solitons to harmonic-Ricci solitons, including non-gradient cases in compact manifolds and specific cases in non-compact manifolds.

Findings

Rigidity characterized by vanishing of modified curvature tensors.

Extension of known Ricci soliton results to harmonic-Ricci solitons.

Applicable to both compact and certain non-compact manifolds.

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.