Conformal diffeomorphisms of gradient Ricci solitons and generalized quasi-Einstein manifolds

TL;DR

This paper extends rigidity results for conformal transformations of Einstein metrics to generalized quasi-Einstein metrics, including gradient Ricci solitons, and classifies structure-preserving conformal diffeomorphisms.

Contribution

It introduces notions of conformal diffeomorphisms preserving GQE structures and classifies such transformations, extending known results to a broader class of metrics.

Findings

Complete GQE metrics admit structure-preserving conformal vector fields only on the sphere.

In the compact case, GQE metrics with such diffeomorphisms are conformal to the sphere.

In the complete case, only sphere conformal transformations and stereographic projections occur.

Abstract

In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. We show that a complete GQE metric admits a structure-preserving, non-homothetic complete conformal vector field if and only if it is a round sphere. We also classify the structure-preserving conformal diffeomorphisms. In the compact case, if a GQE metric admits a structure-preserving, non-homothetic conformal diffeomorphism, then the metric is conformal to the sphere, and isometric to the sphere in the case of a gradient Ricci soliton. In the complete case, the only structure-preserving non-homothetic conformal diffeomorphisms from a shrinking or steady gradient Ricci soliton to another soliton are the conformal transformations of spheres and stereographic projection.