  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.
