Generic properties of homogeneous Ricci solitons

Abstract

Abstract We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and non-compact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of left invariant vector fields. We prove that a left invariant soliton of gradient type must be a Riemannian product with non-trivial Euclidean de Rham factor. As an application of our results we prove that any generalized metric Heisenberg Lie group is a non-gradient left invariant Ricci soliton of expanding type.