Generic Properties of Homogeneous Ricci Solitons

TL;DR

This paper investigates the geometric properties of homogeneous Ricci solitons, establishing nonexistence results for certain types and characterizing gradient and non-gradient cases, especially on Lie groups.

Contribution

It provides new nonexistence results for compact and steady homogeneous Ricci solitons and characterizes gradient solitons as Riemannian products, with applications to metric Heisenberg groups.

Findings

No compact or steady homogeneous Ricci solitons exist.

Gradient solitons are Riemannian products with Euclidean factors.

Generalized metric Heisenberg groups are nongradient expanding Ricci solitons.

Abstract

We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact 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 a nontrivial Euclidean de Rham factor. As an application of our results we prove that any generalized metric Heisenberg Lie group is a nongradient left invariant Ricci soliton of expanding type.