3-symmetric spaces, Ricci solitons, and homogeneous structures

TL;DR

This paper classifies 3-symmetric spaces, describes their moduli spaces, and constructs explicit expanding Ricci solitons, revealing new homogeneous Ricci solitons and analyzing their symmetry properties.

Contribution

It provides a full classification of 3-symmetric spaces, describes their moduli spaces, and constructs new homogeneous Ricci solitons, including a general method for spaces of Type III.

Findings

Classification of 3-symmetric spaces up to Riemannian products.

Explicit description of moduli spaces for Type III spaces.

Construction of expanding Ricci solitons on various homogeneous spaces.

Abstract

The full classification of Riemannian $3$-symmetric spaces is presented. Up to Riemannian products the main building blocks consist in (possibly symmetric) spaces with semisimple isometry group, nilpotent Lie groups of step at most $2$ and spaces of type III and IV. For the most interesting family of examples, the Type III spaces, we produce an explicit description including results concerning the moduli space of all $3$-symmetric metrics living on a given Type III space. Each moduli space contains a unique distinguished point corresponding to an (almost-K\"ahler) expanding Ricci soliton metric. For certain classes of 3-symmetric metrics there are many different groups acting transitively and isometrically on a fixed Riemannian 3-symmetric space. The construction of expanding Ricci solitons on spaces of Type III is also shown to generalize to \emph{any} effective representation of a simple Lie group of non-compact type, yielding a very general construction of homogeneous Ricci solitons. We also give a procedure to compute the isometry group of any Ambrose--Singer space.