3-symmetric spaces, Ricci solitons, and homogeneous structures
Thomas Murphy, Paul-Andi Nagy
TL;DR
This work provides a complete Lie-algebraic framework for classifying and constructing Riemannian 3-symmetric spaces. By analyzing the transvection algebra $$, its radical, and admissible representations, the authors obtain a canonical decomposition into Type I–IV factors and prove regularity results linking infinitesimal data to global homogeneous spaces. They reveal rich moduli for Type III metrics, including unique almost-Kähler expanding Ricci solitons and a broad mechanism to generate homogeneous Ricci solitons from arbitrary representations of non-compact simple Lie groups. The paper also develops tools to compute the full isometry group of Ambrose–Singer spaces and to understand isotropy-invariant metrics and polar foliations, significantly advancing the structural understanding and applications of Riemannian 3-symmetric geometry.
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-Kähler) 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.