Cohomogeneity one actions on symmetric spaces of noncompact type

TL;DR

This paper develops a conceptual framework to classify cohomogeneity one isometric actions on noncompact symmetric spaces, leading to new examples and explicit classifications of such actions.

Contribution

It introduces a new conceptual approach to classify cohomogeneity one actions on noncompact symmetric spaces using orbit equivalence, resulting in new examples and explicit classifications.

Findings

Many new cohomogeneity one actions identified

Explicit classifications achieved for specific symmetric spaces

A unified conceptual approach developed for classification

Abstract

An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.