Cohomogeneity one actions on unit spheres

TL;DR

This paper classifies all isometric cohomogeneity one actions on unit spheres and extends the classification to certain Riemannian symmetric spaces, providing a comprehensive understanding of these symmetries.

Contribution

It offers a complete classification of cohomogeneity one actions on spheres and applies this to classify actions on complex and quaternionic projective spaces.

Findings

Complete classification of cohomogeneity one actions on spheres

Classification of actions on complex projective spaces

Classification of actions on quaternionic projective spaces

Abstract

In this paper, we provide a complete classification for all the isometric cohomogeneity one actions on unit spheres. Using this theory, we can very easily classify all the isometric cohomogeneity one actions on the Riemannian symmetric spaces $\mathbb{C}\mathrm{P}^{m-1}$ and $\mathbb{H}\mathrm{P}^{m-1}$.