Polar actions on homogeneous 3-spaces

TL;DR

This paper classifies polar isometric actions on simply connected 3D homogeneous spaces, focusing on orbit structures and extrinsically homogeneous surfaces, advancing understanding of their geometric properties.

Contribution

It provides a complete classification of polar actions and extrinsically homogeneous surfaces in 3D homogeneous spaces, up to orbit equivalence.

Findings

Classification of polar actions on 3D homogeneous spaces

Description of orbit foliations for cohomogeneity one actions

Analysis of extrinsically homogeneous surfaces in these spaces

Abstract

We classify polar isometric actions on simply connected 3-dimensional Riemannian homogeneous spaces, up to orbit equivalence. In particular, we classify extrinsically homogeneous surfaces in such spaces and study the geometry of the orbit foliations of the corresponding cohomogeneity one actions.