On homogeneous manifolds whose isotropy actions are polar

TL;DR

This paper classifies and characterizes homogeneous Riemannian spaces with polar isotropy actions, showing that under certain conditions they are symmetric, and providing examples of non-polar cases.

Contribution

It extends the classification of homogeneous spaces with polar isotropy actions to non-compact cases and identifies non-polar examples like Heisenberg groups.

Findings

Simply connected homogeneous spaces with polar isotropy actions are symmetric.

Many non-compact homogeneous spaces do not have polar isotropy actions.

Heisenberg groups and Damek-Ricci spaces have non-polar isotropy actions.

Abstract

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek-Ricci spaces have non-polar isotropy actions.