Polar actions with a fixed point

TL;DR

This paper establishes a criterion for when isometric Lie group actions on Riemannian manifolds are polar, particularly focusing on actions with fixed points, and classifies such actions on symmetric spaces.

Contribution

It provides a new criterion for polar actions with fixed points and classifies these actions on symmetric spaces, advancing understanding of symmetry in Riemannian geometry.

Findings

An action with a fixed point is polar iff the slice representation is polar and the section is a totally geodesic submanifold.

The paper classifies all polar actions with a fixed point on symmetric spaces.

Provides a criterion linking fixed points, slice representations, and sections in polar actions.

Abstract

We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold. We apply this to obtain a classification of polar actions with a fixed point on symmetric spaces.