Topics in polar actions

TL;DR

This paper provides lecture notes on polar actions in Riemannian geometry, covering their characterization, orbifold points, and variationally complete actions, with insights into related geometric structures and criteria.

Contribution

It offers a comprehensive overview of polar actions, including new characterizations and proofs related to their properties and connections to orbifolds and symmetric spaces.

Findings

Polar actions characterized by integrability of normal distributions.

Orbifold points characterized by polarity of slice representations.

Variationally complete actions are hyperpolar on non-negatively curved manifolds.

Abstract

These are the notes for a series of lectures at the Institute of Geometry and Topology of the University of Stuttgart, Germany, in July 13-15, 2022. We assume basic knowledge of isometric actions on Riemannian manifolds, including the normal slice theorem and the principal orbit type theorem. Lecture 1 introduces polar actions and culminates with Heintze, Liu and Olmos's argument to characterize them in terms of integrability of the distribution of normal spaces to the principal orbits. The other two lectures are devoted to two of Lytchak and Thorbergsson's results. In Lecture 2 we briefly review Riemannian orbifolds from the metric point of view, and explain their characterization of orbifold points in the orbit space of a proper and isometric action in terms of polarity of the slice representation above. In Lecture 3 we present their proof of the fact that variationally complete actions in the sense of Bott and Samelson on non-negatively curved manifolds are hyperpolar. The appendix contains explanations of some results used in the lectures, namely: a more or less self-contained derivation of Wilking's transversal Jacobi equation; a discussion of Cartan's and Hermann's criterions for the existence of totally geodesic submanifolds, and a criterion for the polarity of isometric actions on symmetric spaces.