Topics in polar actions
Claudio Gorodski
TL;DR
This work surveys polar actions, detailing how sections intersect all orbits orthogonally and how their geometry reduces the study to discrete Weyl-type symmetries. It connects polarity to the integrability of the principal horizontal distribution, and shows how the orbit space of a polar action forms a Riemannian orbifold, with orbifold points precisely where slice representations are polar. The text then presents foundational results on variational completeness, proving that variationally complete actions on nonnegatively curved spaces are hyperpolar, and outlines the key converse results showing variational completeness implies polarity. Core tools include Cartan–Hermann criteria for totally geodesic submanifolds and Wilking’s transversal Jacobi equation, which together bridge local linear models to global geometric structure. Collectively, these results illuminate how polar and hyperpolar structures organize orbit spaces, slice representations, and orbifold geometry, with implications for classifications on symmetric and non-symmetric spaces alike.
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.