The Riemannian geometry of orbit spaces. The metric, geodesics, and integrable systems

TL;DR

This paper explores the Riemannian geometry of orbit spaces formed by Lie group actions on manifolds, analyzing geodesics, convexity, and integrable systems through examples like matrices and symplectic reductions.

Contribution

It develops the geometric framework for orbit spaces, introduces notions of geodesics and convexity, and generalizes integrable systems such as Calogero-Moser models with spin.

Findings

Minimal geodesic arcs are length minimizing and hit singular strata only at endpoints.

Geodesics in orbit spaces can be characterized via projections of normal geodesics in the original manifold.

Examples include explicit equations for geodesics in matrix spaces and their relation to integrable systems.

Abstract

We investigate the rudiments of Riemannian geometry on orbit spaces $M/G$ for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space $M/G$ and they can hit strata which are more singular only at the end points. This is phrased as convexity result. The geodesic spray, viewed as a (strata-preserving) vector field on $TM/G$, leads to the notion of geodesics in $M/G$ which are projections under $M\to M/G$ of geodesics which are normal to the orbits. It also leads to `ballistic curves' which are projections of the other geodesics. In examples (Hermitian and symmetric matrices, and more generally polar representations) we compute their equations by singular symplectic reductions and obtain generalizations of Calogero-Moser systems with spin.