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

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

2001-02-20

arxiv.org/abs/math/0102159v1

Publ. Math. Debrecen 62 (2003), 247-276

Mathematics

Differential Geometry

AmSTeX, 21 pages

Scientific paper

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.

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