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

By D. Alekseevsky, Andreas Kriegl, M. Losik, Peter W. Michor. Publicationes mathematicae (Debrecen), 2001

65 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1063c33e8dd9734b22d2b0c95073ba3cb6208468

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In Geometry and Topology

  1. Peter W. Michor
  2. B. Clarke

In Mathematical Physics

  1. Peter W. Michor
  2. Tom Leinster
  3. Michel Waldschmidt
  4. B. Clarke

In Statistical and Nonlinear Physics

  1. Peter W. Michor

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