A General Local-to-Global Principle for Convexity of Momentum Maps

Abstract

We extend the Local-to-Global-Principle used in the proof of convexity theorems for momentum maps to not necessarily closed maps whose target space carries a convexity structure which need not be based on a metric. Using a new factorization of the momentum map, convexity of its image is proved without local fiber connectedness, and for almost arbitrary spaces of definition. Geodesics are obtained by straightening rather than shortening of arcs, which allows a unified treatment and extension of previous convexity results.Comment: 19 pages LaTeX2e, Preprint 2009, see also: Convexity of Momentum Maps: A Topological Analysis, several parts of the content were presented at the Young Topologists Meeting 2010 in Copenhagen, Denmark, June 16-20, 2010, and at Geometry, Mechanics, and Dynamics: A workshop celebrating the 60th birthday of Tudor Ratiu at CIRM, Luminy, France, July 12-16, 201