Openness and convexity for momentum maps

Details Openness and convexity for momentum maps Openness and convexity for momentum maps

2005-11-23

arxiv.org/abs/math/0511576v3

Mathematics

Symplectic Geometry

25 pages, 2 figures. Updated version with added details and minor corrections

Scientific paper

The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the "Lokal-global-Prinzip" that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to non-compact manifolds.

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