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Transactions of the American Mathematical Society

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Openness and convexity for momentum maps

Authors: Petre Birtea, Juan-Pablo Ortega and Tudor S. Ratiu
Journal: Trans. Amer. Math. Soc. 361 (2009), 603-630
MSC (2000): Primary 53D20
Published electronically: September 9, 2008
MathSciNet review: 2452817
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Abstract: The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a 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 so-called Local-to-Global Principle 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 noncompact manifolds.

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Additional Information

Petre Birtea
Affiliation: Departamentul de Matematică, Universitatea de Vest, RO–1900 Timişoara, Romania

Juan-Pablo Ortega
Affiliation: Département de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16 route de Gray, F–25030 Besançon cédex, France

Tudor S. Ratiu
Affiliation: Centre Bernoulli, École Polytechnique Fédérale de Lausanne, CH–1015 Lausanne, Switzerland

Received by editor(s): July 11, 2006
Published electronically: September 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society