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Openness and convexity for momentum maps
Author(s):
Petre
Birtea;
Juan-Pablo
Ortega;
Tudor
S.
Ratiu
Journal:
Trans. Amer. Math. Soc.
361
(2009),
603-630.
MSC (2000):
Primary 53D20
Posted:
September 9, 2008
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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 Matematica, Universitatea de Vest, RO-1900 Timisoara, Romania
Email:
birtea@math.uvt.ro
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
Email:
Juan-Pablo.Ortega@math.univ-fcomte.fr
Tudor
S.
Ratiu
Affiliation:
Centre Bernoulli, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Email:
tudor.ratiu@epfl.ch
DOI:
10.1090/S0002-9947-08-04689-8
PII:
S 0002-9947(08)04689-8
Received by editor(s):
July 11, 2006
Posted:
September 9, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
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