The Hamiltonian tube of a cotangent-lifted action

The Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawb...

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Bibliographic Details
Published in:Journal of symplectic geometry 2017, Vol.15 (3), p.803-852
Main Authors: Rodríguez-Olmos, Miguel, Teixidó-Román, Miguel
Format: Article
Language:English
Subjects:
53 Differential geometry
53D Symplectic geometry, contact geometry
Classificació AMS
Geometria
Geometria diferencial
Matemàtiques i estadística
Àrees temàtiques de la UPC
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Summary:The Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawback of the MGS form is that it does not have an explicit expression. We will obtain a MGS form for cotangent-lifted actions on cotangent bundles that, in addition to its defining features, respects the additional fibered structure present. This model generalizes previous results obtained by T. Schmah for orbits with fully-isotropic momentum. In addition, our construction is explicit up to the integration of a differential equation on G. This equation can be easily solved for the groups SO(3) or SL(2), thus giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle. Peer Reviewed
ISSN:1527-5256
1540-2347
DOI:10.4310/JSG.2017.v15.n3.a7