A cotangent bundle Hamiltonian tube theorem and its applications in reduction theory
The Marle-Guillemin-Sternberg (MGS) model is an extremely important tool for the theory of Hamiltonian actions on symplectic manifolds. It has been extensively used to prove many local results both in symplectic geometry and in symmetric Hamiltonian systems theory. It provides a model for a tubular...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/288376 |
| Acceso en línea: | http://hdl.handle.net/10803/288376 https://dx.doi.org/10.5821/dissertation-2117-95674 |
| Access Level: | acceso abierto |
| Palabra clave: | Cotangent bundles Normal forms Stratified spaces Singular reduction Momentum maps 514 |
| Sumario: | The Marle-Guillemin-Sternberg (MGS) model is an extremely important tool for the theory of Hamiltonian actions on symplectic manifolds. It has been extensively used to prove many local results both in symplectic geometry and in symmetric Hamiltonian systems theory. It provides a model for a tubular neighborhood of a group orbit and puts in normal form the group action and the symplectic structure. The main drawback of the MGS model is that it is not explicit. Only it existence and main properties can be proved. Moreover, for cotangent bundles, this model does not respect the natural fibration. In the first part of the thesis we build an MGS model specially adapted to the cotangent bundle geometry. 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), hence giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle. In the second part of the thesis we apply this adapted MGS model to describe the structure of the symplectic reduction of a cotangent bundle. We show that the base projection of any momentum leaf is a Whitney stratified space. Moreover, we can refine the orbit-type stratification of the symplectic reduced space so that each piece is a fibered space. We prove that each of those pieces is endowed with a constant rank presymplectic form and that there is always one unique piece which is open and dense. Furthermore, this maximal piece is symplectomorphic to a vector subbundle of a certain cotangent bundle. |
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