Orbits of families of vector fields on subcartesian spaces

[Orbites d'ensembles de champs de vecteurs sur des espaces sous-cartésiens]

Śniatycki, Jedrzej

Annales de l'Institut Fourier, Tome 53 (2003), p. 2257-2296 / Harvested from Numdam

Résumé

Nous démontrons que les orbites d’un ensemble complet de champs de vecteurs sur des espaces sous-cartésiens sont des variétés différentielles. Ce résultat permet de décrire la structure de l’espace de phase réduite d’un système hamiltonien à l’aide de l’algèbre de Poisson réduite. De plus, nous pouvons donner une description globale des structures géométriques de classe $C^\infty $ sur une famille de variétés formant un feuilletage singulier d’un espace sous-cartésien, en fonction d’objets définis par l’ensemble des champs de vecteurs correspondants.

Abstract

Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified spaces, Poisson spaces, and almost complex spaces are discussed as examples.

Détail

Publié le : 2003-01-01

MR 2044173 | Zbl 1048.53060

DOI : https://doi.org/10.5802/aif.2006
Classification:  58A40,  70H33,  32C15
Mots clés: structure presque complexe, espace différentiel, espace kählérien, réduction de Poisson, réduction singulière, espace stratifié

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