Orbits of families of vector fields on subcartesian spaces  [ Orbites d'ensembles de champs de vecteurs sur des espaces sous-cartésiens ]
Annales de l'Institut Fourier, Tome 53 (2003) no. 7, p. 2257-2296
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 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.
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.
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
@article{AIF_2003__53_7_2257_0,
     author = {\'Sniatycki, Jedrzej},
     title = {Orbits of families of vector fields on subcartesian spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     pages = {2257-2296},
     doi = {10.5802/aif.2006},
     zbl = {1048.53060},
     mrnumber = {2044173},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_7_2257_0}
}
Śniatycki, Jedrzej. Orbits of families of vector fields on subcartesian spaces. Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2257-2296. doi : 10.5802/aif.2006. http://www.numdam.org/item/AIF_2003__53_7_2257_0/

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