[Orbites d'ensembles de champs de vecteurs sur des espaces sous-cartésiens]
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 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.
Keywords: almost complex structure, differential spoace, Kähler space, Poisson reduction, singular reduction, stratified space
Mot 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},
pages = {2257--2296},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {53},
number = {7},
year = {2003},
doi = {10.5802/aif.2006},
mrnumber = {2044173},
zbl = {1048.53060},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2006/}
}
TY - JOUR AU - Śniatycki, Jedrzej TI - Orbits of families of vector fields on subcartesian spaces JO - Annales de l'Institut Fourier PY - 2003 SP - 2257 EP - 2296 VL - 53 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2006/ DO - 10.5802/aif.2006 LA - en ID - AIF_2003__53_7_2257_0 ER -
%0 Journal Article %A Śniatycki, Jedrzej %T Orbits of families of vector fields on subcartesian spaces %J Annales de l'Institut Fourier %D 2003 %P 2257-2296 %V 53 %N 7 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2006/ %R 10.5802/aif.2006 %G en %F 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://archive.numdam.org/articles/10.5802/aif.2006/
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