[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://www.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://www.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://www.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://www.numdam.org/articles/10.5802/aif.2006/
[1] Subcartesian and subriemannian spaces, Notices Amer. Math. Soc, Volume 14 (1967), pp. 111
[2] The theory of Bessel potentials. IV., Ann. Inst. Fourier (Grenoble), Volume 25 (1975) no. 3/4, pp. 27-69 | DOI | Numdam | MR | Zbl
[3] Subcartesian spaces, J. Differential Geom, Volume 15 (1980), pp. 393-416 | MR | Zbl
[4] Global aspects of classical integrable systems, Birkhäuser, Basel, 1997 | MR | Zbl
[5] Proper group actions and symplectic stratified spaces, Pacific J. Math, Volume 181 (1997), pp. 201-229 | DOI | MR | Zbl
[6] Lifting isotopies from orbit spaces, Topology, Volume 14 (1975), pp. 245-272 | DOI | MR | Zbl
[7] The Structure of orbit spaces and the singularities of equivariant mappings (Monografias de Matemática), Volume vol. 35 (1980), pp. Rio de Janeiro | Zbl
[8] Differential structure of orbit spaces, Canad. J. Math, Volume 53 (2001), pp. 715-755 | DOI | MR | Zbl
[9] Lie groups, Springer Verlag, New York, 1999 | MR | Zbl
[10] Stratified Morse theory, Springer Verlag, New York, 1988 | MR | Zbl
[11] Kähler spaces, nilpotent orbits, and singular reduction (e-print, Mathematics ArXiv DG/0104213)
[12] Symplectic geometry and analytical mechanics, D. Reidel Publishing Company, Dordrecht, 1987 | MR | Zbl
[13] Calculus on subcartesian spaces, J. Differential Geom, Volume 10 (1975), pp. 551-573 | MR | Zbl
[14] The de Rham cohomology on subcartesian spaces, J. Differential Geom, Volume 10 (1975), pp. 575-588 | MR | Zbl
[15] Complex analytic coordinates in almost complex manifolds, Ann. of Math., Volume 65 (1957), pp. 391-404 | DOI | MR | Zbl
[16] Analytic and geometric study of study of stratified spaces, Lecture Notes in Mathematics, vol. 1768, Springer Verlag, Berlin, 2001 | MR | Zbl
[17] Smooth functions invariant under the action of a compact Lie group, Topology, Volume 14 (1975), pp. 63-68 | DOI | MR | Zbl
[18] Abstract covariant derivative, Colloq. Math, Volume 18 (1967), pp. 251-272 | MR | Zbl
[19] Differential modules, Colloq. Math, Volume 24 (1971), pp. 45-79 | MR | Zbl
[20] Wstȩp do Geometrii Ró\. zniczkowej, vol. 42, PWN, Warszawa, 1972 | MR | Zbl
[21] Stratified symplectic spaces and reduction, Ann. Math, Volume 134 (1991), pp. 375-422 | DOI | MR | Zbl
[22] Almost Poisson structures and nonholonomic singular reduction, Rep. Math. Phys, Volume 48 (2001), pp. 235-248 | DOI | Zbl
[23] Integral curves of derivations on locally semi-algebraic differential spaces, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24--27 (2002), pp. 825-831 | Zbl
[24] Differenzierbare Räume, Math. Ann., Volume 180 (1969), pp. 269-296 | DOI | MR | Zbl
[25] Differential forms on differentiable spaces, Rend. Mat. (2), Volume 6 (1971), pp. 237-258 | MR | Zbl
[26] Acessible sets, orbits and foliations with singularities, Proc. London Math. Soc., Volume 29 (1974), pp. 699-713 | DOI | MR | Zbl
[27] Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc, Volume 180 (1973), pp. 171-188 | DOI | MR | Zbl
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