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, pp. 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 : 10.5802/aif.2006
Classification : 58A40, 70H33, 32C15
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
Śniatycki, Jedrzej 1

1 University of Calgary, Department of Mathematics and Statistics, 2500 University Drive NW, Calgary, Alberta T2N 1N4 (Canada)
@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] N. Aronszajn Subcartesian and subriemannian spaces, Notices Amer. Math. Soc, Volume 14 (1967), pp. 111

[2] N. Aronszajn; P. Szeptycki The theory of Bessel potentials. IV., Ann. Inst. Fourier (Grenoble), Volume 25 (1975) no. 3/4, pp. 27-69 | DOI | Numdam | MR | Zbl

[3] N. Aronszajn; P. Szeptycki Subcartesian spaces, J. Differential Geom, Volume 15 (1980), pp. 393-416 | MR | Zbl

[4] R. Cushman; L. Bates Global aspects of classical integrable systems, Birkhäuser, Basel, 1997 | MR | Zbl

[5] L. Bates; E. Lerman Proper group actions and symplectic stratified spaces, Pacific J. Math, Volume 181 (1997), pp. 201-229 | DOI | MR | Zbl

[6] E. Bierstone Lifting isotopies from orbit spaces, Topology, Volume 14 (1975), pp. 245-272 | DOI | MR | Zbl

[7] E. Bierstone The Structure of orbit spaces and the singularities of equivariant mappings (Monografias de Matemática), Volume vol. 35 (1980), pp. Rio de Janeiro | Zbl

[8] R. Cushman; J. {#x015A;}niatycki Differential structure of orbit spaces, Canad. J. Math, Volume 53 (2001), pp. 715-755 | DOI | MR | Zbl

[9] J.J. Duistermaat; J.A.C. Kolk Lie groups, Springer Verlag, New York, 1999 | MR | Zbl

[10] M. Goresky; R. MacPherson Stratified Morse theory, Springer Verlag, New York, 1988 | MR | Zbl

[11] J. Huebschmann Kähler spaces, nilpotent orbits, and singular reduction (e-print, Mathematics ArXiv DG/0104213)

[12] P. Libermann; C.-M. Marle Symplectic geometry and analytical mechanics, D. Reidel Publishing Company, Dordrecht, 1987 | MR | Zbl

[13] C.D. Marshall Calculus on subcartesian spaces, J. Differential Geom, Volume 10 (1975), pp. 551-573 | MR | Zbl

[14] C.D. Marshall The de Rham cohomology on subcartesian spaces, J. Differential Geom, Volume 10 (1975), pp. 575-588 | MR | Zbl

[15] A. Newlander; L. Nirenberg Complex analytic coordinates in almost complex manifolds, Ann. of Math., Volume 65 (1957), pp. 391-404 | DOI | MR | Zbl

[16] M.J. Pflaum Analytic and geometric study of study of stratified spaces, Lecture Notes in Mathematics, vol. 1768, Springer Verlag, Berlin, 2001 | MR | Zbl

[17] G.W. Schwarz Smooth functions invariant under the action of a compact Lie group, Topology, Volume 14 (1975), pp. 63-68 | DOI | MR | Zbl

[18] R. Sikorski Abstract covariant derivative, Colloq. Math, Volume 18 (1967), pp. 251-272 | MR | Zbl

[19] R. Sikorski Differential modules, Colloq. Math, Volume 24 (1971), pp. 45-79 | MR | Zbl

[20] R. Sikorski Wstȩp do Geometrii Ró\. zniczkowej, vol. 42, PWN, Warszawa, 1972 | MR | Zbl

[21] R. Sjamaar; E. Lerman Stratified symplectic spaces and reduction, Ann. Math, Volume 134 (1991), pp. 375-422 | DOI | MR | Zbl

[22] J. {#x015A;}niatycki Almost Poisson structures and nonholonomic singular reduction, Rep. Math. Phys, Volume 48 (2001), pp. 235-248 | DOI | Zbl

[23] J. {#x015A;}niatycki 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] K. Spallek Differenzierbare Räume, Math. Ann., Volume 180 (1969), pp. 269-296 | DOI | MR | Zbl

[25] K. Spallek Differential forms on differentiable spaces, Rend. Mat. (2), Volume 6 (1971), pp. 237-258 | MR | Zbl

[26] P. Stefan Acessible sets, orbits and foliations with singularities, Proc. London Math. Soc., Volume 29 (1974), pp. 699-713 | DOI | MR | Zbl

[27] H. J. Sussmann Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc, Volume 180 (1973), pp. 171-188 | DOI | MR | Zbl

Cité par Sources :