Śniatycki, Jedrzej
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
Zbl 1048.53060 | MR 2044173
doi : 10.5802/aif.2006
URL stable : http://www.numdam.org/item?id=AIF_2003__53_7_2257_0

Classification:  58A40,  70H33,  32C15
Mots clés: structure presque complexe, espace différentiel, espace kählérien, réduction de Poisson, réduction singulière, espace stratifié
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.

Bibliographie

[1] N. Aronszajn Subcartesian and subriemannian spaces, Notices Amer. Math. Soc, 14 (1967), p. 111

[2] N. Aronszajn; P. Szeptycki The theory of Bessel potentials. IV., Ann. Inst. Fourier (Grenoble), 25 (1975) no. 3/4, p. 27 -69 Article  Numdam | MR 435824 | Zbl 0304.31010

[3] N. Aronszajn; P. Szeptycki Subcartesian spaces, J. Differential Geom, 15 (1980), p. 393 -416 MR 620895 | Zbl 0451.58006

[4] R. Cushman; L. Bates Global aspects of classical integrable systems, Birkhäuser, Basel (1997) MR 1438060 | Zbl 0882.58023

[5] L. Bates; E. Lerman Proper group actions and symplectic stratified spaces, Pacific J. Math, 181 (1997), p. 201 -229 Article  MR 1486529 | Zbl 0902.58008

[6] E. Bierstone Lifting isotopies from orbit spaces, Topology, 14 (1975), p. 245 -272 Article  MR 375356 | Zbl 0317.57015

[7] E. Bierstone The Structure of orbit spaces and the singularities of equivariant mappings, Instituto de Matemática Pura e Applicada, Rio de Janeiro (Monografias de Matemática) vol. 35 (1980), p. Rio de Janeiro Zbl 0501.57001

[8] R. Cushman; J. {#X015A;}Niatycki Differential structure of orbit spaces, Canad. J. Math, 53 (2001), p. 715 -755 Article  MR 1848504 | Zbl 01688933

[9] J.J. Duistermaat; J.A.C. Kolk Lie groups, Springer Verlag, New York (1999) MR 1738431 | Zbl 0955.22001

[10] M. Goresky; R. Macpherson Stratified Morse theory, Springer Verlag, New York (1988) MR 932724 | Zbl 0639.14012

[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 882548 | Zbl 0643.53002

[13] C.D. Marshall Calculus on subcartesian spaces, J. Differential Geom, 10 (1975), p. 551 -573 MR 394742 | Zbl 0317.58007

[14] C.D. Marshall The de Rham cohomology on subcartesian spaces, J. Differential Geom, 10 (1975), p. 575 -588 MR 394743 | Zbl 0319.58003

[15] A. Newlander; L. Nirenberg Complex analytic coordinates in almost complex manifolds, Ann. of Math., 65 (1957), p. 391 -404 Article  MR 88770 | Zbl 0079.16102

[16] M.J. Pflaum Analytic and geometric study of study of stratified spaces, Springer Verlag, Berlin, Lecture Notes in Mathematics, vol. 1768 (2001) MR 1869601 | Zbl 0988.58003

[17] G.W. Schwarz Smooth functions invariant under the action of a compact Lie group, Topology, 14 (1975), p. 63 -68 Article  MR 370643 | Zbl 0297.57015

[18] R. Sikorski Abstract covariant derivative, Colloq. Math, 18 (1967), p. 251 -272 MR 222799 | Zbl 0162.25101

[19] R. Sikorski Differential modules, Colloq. Math, 24 (1971), p. 45 -79 MR 482794 | Zbl 0226.53004

[20] R. Sikorski Wstȩp do Geometrii Ró\. zniczkowej, PWN, Warszawa vol. 42 (1972) MR 467544 | Zbl 0255.53001

[21] R. Sjamaar; E. Lerman Stratified symplectic spaces and reduction, Ann. Math, 134 (1991), p. 375 -422 Article  MR 1127479 | Zbl 0759.58019

[22] J. {#X015A;}Niatycki Almost Poisson structures and nonholonomic singular reduction, Rep. Math. Phys, 48 (2001), p. 235 -248 Article  Zbl 1015.53051

[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, Wilmington, NC, USA (2002), p. 825 -831 Zbl 1086.53100

[24] K. Spallek Differenzierbare Räume, Math. Ann., 180 (1969), p. 269 -296 Article  MR 261035 | Zbl 0169.52901

[25] K. Spallek Differential forms on differentiable spaces, Rend. Mat. (2), 6 (1971), p. 237 -258 MR 304706 | Zbl 0221.58003

[26] P. Stefan Acessible sets, orbits and foliations with singularities, Proc. London Math. Soc., 29 (1974), p. 699 -713 Article  MR 362395 | Zbl 0342.57015

[27] H. J. Sussmann Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc, 180 (1973), p. 171 -188 Article  MR 321133 | Zbl 0274.58002