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Stratified Subcartesian Spaces

  • Tsasa Lusala (a1) and Jędrzej Śniatycki (a1)
Abstract

We show that if the family of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in is locally closed, then defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space of orbits M/G of a proper action of a Lie group G on a smooth manifold M is given by orbits of the family of all vector fields on M/G.

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References
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[1] Aronszajn, N., Subcartesian and subriemannian spaces. Notices Amer. Math. Soc. 14(1967), 111.
[2] Bierstone, E., Lifting isotopies from orbit spaces. Topology 14(1975), no. 3, 245252. doi:10.1016/0040-9383(75)90005-1
[3] Bierstone, E., The structure of orbit spaces and the singularities of equivariant mappings. Monografías de Matemática, 35, Instituto de Matemática Pura e Applicada, Rio de Janeiro, 1980.
[4] Cushman, R. and Śniatycki, J., Differential structure of orbit spaces. Canad. J. Math. 53(2001), no. 4, 715755. doi:10.4153/CJM-2001-029-1
[5] Duistermaat, J. J., Dynamical systems with symmetries. http://www.math.uu.nl/people/duis/homepageHD/sym.pdf
[6] Duistermaat, J. J. and Kolk, J. A. C., Lie groups. Springer-Verlag, Berlin, 2000.
[7] Goresky, M. and MacPherson, R., Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 14, Springer Verlag, Berlin, 1988.
[8] Mather, J. N., Stratifications and mappings. In: Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, pp. 195232.
[9] Sikorski, R., Wstęp do geometrii Różniczkowej. Biblioteka Matematyczna, 42, Państwowe Wydawnictwo Naukowe, Warsaw, 1972.
[10] Śniatycki, J., Orbits of families of vector fields on subcartesian spaces. Ann. Inst. Fourier (Grenoble) 53(2003), no. 7, 22572296.
[11] Whitney, H., Local properties of analytic varieties. In: Differentiable and combinatorial topology, Princeton University Press, Princeton, NJ, 1965.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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