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Generalizations of Frobenius’ Theorem on Manifolds and Subcartesian Spaces

Published online by Cambridge University Press:  20 November 2018

Jędrzej Śniatycki
Jędrzej Śniatycki*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4
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Abstract

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Let $\mathcal{F}$ be a family of vector fields on a manifold or a subcartesian space spanning a distribution $D.$ We prove that an orbit $O$ of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$ and it has constant rank on $O$. This result implies Frobenius’ theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

Keywords

58A3058A40differential spacesgeneralized distributionsorbitsFrobenius’ theoremSussmann's theorem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 3 , 01 September 2007 , pp. 447 - 459
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Aronszajn, N., Subcartesian and subriemannian spaces. Notices Amer. Math. Soc. 14(1967), 111.Google Scholar
[2] Kolář, I., Michor, P. W. and Slovák, J., Natural Operators in Differential Geometry. Springer-Verlag, Berlin, 1993.Google Scholar
[3] Kowalczyk, A., The open immersion invariance of differential spaces of class D 0 . Demonstratio Math. 13(1980), no. 2, 539–550.Google Scholar
[4] Sikorski, R., Abstract covariant derivative. Colloq. Math. 18(1967), 251–272.Google Scholar
[5] Sikorski, R., Differential modules. Colloq. Math. 24(1971/72), 45–79.Google Scholar
[6] Sikorski, R., Wstęp do Geometrii Różniczkowej, PWN, Warszawa, 1972.Google Scholar
[7] Śniatycki, J., Integral curves of derivations on locally semi-algebraic differential spaces. In: Dynamical Systems and Differential Equations, American Institute of Mathematical Sciences Press, Springfield, MO, 2003, pp. 825–831.Google Scholar
[8] Śniatycki, J., Orbits of families of vector fields on subcartesian spaces. Ann. Inst. Fourier (Grenoble), 53(2003), no. 7, 2257–2296.Google Scholar
[9] Spallek, K., Differenzierbare Räume. Math. Ann 180(1969), 269–296.Google Scholar
[10] Sussmann, H. J., Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180(1973), 171–188.Google Scholar
[11] Walczak, P. G., A theorem on diffeomorphisms in the category of differential spaces. Bull. Acad. Polon. Sci., Sér. Sci. Math Astronom. Phys. 21(1973), 325–329.Google Scholar
[12] Warner, F. W., Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics 94, Springer-Verlag, New York, 1983.Google Scholar