The neighborhood of a singular leaf

Laurent-Gengoux, Camille; Ryvkin, Leonid

doi:10.5802/jep.165

The neighborhood of a singular leaf
[Le voisinage d’une feuille singulière]

Laurent-Gengoux, Camille ¹ ; Ryvkin, Leonid ^2,^3,⁴

¹ Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine Metz, France
² Institut Mathématiques de Jussieu, Université Paris Diderot Paris, France
³ Faculty of Mathematics, Universität Duisburg-Essen Essen, Germany
⁴ Institut für Mathematik, Georg-August-Universität Göttingen Göttingen, Germany

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1037-1064.

Résumé
Abstract

Un résultat important concernant les feuilletages réguliers est leur trivialité semi-locale formelle au voisinage des feuilles simplement connexes. Nous étendons ce résultat aux feuilletages singuliers pour toute feuille $2$ -connexe et pour une classe importante de feuilles $1$ -connexes en démontrant un théorème de Levi-Malcev semi-local pour la partie semi-simple de leur algébroïde de Lie d’holonomie.

An important result for regular foliations is their formal semi-local triviality near simply connected leaves. We extend this result to singular foliations for all $2$ -connected leaves and a wide class of $1$ -connected leaves by proving a semi-local Levi-Malcev theorem for the semi-simple part of their holonomy Lie algebroid.

Reçu le : 2020-06-27
Accepté le : 2021-05-03
Publié le : 2021-05-20

DOI : 10.5802/jep.165

Classification : 53C12, 57R30, 93B18
Keywords: Singular foliations, singular leaves, linearizations in control theory, Levi-Malcev theorems, Lie algebroids
Mot clés : Feuilletages singuliers, feuilles singulières, théorie du contrôle et linéarisation, théorèmes de Levi-Malcev, algébroïdes de Lie

Affiliations des auteurs :

Laurent-Gengoux, Camille ¹ ; Ryvkin, Leonid ^{2, 3, 4}

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Laurent-Gengoux, Camille; Ryvkin, Leonid. The neighborhood of a singular leaf. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1037-1064. doi : 10.5802/jep.165. http://www.numdam.org/articles/10.5802/jep.165/

Bibliographie
Cité par

[AM69] Atiyah, M. F.; Macdonald, I. G. Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, MA, 1969 | Zbl

[And17] Androulidakis, Iakovos Personal communication, 2017

[AS09] Androulidakis, Iakovos; Skandalis, Georges The holonomy groupoid of a singular foliation, J. reine angew. Math., Volume 626 (2009), pp. 1-37 | DOI | MR | Zbl

[AZ13] Androulidakis, Iakovos; Zambon, Marco Smoothness of holonomy covers for singular foliations and essential isotropy, Math. Z., Volume 275 (2013) no. 3-4, pp. 921-951 | DOI | MR | Zbl

[AZ14] Androulidakis, Iakovos; Zambon, Marco Holonomy transformations for singular foliations, Adv. Math., Volume 256 (2014), pp. 348-397 | DOI | MR | Zbl

[BLM19] Bursztyn, Henrique; Lima, Hudson; Meinrenken, Eckhard Splitting theorems for Poisson and related structures, J. reine angew. Math., Volume 754 (2019), pp. 281-312 | DOI | MR | Zbl

[Cer79] Cerveau, Dominique Distributions involutives singulières, Ann. Inst. Fourier (Grenoble), Volume 29 (1979) no. 3, pp. 261-294 | DOI | Numdam | MR | Zbl

[Daz85] Dazord, Pierre Feuilletages à singularités, Indag. Math., Volume 47 (1985), pp. 21-39 | DOI | Zbl

[Deb01] Debord, Claire Holonomy groupoids of singular foliations, J. Differential Geom., Volume 58 (2001) no. 3, pp. 467-500 http://projecteuclid.org/euclid.jdg/1090348356 | MR | Zbl

[Duf01] Dufour, Jean-Paul Normal forms for Lie algebroids, Lie algebroids and related topics in differential geometry (Warsaw, 2000) (Banach Center Publ.), Volume 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, pp. 35-41 | DOI | MR | Zbl

[GG20] Grabowska, Katarzyna; Grabowski, Janusz Solvable Lie algebras of vector fields and a Lie’s conjecture, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), 065, 14 pages | DOI | MR | Zbl

[Her63] Hermann, Robert On the accessibility problem in control theory, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 325-332 | DOI | Zbl

[Hoc54] Hochschild, G. Cohomology classes of finite type and finite dimensional kernels for Lie algebras, Amer. J. Math., Volume 76 (1954), pp. 763-778 | DOI | MR | Zbl

[LGLS20] Laurent-Gengoux, Camille; Lavau, Sylvain; Strobl, Thomas The universal Lie $\infty$ -algebroid of a singular foliation, Doc. Math., Volume 25 (2020), pp. 1571-1652 | MR | Zbl

[LGR19] Laurent-Gengoux, Camille; Ryvkin, Leonid The holonomy of a singular leaf, 2019 | arXiv

[Mac87] Mackenzie, K. Lie groupoids and Lie algebroids in differential geometry, London Math. Society Lect. Note Series, 124, Cambridge University Press, Cambridge, 1987 | DOI | Numdam | MR | Zbl

[Mac05] Mackenzie, K. General theory of Lie groupoids and Lie algebroids, London Math. Society Lect. Note Series, 213, Cambridge University Press, Cambridge, 2005 | DOI | MR | Zbl

[Mal67] Malgrange, Bernard Ideals of differentiable functions, TIFR Studies in Math., 3, Tata Institute of Fundamental Research/Oxford University Press, Bombay/London, 1967 | MR

[MZ04] Monnier, Philippe; Zung, Nguyen Tien Levi decomposition for smooth Poisson structures, J. Differential Geom., Volume 68 (2004) no. 2, pp. 347-395 http://projecteuclid.org/euclid.jdg/1115669514 | MR | Zbl

[Ree52] Reeb, Georges Sur certaines propriétés topologiques des variétés feuilletées, Actualités Sci. Ind., 1183, Hermann & Cie., Paris, 1952 | Zbl

[Tou68] Tougeron, Jean-Claude Idéaux de fonctions différentiables. I, Ann. Inst. Fourier (Grenoble), Volume 18 (1968) no. 1, pp. 177-240 | DOI | Zbl

[Wei00] Weinstein, Alan Linearization problems for Lie algebroids and Lie groupoids, Lett. Math. Phys., Volume 52 (2000) no. 1, pp. 93-102 | DOI | MR

[Zam19] Zambon, Marco Personal communication, 2019

[Zun03] Zung, Nguyen Tien Levi decomposition of analytic Poisson structures and Lie algebroids, Topology, Volume 42 (2003) no. 6, pp. 1403-1420 | DOI | MR | Zbl

Cité par Sources :