The neighborhood of a singular leaf

Camille Laurent-Gengoux; Leonid Ryvkin

doi:10.5802/jep.165

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

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

¹ Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine Metz, France
² Faculty of Mathematics, Universität Duisburg-Essen Essen, Germany
³ Institut Mathématiques de Jussieu, Université Paris Diderot Paris, France
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 :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Camille Laurent-Gengoux and Leonid Ryvkin},
     title = {The neighborhood of a singular leaf},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1037--1064},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.165},
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Camille Laurent-Gengoux; Leonid Ryvkin. The neighborhood of a singular leaf. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1037-1064. doi : 10.5802/jep.165. https://jep.centre-mersenne.org/articles/10.5802/jep.165/

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