Local reduction theorems and invariants for singular contact structures

Jakubczyk, Bronislaw; Zhitomirskii, Michail

doi:10.5802/aif.1823

Local reduction theorems and invariants for singular contact structures
[Théorèmes de réduction et invariants locaux des structures singulières de contact]

Jakubczyk, Bronislaw ¹ ; Zhitomirskii, Michail ²

¹ Polish Academy of Sciences, Institute of Mathematics, Sniadeckich 8, 00-950 Warsaw (Pologne)
² Technion, Department of Mathematics, 32000 Haifa (Israël)

Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 237-295.

Résumé
Abstract

Soit $ω$ une 1-forme différentielle locale sur une variété $M$ de dimension $2 k + 1$ . Par définition, elle définit une structure locale singulière de contact si le lieu $S$ de ses points singuliers $S = {p \in M : (ω \land {(d ω)}^{k}) (p) = 0}$ est nulle part dense. Dans un tel cas on peut définir la restriction (pullback) ${ω |}_{S}$ de $ω$ sur l’hypersurface singulière $S$ . Nos théorèmes disent que, dans les catégories holomorphe, analytique réelle et $C^{\infty}$ , l’équation locale de Pfaff ${ω |}_{S} = 0$ sur $S$ détermine l’équation locale de Pfaff $ω = 0$ sur $M$ , à un difféomorphisme près, si on exclut certaines dégénérescences de codimension infinie de $ω$ . De plus, si $S$ est lisse, l’équation locale de Pfaff $ω = 0$ sur $M$ est déterminée, à un difféomorphisme près, par sa restriction sur $S$ et deux invariants complémentaires: une orientation et une connexion partielle. Ces invariants sont en général indépendants. Nos résultats impliquent une classification des singularités des équations de Pfaff locales en dimension 3.

A differential 1-form on a $(2 k + 1)$ -dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S = {p \in M : (ω \land {(d ω)}^{k} (p) = 0}$ , is nowhere dense in $M$ . Then $S$ is a hypersurface with singularities and the restriction of $ω$ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $(ω)$ generated by $ω$ is determined, up to a diffeomorphism, by its restriction to $S$ , if we eliminate certain degenerated singularities of $ω$ (in the holomorphic case they form a set of infinite codimension). We also define other invariants of local singular contact structures: orientations, a line bundle, and a partial connection. We study the problem when these invariants, together with the hypersurface $S$ and the restriction of the Pfaffian equation $(ω)$ to $S$ , form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where $S$ has no singularities.

MR Zbl

DOI : 10.5802/aif.1823

Classification : 58A17, 53B99
Keywords: contact structure, singularity, pfaffian equation, equivalence, local invariants, reduction theorems, homotopy method
Mot clés : structure de contact, singularité, équation de Pfaff, équivalence, invariants locaux, théorèmes de réduction, méthode homotopique

Affiliations des auteurs :

Jakubczyk, Bronislaw ¹ ; Zhitomirskii, Michail ²

¹ Polish Academy of Sciences, Institute of Mathematics, Sniadeckich 8, 00-950 Warsaw (Pologne)
² Technion, Department of Mathematics, 32000 Haifa (Israël)

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     title = {Local reduction theorems and invariants for singular contact structures},
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Jakubczyk, Bronislaw; Zhitomirskii, Michail. Local reduction theorems and invariants for singular contact structures. Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 237-295. doi : 10.5802/aif.1823. http://www.numdam.org/articles/10.5802/aif.1823/

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