Local reduction theorems and invariants for singular contact structures
Annales de l'Institut Fourier, Volume 51 (2001) no. 1, p. 237-295

A differential 1-form on a (2k+1)-dimensional manifolds M defines a singular contact structure if the set S of points where the contact condition is not satisfied, S={pM:(ω(dω) 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.

Soit ω une 1-forme différentielle locale sur une variété M de dimension 2k+1. Par définition, elle définit une structure locale singulière de contact si le lieu S de ses points singuliers S={pM:(ω(dω) 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 , 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.

DOI : https://doi.org/10.5802/aif.1823
Classification:  58A17,  53B99
Keywords: contact structure, singularity, pfaffian equation, equivalence, local invariants, reduction theorems, homotopy method
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     author = {Jakubczyk, Bronislaw and Zhitomirskii, Michail},
     title = {Local reduction theorems and invariants for singular contact structures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {1},
     year = {2001},
     pages = {237-295},
     doi = {10.5802/aif.1823},
     zbl = {1047.53051},
     mrnumber = {1821076},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2001__51_1_237_0}
}
Jakubczyk, Bronislaw; Zhitomirskii, Michail. Local reduction theorems and invariants for singular contact structures. Annales de l'Institut Fourier, Volume 51 (2001) no. 1, pp. 237-295. doi : 10.5802/aif.1823. http://www.numdam.org/item/AIF_2001__51_1_237_0/

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