A differential 1-form on a -dimensional manifolds defines a singular contact structure if the set of points where the contact condition is not satisfied, , is nowhere dense in . Then is a hypersurface with singularities and the restriction of to can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation generated by is determined, up to a diffeomorphism, by its restriction to , if we eliminate certain degenerated singularities of (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 and the restriction of the Pfaffian equation to , form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where has no singularities.
Soit une 1-forme différentielle locale sur une variété de dimension . Par définition, elle définit une structure locale singulière de contact si le lieu de ses points singuliers est nulle part dense. Dans un tel cas on peut définir la restriction (pullback) de sur l’hypersurface singulière . Nos théorèmes disent que, dans les catégories holomorphe, analytique réelle et , l’équation locale de Pfaff sur détermine l’équation locale de Pfaff sur , à un difféomorphisme près, si on exclut certaines dégénérescences de codimension infinie de . De plus, si est lisse, l’équation locale de Pfaff sur est déterminée, à un difféomorphisme près, par sa restriction sur 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.
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
Jakubczyk, Bronislaw 1; Zhitomirskii, Michail 2
@article{AIF_2001__51_1_237_0, author = {Jakubczyk, Bronislaw and Zhitomirskii, Michail}, title = {Local reduction theorems and invariants for singular contact structures}, journal = {Annales de l'Institut Fourier}, pages = {237--295}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {51}, number = {1}, year = {2001}, doi = {10.5802/aif.1823}, zbl = {1047.53051}, mrnumber = {1821076}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1823/} }
TY - JOUR AU - Jakubczyk, Bronislaw AU - Zhitomirskii, Michail TI - Local reduction theorems and invariants for singular contact structures JO - Annales de l'Institut Fourier PY - 2001 SP - 237 EP - 295 VL - 51 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1823/ DO - 10.5802/aif.1823 LA - en ID - AIF_2001__51_1_237_0 ER -
%0 Journal Article %A Jakubczyk, Bronislaw %A Zhitomirskii, Michail %T Local reduction theorems and invariants for singular contact structures %J Annales de l'Institut Fourier %D 2001 %P 237-295 %V 51 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1823/ %R 10.5802/aif.1823 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1823/
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