Local reduction theorems and invariants for singular contact structures
Annales de l'Institut Fourier, Volume 51 (2001) no. 1, pp. 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: 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
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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     pages = {237--295},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
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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/

[A] A. Agrachev Methods of Control Theory in Nonholonomic Geometry, Proc. Int. Congress of Math. Zurich 1994, Volume Vol. 2 (1995), pp. 1473-1483 | Zbl

[AG] V.I. Arnold; A.B. Givental Symplectic geometry, Encyclopaedia of Mathematical Sciences, Vol. 4, Springer, Berlin, 1990 | MR | Zbl

[AI] V.I. Arnold; Yu. S. Ilyashenko Ordinary differential equations, Encyclopaedia of Mathematical Sciences, Vol. 1, Springer, Berlin, 1988 | MR | Zbl

[BC3G] R.L. Bryant; S.S. Chern; R.B. Gardner; H.L. Goldschmidt; P.A. Griffiths Exterior Differential Systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, 1991 | MR | Zbl

[BH] R.L. Bryant; L. Hsu Rigidity of integral curves of rank 2 distributions, Inventiones Math., Volume 114 (1993), pp. 435-461 | DOI | MR | Zbl

[BJ] S. Balcerzyk; T. Józefiak Commutative Rings; Dimension, Multiplicity and Homological Methods, Polish Scientific Publishers, Warsaw, 1989 | MR | Zbl

[Bo] R.I. Bogdanov Moduli of C normal forms of singular points of vector fields on a plane, Functional Anal. Appl., Volume 11 (1977) no. 1, pp. 57-58 | MR | Zbl

[E] D. Eisenbud Commutative Algebra, Springer-Verlag, 1994 | MR | Zbl

[JP] B. Jakubczyk; F. Przytycki Singularities of k-tuples of vector fields, Dissertationes Mathematicae, Warsaw, Volume 213 (1984), pp. 1-64 | MR | Zbl

[JZh1] B. Jakubczyk; M. Zhitomirskii Singularities and normal forms of generic 2-distributions on 3-manifolds, Studia Math., Volume 113 (1995), pp. 223-248 | MR | Zbl

[JZh2] B. Jakubczyk; M. Zhitomirskii Odd-dimensional Pfaffian equations; reduction to the hypersurface of singular points, Comptes Rendus Acad. Sci. Paris, Série I, Volume t. 325 (1997), pp. 423-428 | MR | Zbl

[Lo] S. Łojasiewicz Introduction to Complex Analytic Geometry, Birkhäuser, Basel, 1991 | Zbl

[LS] W. Liu; H. Sussmann Shortest paths for sub-Riemannian metrics on rank 2 distributions, Mem. Amer. Math. Soc., Volume 118 (1995) no. 564 | Zbl

[Ma1] J. Martinet Sur les singularites des formes differentielles, Ann. Inst. Fourier, Volume 20 (1970) no. 1, pp. 95-178 | DOI | Numdam | MR | Zbl

[Ma2] J. Martinet A letter to M. Zhitomirskii (1989)

[Mlg] B. Malgrange Ideals of differentiable functions, Oxford University Press, 1966 | MR | Zbl

[Mon] R. Montgomery A Survey on Singular Curves in Sub-Riemannian Geometry, J. Dynamical and Control Systems, Volume 1 (1995) no. 1, pp. 49-90 | DOI | MR | Zbl

[Mou] R. Moussu Sur l'existence d'intégrales premières pour un germe de forme de Pfaff, Ann. Inst. Fourier, Volume 26 (1976) no. 2, pp. 171-220 | DOI | Numdam | MR | Zbl

[MR] J. Martinet; J.-P. Ramis Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. Ecole Norm. Sup., Volume 16 (1983), pp. 571-621 | Numdam | MR | Zbl

[MZh] P. Mormul; M. Zhitomirskii Modules of vector fields, differential forms and degenerations of differential systems, Israel J. of Mathematics, Volume 95 (1996), pp. 411-428 | DOI | MR | Zbl

[P] F. Pelletier Singularités d'ordre supérieur de 1-formes, 2-formes et équations de Pfaff, Publications Mathématiques IHES, Bures-sur-Yvette (1985) no. 61, pp. 129-169 | Numdam | MR | Zbl

[Ro] R. Roussarie Modèles locaux de champs et de formes, Astérisque, Volume 30 (1975), pp. 1-181 | MR | Zbl

[Ru] J.M. Ruiz The Basic Theory of Power Series, Advanced Lectures in Mathematics, Vieveg, Wiesbaden, 1993 | MR

[T] J.-C. Tougeron Idéaux des fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 71, Springer, 1972 | MR | Zbl

[VKL] A.M. Vinogradov; I.C. Krasilshchik; V.V. Lychagin Introduction to Geometry of Nonlinear Differential Equations (in Russian), Nauka, Moscow, 1986 | MR | Zbl

[Zh1] M. Zhitomirskii Typical singularities of differential 1-forms and Pfaffian equations, Translations of Math. Monographs, Vol. 113, AMS, Providence, 1992 | MR | Zbl

[Zh2] M. Zhitomirskii Singularities and normal forms of odd-dimensional Pfaff equations, Functional Anal. Applic., Volume 23 (1989), pp. 59-61 | DOI | MR | Zbl

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