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
     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/

[A] A. Agrachev Methods of Control Theory in Nonholonomic Geometry, Proc. Int. Congress of Math. Zurich 1994, Birkhäuser, Basel, Tome Vol. 2 (1995), pp. 1473-1483 | Zbl 0848.93012

[AG] V.I. Arnold; A.B. Givental Symplectic geometry, Springer, Berlin, Encyclopaedia of Mathematical Sciences, Tome Vol. 4 (1990) | MR 1042758 | Zbl 0780.58016

[AI] V.I. Arnold; Yu. S. Ilyashenko Ordinary differential equations, Springer, Berlin, Encyclopaedia of Mathematical Sciences, Tome Vol. 1 (1988) | MR 970794 | Zbl 0659.58012

[BC3G] R.L. Bryant; S.S. Chern; R.B. Gardner; H.L. Goldschmidt; P.A. Griffiths Exterior Differential Systems, Springer-Verlag, Mathematical Sciences Research Institute Publications, Tome Vol. 18 (1991) | MR 1083148 | Zbl 0726.58002

[BH] R.L. Bryant; L. Hsu Rigidity of integral curves of rank 2 distributions, Inventiones Math., Tome 114 (1993), pp. 435-461 | Article | MR 1240644 | Zbl 0807.58007

[BJ] S. Balcerzyk; T. Józefiak Commutative Rings; Dimension, Multiplicity and Homological Methods, Polish Scientific Publishers, Warsaw (1989) | MR 1084368 | Zbl 0685.13002

[Bo] R.I. Bogdanov Moduli of C normal forms of singular points of vector fields on a plane, Functional Anal. Appl., Tome 11 (1977) no. 1, p. 57-58 | MR 482804 | Zbl 0384.57015

[E] D. Eisenbud Commutative Algebra, Springer-Verlag (1994) | MR 1322960 | Zbl 0819.13001

[JP] B. Jakubczyk; F. Przytycki Singularities of k-tuples of vector fields, Dissertationes Mathematicae, Warsaw, Tome 213 (1984), pp. 1-64 | MR 744876 | Zbl 0565.58007

[JZh1] B. Jakubczyk; M. Zhitomirskii Singularities and normal forms of generic 2-distributions on 3-manifolds, Studia Math., Tome 113 (1995), pp. 223-248 | MR 1330209 | Zbl 0829.58007

[JZh2] B. Jakubczyk; M. Zhitomirskii Odd-dimensional Pfaffian equations; reduction to the hypersurface of singular points, Comptes Rendus Acad. Sci. Paris, Série I, Tome t. 325 (1997), pp. 423-428 | MR 1467099 | Zbl 0889.58006

[Lo] S. Łojasiewicz Introduction to Complex Analytic Geometry, Birkhäuser, Basel (1991) | Zbl 0747.32001

[LS] W. Liu; H. Sussmann Shortest paths for sub-Riemannian metrics on rank 2 distributions, Mem. Amer. Math. Soc., Tome 118 (1995) no. 564 | Zbl 0843.53038

[Ma1] J. Martinet Sur les singularites des formes differentielles, Ann. Inst. Fourier, Tome 20 (1970) no. 1, pp. 95-178 | Article | Numdam | MR 286119 | Zbl 0189.10001

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

[Mlg] B. Malgrange Ideals of differentiable functions, Oxford University Press (1966) | MR 212575 | Zbl 0177.17902

[Mon] R. Montgomery A Survey on Singular Curves in Sub-Riemannian Geometry, J. Dynamical and Control Systems, Tome 1 (1995) no. 1, pp. 49-90 | Article | MR 1319057 | Zbl 0941.53021

[Mou] R. Moussu Sur l'existence d'intégrales premières pour un germe de forme de Pfaff, Ann. Inst. Fourier, Tome 26 (1976) no. 2, pp. 171-220 | Article | Numdam | MR 415657 | Zbl 0328.58002

[MR] J. Martinet; J.-P. Ramis Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. Ecole Norm. Sup., Tome 16 (1983), pp. 571-621 | Numdam | MR 740592 | Zbl 0534.34011

[MZh] P. Mormul; M. Zhitomirskii Modules of vector fields, differential forms and degenerations of differential systems, Israel J. of Mathematics, Tome 95 (1996), pp. 411-428 | Article | MR 1418303 | Zbl 0866.58003

[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 783350 | Zbl 0568.58001

[Ro] R. Roussarie Modèles locaux de champs et de formes, Astérisque, Tome 30 (1975), pp. 1-181 | MR 440570 | Zbl 0327.57017

[Ru] J.M. Ruiz The Basic Theory of Power Series, Vieveg, Wiesbaden, Advanced Lectures in Mathematics (1993) | MR 1234937

[T] J.-C. Tougeron Idéaux des fonctions différentiables, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, Tome 71 (1972) | MR 440598 | Zbl 0251.58001

[VKL] A.M. Vinogradov; I.C. Krasilshchik; V.V. Lychagin Introduction to Geometry of Nonlinear Differential Equations (in Russian), Nauka, Moscow (1986) | MR 855844 | Zbl 0592.35002

[Zh1] M. Zhitomirskii Typical singularities of differential 1-forms and Pfaffian equations, AMS, Providence, Translations of Math. Monographs, Tome Vol. 113 (1992) | MR 1195792 | Zbl 0771.58001

[Zh2] M. Zhitomirskii Singularities and normal forms of odd-dimensional Pfaff equations, Functional Anal. Applic., Tome 23 (1989), pp. 59-61 | Article | MR 998435 | Zbl 0687.58001