Local reduction theorems and invariants for singular contact structures

Jakubczyk, Bronislaw; Zhitomirskii, Michail

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

Soit $\omega$ 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 $S=\{p\in M:(\omega \wedge {\left(d\omega \right)}^k)\left(p\right)=0\}$ est nulle part dense. Dans un tel cas on peut définir la restriction (pullback) ${\omega |}_S$ de $\omega$ sur l’hypersurface singulière $S$. Nos théorèmes disent que, dans les catégories holomorphe, analytique réelle et $C^{\infty }$, l’équation locale de Pfaff ${\omega |}_S=0$ sur $S$ détermine l’équation locale de Pfaff $\omega =0$ sur $M$, à un difféomorphisme près, si on exclut certaines dégénérescences de codimension infinie de $\omega$. De plus, si $S$ est lisse, l’équation locale de Pfaff $\omega =0$ 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 $(2k+1)$-dimensional manifolds $M$ defines a singular contact structure if the set $S$ of points where the contact condition is not satisfied, $S=\{p\in M:(\omega \wedge {\left(d\omega \right)}^k\left(p\right)=0\}$, is nowhere dense in $M$. Then $S$ is a hypersurface with singularities and the restriction of $\omega$ to $S$ can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation $\left(\omega \right)$ generated by $\omega$ is determined, up to a diffeomorphism, by its restriction to $S$, if we eliminate certain degenerated singularities of $\omega$ (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 $\left(\omega \right)$ 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.

[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 $C^{\infty }$ 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 | Numdam | 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