Distributions of corank 1 and their characteristic vector fields

Authors:
B. Jakubczyk and M. Zhitomirskii

Journal:
Trans. Amer. Math. Soc. **355** (2003), 2857-2883

MSC (2000):
Primary 58A17; Secondary 53B99

Published electronically:
March 19, 2003

MathSciNet review:
1975403

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold is trivializable, i.e., transformable to a constant family by a family of diffeomorphisms, if all distributions of the family have the same characteristic line field. The characteristic line field is a field of tangent lines which is invariantly assigned to a corank one distribution. It is defined on , if , or on a subset of called the Martinet hypersurface, if . Our second main result states that if two corank one distributions have the same characteristic line field and are close to each other, then they are equivalent via a diffeomorphism. This holds under a weak assumption on the singularities of the distributions. The second result implies that the abnormal curves of a distribution determine the equivalence class of the distribution, among distributions close to a given one.

**[A]**Andrei A. Agrachev,*Methods of control theory in nonholonomic geometry*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1473–1483. MR**1404051****[ArIl]**V. I. Arnol′d and Yu. S. Il′yashenko,*Ordinary differential equations*, Current problems in mathematics. Fundamental directions, Vol. 1, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 7–149, 244 (Russian). MR**823489**

D. V. Anosov and V. I. Arnol′d (eds.),*Dynamical systems. I*, Encyclopaedia of Mathematical Sciences, vol. 1, Springer-Verlag, Berlin, 1988. Ordinary differential equations and smooth dynamical systems; Translated from the Russian [ MR0823488 (86i:58037)]. MR**970793****[BS]**Edward Bierstone and Gerald W. Schwarz,*Continuous linear division and extension of \cal𝐶^{∞} functions*, Duke Math. J.**50**(1983), no. 1, 233–271. MR**700140**, 10.1215/S0012-7094-83-05011-1**[BH]**Robert L. Bryant and Lucas Hsu,*Rigidity of integral curves of rank 2 distributions*, Invent. Math.**114**(1993), no. 2, 435–461. MR**1240644**, 10.1007/BF01232676**[C]**Henri Cartan,*Variétés analytiques réelles et variétés analytiques complexes*, Bull. Soc. Math. France**85**(1957), 77–99 (French). MR**0094830****[DJ]**P. Domanski and B. Jakubczyk, Linear continuous division for exterior and interior products (accepted to Proc. Amer. Math. Soc.).**[E]**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****[Gol]**Alexander Golubev,*On the global stability of maximally nonholonomic two-plane fields in four dimensions*, Internat. Math. Res. Notices**11**(1997), 523–529. MR**1448335**, 10.1155/S1073792897000342**[G]**John W. Gray,*Some global properties of contact structures*, Ann. of Math. (2)**69**(1959), 421–450. MR**0112161****[JP]**Bronisław Jakubczyk and Feliks Przytycki,*On J. Martinet’s conjecture*, Bull. Acad. Polon. Sci. Sér. Sci. Math.**27**(1979), no. 9, 731–735 (English, with Russian summary). MR**600727****[JZh1]**Bronisław Jakubczyk and Michail Zhitomirskii,*Odd-dimensional Pfaffian equations: reduction to the hypersurface of singular points*, C. R. Acad. Sci. Paris Sér. I Math.**325**(1997), no. 4, 423–428 (English, with English and French summaries). MR**1467099**, 10.1016/S0764-4442(97)85629-7**[JZh2]**B. Jakubczyk and M. Zhitomirskii,*Local reduction theorems and invariants for singular contact structures*, Ann. Inst. Fourier (Grenoble)**51**(2001), no. 1, 237–295 (English, with English and French summaries). MR**1821076****[LS]**Wensheng Liu and Héctor J. Sussman,*Shortest paths for sub-Riemannian metrics on rank-two distributions*, Mem. Amer. Math. Soc.**118**(1995), no. 564, x+104. MR**1303093****[Mlg]**B. Malgrange,*Ideals of differentiable functions*, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR**0212575****[Mar]**Jean Martinet,*Sur les singularités des formes différentielles*, Ann. Inst. Fourier (Grenoble)**20**(1970), no. fasc. 1, 95–178 (French, with English summary). MR**0286119****[Mon]**R. Montgomery,*A survey of singular curves in sub-Riemannian geometry*, J. Dynam. Control Systems**1**(1995), no. 1, 49–90. MR**1319057**, 10.1007/BF02254656**[MZh]**Richard Montgomery and Michail Zhitomirskii,*Geometric approach to Goursat flags*, Ann. Inst. H. Poincaré Anal. Non Linéaire**18**(2001), no. 4, 459–493 (English, with English and French summaries). MR**1841129**, 10.1016/S0294-1449(01)00076-2**[Ru]**Jesús M. Ruiz,*The basic theory of power series*, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1993. MR**1234937****[TEG]**C. B. Thomas, Y. Eliashberg, and E. Giroux,*3-dimensional contact geometry*, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 48–65. MR**1432458****[Zh1]**M. Ya. Zhitomirskiĭ,*Singularities and normal forms of odd-dimensional Pfaffian equations*, Funktsional. Anal. i Prilozhen.**23**(1989), no. 1, 70–71 (Russian); English transl., Funct. Anal. Appl.**23**(1989), no. 1, 59–61. MR**998435**, 10.1007/BF01078579**[Zh2]**Michail Zhitomirskiĭ,*Typical singularities of differential 1-forms and Pfaffian equations*, Translations of Mathematical Monographs, vol. 113, American Mathematical Society, Providence, RI; in cooperation with Mir Publishers, Moscow, 1992. Translated from the Russian. MR**1195792**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
58A17,
53B99

Retrieve articles in all journals with MSC (2000): 58A17, 53B99

Additional Information

**B. Jakubczyk**

Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw, Poland and Institute of Applied Mathematics, University of Warsaw, Poland

Email:
B.Jakubczyk@impan.gov.pl

**M. Zhitomirskii**

Affiliation:
Department of Mathematics, Technion, 32000 Haifa, Israel

Email:
mzhi@techunix.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-03-03248-3

Keywords:
Pfaff equation,
equivalence,
contact structure,
quasi-contact structure,
singularity,
invariants,
line field,
homotopy method

Received by editor(s):
January 9, 2002

Received by editor(s) in revised form:
September 4, 2002

Published electronically:
March 19, 2003

Additional Notes:
The first author was supported by the Committee for Scientific Research (KBN), Poland, grant 2P03A 03516

The second author was supported by the Fund for the Promotion of Research at the Technion

Article copyright:
© Copyright 2003
American Mathematical Society