Distributions of corank 1 and their characteristic vector fields

Jakubczyk, B.; Zhitomirskii, M.

doi:10.1090/S0002-9947-03-03248-3

Distributions of corank 1 and their characteristic vector fields
HTML articles powered by AMS MathViewer

by B. Jakubczyk and M. Zhitomirskii PDF

Trans. Amer. Math. Soc. 355 (2003), 2857-2883 Request permission

Abstract:

We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold $M^{n}$ 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 $M^{n}$, if $n=2k$, or on a subset of $M^{n}$ called the Martinet hypersurface, if $n=2k+1$. 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.

References

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
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Edward Bierstone and Gerald W. Schwarz, Continuous linear division and extension of ${\cal C}^{\infty }$ functions, Duke Math. J. 50 (1983), no. 1, 233–271. MR 700140, DOI 10.1215/S0012-7094-83-05011-1
Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank $2$ distributions, Invent. Math. 114 (1993), no. 2, 435–461. MR 1240644, DOI 10.1007/BF01232676
Henri Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77–99 (French). MR 94830, DOI 10.24033/bsmf.1481
P. Domański and B. Jakubczyk, Linear continuous division for exterior and interior products (accepted to Proc. Amer. Math. Soc.).
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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, DOI 10.1155/S1073792897000342
John W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR 112161, DOI 10.2307/1970192
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
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, DOI 10.1016/S0764-4442(97)85629-7
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, DOI 10.5802/aif.1823
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, DOI 10.1090/memo/0564
B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
Jean Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 95–178 (French, with English summary). MR 286119, DOI 10.5802/aif.340
R. Montgomery, A survey of singular curves in sub-Riemannian geometry, J. Dynam. Control Systems 1 (1995), no. 1, 49–90. MR 1319057, DOI 10.1007/BF02254656
Richard Montgomery and Michail Zhitomirskii, Geometric approach to Goursat flags, Ann. Inst. H. Poincaré C Anal. Non Linéaire 18 (2001), no. 4, 459–493 (English, with English and French summaries). MR 1841129, DOI 10.1016/S0294-1449(01)00076-2
Jesús M. Ruiz, The basic theory of power series, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1993. MR 1234937, DOI 10.1007/978-3-322-84994-6
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
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, DOI 10.1007/BF01078579
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, DOI 10.1090/mmono/113