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Transactions of the American Mathematical Society

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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
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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.

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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

M. Zhitomirskii
Affiliation: Department of Mathematics, Technion, 32000 Haifa, Israel

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

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