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

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

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 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.

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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

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