Distributions of corank 1 and their characteristic vector fields

Abstract

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