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In a expository text on differential geometry I am reading about the geometry of distributions of a corank one.
Here the first properties are reported without proof, and no reference is given.

I would know some reference for the study of distributions $D$ of corank 1 on a smooth manifold $M$.
To give an idea of what I need it should start from:

  • the definition of the class of $D$,
  • the relation of the class of $D$ in a point $m$ with the maximal dimension of submanifolds tangent to $D$ through $m$,
  • the Darboux theorem for the $1$-form locally defining $D$,
  • the definition of the associated characteristic distribution.

I would prefer if it was a textbook, but any suggestion is welcome.

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I do not have a reference in mind, but I can suggest keywords: contact topology (which is the least integrable case), sub-Riemannian geometry (which is the case when the distribution is endowed with a metric). Books carrying these words in their title can be a starting place. You can also try to look at the introduction and references of the short book Confoliations by Thurston and Eliashberg math.stanford.edu/~eliash/Public/conf-m.pdf (which considers a case including both contact structures and foliations). –  Benoît Kloeckner Aug 7 '11 at 12:34
    
Dear Benoît Kloeckner, thanks for the reference to the book of Thurston and Eliashberg. Following your suggestion to refine my serch, I found a paper of R.Montgomery and Zhitomirskij ``Geometric approach to Goursat flags'', in an appendix there is the reference to Cartan and Frobenius and their work on Pfaffian systems. I''l give a look. –  Giuseppe Tortorella Aug 10 '11 at 12:53
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