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.

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 bookConfoliationsby 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