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.