Completely integrable distribution

Given a $k$ -dimensional distribution $D \subset T M$ , let us say that a coordinate chart $(U, φ)$ on $M$ is flat for D if at points of $U$ , $D$ is spanned by the first $k$ coordinate vector fields $\partial / \partial x^{1}, \dots, \partial / \partial x^{k}$ . It is obvious that each slice of the form $x^{k + 1} = c^{k + 1}, \dots, x^{n} = c^{n}$ for constants $c^{k + 1}, \dots, c^{n}$ is an integral manifold of $D$ . This is the nicest possible local situation for integral manifolds.

We say that a distribution $D \subset T M$ is completely integrable if there exists a flat chart for $D$ in a neighborhood of every point of $M$ . Obviously every completely integrable distribution is integrable and therefore involutive.

Completely integrable imply, obviously, having integral submanifolds. Therefore, it is an involutive distribution.

Moreover, completely integrable distributions are also characterized by the fact that every maximal integral manifold has just dimension $\dim (D)$ ([Lychagin 1991] page 4). The manifold is then decomposed in leaves of a foliation.