Frobenius theorem

Local version

For a smooth manifold , let be an involutive distribution of rank . Then is integrable.

Furthermore, for each there exist a chart centered at with

such that

Thus, for each submanifold ,

and is an integral manifold of .

Notation

The charts from the theorem are called distinguished charts or foliated charts.

The submanifolds are called plaques.

Proof

See the following texts:

@lee2013 - Page 497

@michor2008 - Page 32

Global version

Let be an involutive regular distribution on a smooth manifold . The collection of all maximal connected integral manifolds of form a foliation of .