Foliation

Definition

A $k$-dimensional foliation of a connected differential manifold $M$ of dimension $n$ is a partition of $M$ into connected immersed submanifolds, called leaves such that:

• The collection of tangent spaces to the leaves (at each point) forms a distribution over $M$
• Any connected integral manifold for the distribution is contained in some leaf

We say that the distribution is induced by the foliation, and we also call the leaf a maximal integral manifold of the foliation.

Facts

Frobenius theorem

Further information: Frobenius theorem

This states that every integrable manifold is induced by a foliation. In other words, if for every point we can get an integral manifold of the distribution containing that point, we can partition the whole manifold into maximal integral manifolds.