# Foliation

From Diffgeom

## Definition

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

- The collection of tangent spaces to the leaves (at each point) forms a distribution over
- 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.