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.
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.