Foliation

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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.