Foliation

Summary

A foliation of a manifold is a decomposition of it into disjoint connected submanifolds of one fixed dimension, the leaves. Locally the leaves fit together like the parallel slices $\R^p \times \set{y}$ of $\R^p \times \R^q$. Foliations arise from integrable plane fields by the Frobenius theorem and from submersions. They also arise from locally free group actions and from the orbits of flows without fixed points. They are a basic object of differential topology and dynamics, because individual leaves can be dense or wind around in complicated ways.