---
tag: 98HTB7MSPX
kind: concept
name: Foliation
msc: [57R30, 53C12]
sources:
  - url: https://en.wikipedia.org/wiki/Foliation
same_as:
  wikidata: Q684220
  wikipedia: 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|5Q4BHBFXJY]] 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.

## Explanations

### L1 human

Think of the pages of a book, or the layers of an onion without its centre.
Near any point the space looks like a stack of parallel sheets, but globally a single sheet can wander through the whole space.
On a torus, take all lines of a fixed slope.
If the slope is rational, each line closes into a circle; if it is irrational, each line wraps around forever and comes close to every point.
Both families are foliations of the torus by curves.