---
tag: Z93KMZ9R37
kind: statement
role: theorem
name: Reeb stability theorem
msc: [57R30]
uses: [98HTB7MSPX, S1XK3HXHFX, 0TVJZYB6SH]
about: [98HTB7MSPX]
sources:
  - url: https://en.wikipedia.org/wiki/Reeb_stability_theorem
same_as:
  wikidata: Q4455018
  wikipedia: Reeb stability theorem
---

## Statement

Let $M$ be a closed connected manifold and let $\mathcal{F}$ be a transversely orientable [[foliation|98HTB7MSPX]] of $M$ of class $C^1$ and codimension $1$.
Assume that $\mathcal{F}$ has a [[compact|0TVJZYB6SH]] leaf $L$ whose [[fundamental group|S1XK3HXHFX]] is finite.
Then every leaf of $\mathcal{F}$ is compact and diffeomorphic to $L$, and there is a fibre bundle $M \to S^1$ whose fibres are the leaves of $\mathcal{F}$.

## Remarks

The theorem is a global consequence of the local Reeb stability theorem.
The local theorem says that a compact leaf with finite holonomy group has a neighbourhood made of leaves that are finite covers of it.
Idea of the proof: since $\pi_1(L)$ is finite and its holonomy acts on a transverse interval preserving orientation, the holonomy is trivial, so nearby leaves are copies of $L$.
The set of points on compact leaves diffeomorphic to $L$ is then open, and a limit argument shows it is closed, so it is all of $M$.

Example: if a closed $3$-manifold $M$ has a transversely orientable codimension one foliation with a leaf homeomorphic to $S^2$, the theorem shows that $M$ fibres over $S^1$ with fibre $S^2$.
The hypothesis on the fundamental group is needed: the [[Reeb foliation|W58NYR1DGR]] of $S^3$ has a compact torus leaf, with infinite fundamental group, while all its other leaves are planes.

## Explanations

### L1 human

A foliation slices a manifold into sheets of one lower dimension.
The theorem says that one closed sheet with almost no loops forces all sheets to be copies of it.
Then the manifold is a stack of these copies arranged around a circle.

### L2 human

The theorem shows that spheres and other leaves with finite fundamental group are rigid inside codimension one foliations.
It is a first step in the study of compact leaves, which leads to Novikov's compact leaf theorem.
The key notion is holonomy, the return map of a transverse interval along loops in a leaf, see the holonomy groupoid of a foliation.
A corollary is that $S^3$ carries no codimension one foliation with a leaf diffeomorphic to $S^2$, since $S^3$ does not fibre over the circle.