Reeb stability theorem

In mathematics, Reeb stability theorem, named after Georges Reeb, asserts that if one leaf of a codimension-one foliation is closed and has finite fundamental group, then all the leaves are closed and have finite fundamental group.

Reeb local stability theorem

Theorem:^[1] Let ${\displaystyle F}$ be a ${\displaystyle C^{1}}$, codimension ${\displaystyle k}$ foliation of a manifold ${\displaystyle M}$ and ${\displaystyle L}$ a compact leaf with finite holonomy group. There exists a neighborhood ${\displaystyle U}$ of ${\displaystyle L}$, saturated in ${\displaystyle F}$ (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a retraction ${\displaystyle \pi :U\to L}$ such that, for every leaf ${\displaystyle L'\subset U}$, ${\displaystyle \pi |_{L'}:L'\to L}$ is a covering map with a finite number of sheets and, for each ${\displaystyle y\in L}$, ${\displaystyle \pi ^{-1}(y)}$ is homeomorphic to a disk of dimension k and is transverse to ${\displaystyle F}$. The neighborhood ${\displaystyle U}$ can be taken to be arbitrarily small.

The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf with finite holonomy, the space of leaves is Hausdorff. Under certain conditions the Reeb local stability theorem may replace the Poincaré–Bendixson theorem in higher dimensions.^[2] This is the case of codimension one, singular foliations ${\displaystyle (M^{n},F)}$, with ${\displaystyle n\geq 3}$, and some center-type singularity in ${\displaystyle Sing(F)}$.

The Reeb local stability theorem also has a version for a noncompact codimension-1 leaf.^[3][4]

Reeb global stability theorem

An important problem in foliation theory is the study of the influence exerted by a compact leaf upon the global structure of a foliation. For certain classes of foliations, this influence is considerable.

Theorem:^[1] Let ${\displaystyle F}$ be a ${\displaystyle C^{1}}$, codimension one foliation of a closed manifold ${\displaystyle M}$. If ${\displaystyle F}$ contains a compact leaf ${\displaystyle L}$ with finite fundamental group, then all the leaves of ${\displaystyle F}$ are compact, with finite fundamental group. If ${\displaystyle F}$ is transversely orientable, then every leaf of ${\displaystyle F}$ is diffeomorphic to ${\displaystyle L}$; ${\displaystyle M}$ is the total space of a fibration ${\displaystyle f:M\to S^{1}}$ over ${\displaystyle S^{1}}$, with fibre ${\displaystyle L}$, and ${\displaystyle F}$ is the fibre foliation, ${\displaystyle \{f^{-1}(\theta )|\theta \in S^{1}\}}$.

This theorem holds true even when ${\displaystyle F}$ is a foliation of a manifold with boundary, which is, a priori, tangent on certain components of the boundary and transverse on other components.^[5] In this case it implies Reeb sphere theorem.

Reeb Global Stability Theorem is false for foliations of codimension greater than one.^[6] However, for some special kinds of foliations one has the following global stability results:

• In the presence of a certain transverse geometric structure:

Theorem:^[7] Let ${\displaystyle F}$ be a complete conformal foliation of codimension ${\displaystyle k\geq 3}$ of a connected manifold ${\displaystyle M}$. If ${\displaystyle F}$ has a compact leaf with finite holonomy group, then all the leaves of ${\displaystyle F}$ are compact with finite holonomy group.

• For holomorphic foliations in complex Kähler manifold:

Theorem:^[8] Let ${\displaystyle F}$ be a holomorphic foliation of codimension ${\displaystyle k}$ in a compact complex Kähler manifold. If ${\displaystyle F}$ has a compact leaf with finite holonomy group then every leaf of ${\displaystyle F}$ is compact with finite holonomy group.

References

• C. Camacho, A. Lins Neto: Geometric theory of foliations, Boston, Birkhauser, 1985
• I. Tamura, Topology of foliations: an introduction, Transl. of Math. Monographs, AMS, v.97, 2006, 193 p.

Notes