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 $F$ be a $C^{1}$, codimension $k$ foliation of a manifold $M$ and $L$ a compact leaf with finite holonomy group. There exists a neighborhood $U$ of $L$, saturated in $F$ (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a retraction $\pi :U\to L$ such that, for every leaf $L'\subset U$, $\pi |_{{L'}}:L'\to L$ is a covering map with a finite number of sheets and, for each $y\in L$, $\pi ^{{-1}}(y)$ is homeomorphic to a disk of dimension k and is transverse to $F$. The neighborhood $U$ 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 $(M^{n},F)$, with $n\geq 3$, and some center-type singularity in $Sing(F)$.

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

This theorem holds true even when $F$ 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 $F$ be a complete conformal foliation of codimension $k\ge 3$ of a connected manifold $M$. If $F$ has a compact leaf with finite holonomy group, then all the leaves of $F$ are compact with finite holonomy group.

• For holomorphic foliations in complex Kähler manifold:

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

1. G. Reeb (1952). Sur certaines propriétés toplogiques des variétés feuillétées. Actualités Sci. Indust. 1183. Paris: Hermann.
2. J. Palis, jr., W. de Melo, Geometric theory of dinamical systems: an introduction, — New-York, Springer,1982.
3. T.Inaba, $C^{2}$ Reeb stability of noncompact leaves of foliations,— Proc. Japan Acad. Ser. A Math. Sci., 59:158{160, 1983
4. J. Cantwell and L. Conlon, Reeb stability for noncompact leaves in foliated 3-manifolds, — Proc. Amer.Math.Soc. 33 (1981), no. 2, 408–410.
5. C. Godbillon, Feuilletages, etudies geometriques, — Basel, Birkhauser, 1991
6. W.T.Wu and G.Reeb, Sur les éspaces fibres et les variétés feuillitées, — Hermann, 1952.
7. R.A. Blumenthal, Stability theorems for conformal foliations, — Proc. AMS. 91, 1984, p. 55–63.
8. J.V. Pereira, Global stability for holomorphic foliations on Kaehler manifolds, — Qual. Theory Dyn. Syst. 2 (2001), 381–384. arXiv:math/0002086v2