Reeb stability theorem explained

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 leafwith 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\ge 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]

Notes and References

1. Book: G. Reeb . Sur certaines propriétés toplogiques des variétés feuillétées . Actualités Sci. Indust. . 1183 . Hermann . Paris . 1952 .
2. J. Palis, jr., W. de Melo, Geometric theory of dynamical 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