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.

Contents

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 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\ge 3, and some center-type singularity in Sing(F).

The Reeb Local Stability Theorem also has a version for a noncompact 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:

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.

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

Notes

  1. ^ a b G. Reeb, 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 [1]
  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.[2]
  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. [3]
  8. ^ J.V. Pereira, Global stability for holomorphic foliations on Kaehler manifolds, — Qual. Theory Dyn. Syst. 2 (2001), 381–384. arXiv:math/0002086v2