“Reeb stability theorem”的意思、由来-开放百科全书

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 be a , codimension foliation of a manifold and a compact leaf with finite holonomy group. There exists a neighborhood of , saturated in (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a retraction such that, for every leaf , is a covering map with a finite number of sheets and, for each , is homeomorphic to a disk of dimension k and is transverse to . The neighborhood 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

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

, with

, and some center-type singularity in

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 be a , codimension one foliation of a closed manifold . If contains a compact leaf with finite fundamental group, then all the leaves of are compact, with finite fundamental group. If is transversely orientable, then every leaf of is diffeomorphic to ; is the total space of a fibration over , with fibre , and is the fibre foliation, .

This theorem holds true even when

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 be a complete conformal foliation of codimension of a connected manifold . If has a compact leaf with finite holonomy group, then all the leaves of are compact with finite holonomy group.

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

References

Notes

1. ^¹ {{cite book | author=G. Reeb | title=Sur certaines propriétés toplogiques des variétés feuillétées | series=Actualités Sci. Indust. | volume=1183 | publisher=Hermann | location=Paris | year=1952 }}
2. ^J. Palis, jr., W. de Melo, Geometric theory of dynamical systems: an introduction, — New-York,Springer,1982.
3. ^T.Inaba, 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}}