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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 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 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 $k\geq 3$ 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

^ ^a ^b G. Reeb, G. Reeb (1952). Sur certaines propriétés toplogiques des variétés feuillétées. Actualités Sci. Indust. Vol. 1183. Paris: Hermann.
^ J. Palis, jr., W. de Melo, Geometric theory of dinamical systems: an introduction, — New-York, Springer,1982.
^ T.Inaba, $C^{2}$ Reeb stability of noncompact leaves of foliations,— Proc. Japan Acad. Ser. A Math. Sci., 59:158{160, 1983[1]
^ 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]
^ C. Godbillon, Feuilletages, etudies geometriques, — Basel, Birkhauser, 1991
^ W.T.Wu and G.Reeb, Sur les éspaces fibres et les variétés feuillitées, — Hermann, 1952.
^ R.A. Blumenthal, Stability theorems for conformal foliations, — Proc. AMS. 91, 1984, p. 55–63. [3]
^ J.V. Pereira, Global stability for holomorphic foliations on Kaehler manifolds, — Qual. Theory Dyn. Syst. 2 (2001), 381–384. [4]

[Reeb-1] G. Reeb, G. Reeb (1952). Sur certaines propriétés toplogiques des variétés feuillétées. Actualités Sci. Indust. Vol. 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. [4]

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@@ Line 3: / Line 3: @@
 == Reeb local stability theorem ==
-Theorem<ref name="Reeb">G. Reeb, {{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 }}</ref>: ''Let <math>F</math> be a <math>C^1</math>, codimension <math>k</math> [[foliation]] of a [[manifold]] <math>M</math> and <math>L</math> a [[compact]] leaf with finite [[holonomy group]]. There exists a [[neighborhood]] <math>U</math> of <math>L</math>, saturated in <math>F</math> (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a [[retraction]] <math>\pi: U\to L</math> such that, for every leaf <math>L'\subset U</math>, <math>\pi|_{L'}:L'\to L</math> is a [[covering]] with a finite number of sheets and, for each <math>y\in L</math>, <math>\pi^{-1}(y)</math> is [[homeomorphic]] to a [[disk]] of [[dimension]] k and is [[transverse]] to <math>F</math>. The neighborhood <math>U</math> can be taken to be arbitrarily small.''
+Theorem<ref name="Reeb">G. Reeb, {{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 }}</ref>: ''Let <math>F</math> be a <math>C^1</math>, codimension <math>k</math> [[foliation]] of a [[manifold]] <math>M</math> and <math>L</math> a [[compact]] leaf with finite [[holonomy|holonomy group]]. There exists a [[neighbourhood|neighborhood]] <math>U</math> of <math>L</math>, saturated in <math>F</math> (also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a [[retraction]] <math>\pi: U\to L</math> such that, for every leaf <math>L'\subset U</math>, <math>\pi|_{L'}:L'\to L</math> is a [[covering]] with a finite number of sheets and, for each <math>y\in L</math>, <math>\pi^{-1}(y)</math> is [[homeomorphism|homeomorphic]] to a [[disk (mathematics)|disk]] of [[dimension]] k and is [[transverse]] to <math>F</math>. The neighborhood <math>U</math> 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
@@ Line 16: / Line 16: @@
 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<ref name="Reeb"/>: ''Let <math>F</math> be a <math>C^1</math>, codimension one foliation of a closed manifold <math>M</math>. If <math>F</math> contains a [[Compact space|compact]] leaf <math>L</math> with finite [[fundamental group]], then all the leaves of <math>F</math> are compact, with finite fundamental group. If <math>F</math> is transversely [[orientable]], then every leaf of <math>F</math> is [[diffeomorphic]] to <math>L</math>; <math>M</math> is the [[total space]] of a [[fibration]] <math>f:M\to S^1</math> over <math>S^1</math>, with [[Fiber (mathematics)|fibre]] <math>L</math>, and <math>F</math> is the fibre foliation, <math>\{f^{-1}(\theta)|\theta\in S^1\}</math>.''
+Theorem<ref name="Reeb"/>: ''Let <math>F</math> be a <math>C^1</math>, codimension one foliation of a closed manifold <math>M</math>. If <math>F</math> contains a [[Compact space|compact]] leaf <math>L</math> with finite [[fundamental group]], then all the leaves of <math>F</math> are compact, with finite fundamental group. If <math>F</math> is transversely [[orientability|orientable]], then every leaf of <math>F</math> is [[diffeomorphism|diffeomorphic]] to <math>L</math>; <math>M</math> is the [[fiber bundle|total space]] of a [[fibration]] <math>f:M\to S^1</math> over <math>S^1</math>, with [[Fiber (mathematics)|fibre]] <math>L</math>, and <math>F</math> is the fibre foliation, <math>\{f^{-1}(\theta)|\theta\in S^1\}</math>.''
 This theorem holds true even when <math>F</math> is a foliation of a [[Manifold#Manifold with boundary|manifold with boundary]], which is, a priori, [[Tangent space|tangent]]
 on certain components of the [[Boundary (topology)|boundary]] and [[Transversality|transverse]] on other components <ref>C. Godbillon, ''Feuilletages, etudies geometriques,'' — Basel, Birkhauser, 1991</ref>. In this case it implies [[Reeb sphere theorem]].
 Reeb Global Stability Theorem is false for foliations of codimension greater than one<ref>W.T.Wu and G.Reeb, ''Sur les éspaces fibres et les variétés feuillitées'', — Hermann, 1952.</ref>. However, for some special kinds of foliations  one has the following global stability results:
@@ Line 26: / Line 25: @@
 * In the presence of a certain transverse geometric structure:
-Theorem<ref>R.A. Blumenthal, ''Stability theorems for conformal foliations'', — Proc. AMS. 91, 1984, p. 55&ndash;63. [http://www.ams.org/journals/proc/1984-091-03/S0002-9939-1984-0744654-X/S0002-9939-1984-0744654-X.pdf]</ref>: ''Let <math>F</math> be a [[Complete metric space|complete]] [[Conformal geometry|conformal]] foliation of codimension <math>k\ge 3</math> of a [[connected]] manifold <math>M</math>. If <math>F</math> has a compact leaf with finite [[holonomy group]], then all the leaves of <math>F</math> are compact with finite holonomy group.''
+Theorem<ref>R.A. Blumenthal, ''Stability theorems for conformal foliations'', — Proc. AMS. 91, 1984, p. 55&ndash;63. [http://www.ams.org/journals/proc/1984-091-03/S0002-9939-1984-0744654-X/S0002-9939-1984-0744654-X.pdf]</ref>: ''Let <math>F</math> be a [[Complete metric space|complete]] [[Conformal geometry|conformal]] foliation of codimension <math>k\ge 3</math> of a [[connection|connected]] manifold <math>M</math>. If <math>F</math> has a compact leaf with finite [[holonomy|holonomy group]], then all the leaves of <math>F</math> are compact with finite holonomy group.''
 * For [[Complex manifold|holomorphic]] foliations in complex [[Kähler manifold]]:
-Theorem<ref>J.V. Pereira, ''Global stability for holomorphic foliations on Kaehler manifolds'', — Qual. Theory Dyn. Syst. 2 (2001), 381&ndash;384. [http://arxiv.org/PS_cache/math/pdf/0002/0002086v2.pdf]</ref>: ''Let <math>F</math> be a holomorphic foliation of codimension <math>k</math> in a compact complex [[Kähler manifold]]. If <math>F</math> has a [[Compact space|compact]] leaf with finite [[holonomy group]] then every leaf of <math>F</math> is compact with finite holonomy group.''
+Theorem<ref>J.V. Pereira, ''Global stability for holomorphic foliations on Kaehler manifolds'', — Qual. Theory Dyn. Syst. 2 (2001), 381&ndash;384. [http://arxiv.org/PS_cache/math/pdf/0002/0002086v2.pdf]</ref>: ''Let <math>F</math> be a holomorphic foliation of codimension <math>k</math> in a compact complex [[Kähler manifold]]. If <math>F</math> has a [[Compact space|compact]] leaf with finite [[holonomy|holonomy group]] then every leaf of <math>F</math> is compact with finite holonomy group.''
 == References ==