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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 manifold|closed]] and has finite [[fundamental group]], then all the leaves are closed and have finite fundamental group.
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| == Reeb local stability theorem ==
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| 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 space|compact]] leaf with finite [[holonomy|holonomy group]]. There exists a [[Neighbourhood (mathematics)|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 [[Deformation retract#|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 [[Transversality (mathematics)|transverse]] to <math>F</math>. The neighborhood <math>U</math> can be taken to be arbitrarily small.''
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| The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf
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| with finite holonomy, the space of leaves is [[Hausdorff space|Hausdorff]].
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| Under certain conditions the Reeb Local Stability Theorem may replace the [[Poincaré–Bendixson theorem]] in higher dimensions.<ref>J. Palis, jr., W. de Melo, ''Geometric theory of dinamical systems: an introduction'', — New-York,
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| Springer,1982.</ref> This is the case of codimension one, singular foliations <math>(M^n,F)</math>, with <math>n\ge 3</math>, and some center-type singularity in <math>Sing(F)</math>.
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| The Reeb Local Stability Theorem also has a version for a noncompact leaf.<ref>T.Inaba, ''<math>C^2</math> Reeb stability of noncompact leaves of foliations,''— Proc. Japan Acad. Ser. A Math. Sci., 59:158{160, 1983 [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195515640]</ref><ref>J. Cantwell and L. Conlon, ''Reeb stability for noncompact leaves in foliated 3-manifolds,'' — Proc. Amer.Math.Soc. 33 (1981), no. 2, 408–410.[http://www.ams.org/journals/proc/1981-083-02/S0002-9939-1981-0624942-5/S0002-9939-1981-0624942-5.pdf]</ref>
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| == Reeb global stability theorem ==
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| 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.
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| 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>.''
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| 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]]
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| on certain components of the [[Boundary (topology)|boundary]] and [[Transversality (mathematics)|transverse]] on other components.<ref>C. Godbillon, ''Feuilletages, etudies geometriques,'' — Basel, Birkhauser, 1991</ref> In this case it implies [[Reeb sphere theorem]].
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| 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:
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| * In the presence of a certain transverse geometric structure:
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| Theorem:<ref>R.A. Blumenthal, ''Stability theorems for conformal foliations'', — Proc. AMS. 91, 1984, p. 55–63. [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 [[Connectedness|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.''
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| * For [[Complex manifold|holomorphic]] foliations in complex [[Kähler manifold]]:
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| Theorem:<ref>J.V. Pereira, ''Global stability for holomorphic foliations on Kaehler manifolds'', — Qual. Theory Dyn. Syst. 2 (2001), 381–384. {{arxiv|math/0002086v2}}</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.''
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| == References ==
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| * C. Camacho, A. Lins Neto: Geometric theory of foliations, Boston, Birkhauser, 1985
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| * I. Tamura, Topology of foliations: an introduction, Transl. of Math. Monographs, AMS, v.97, 2006, 193 p.
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| == Notes ==
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| {{reflist}}
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| [[Category:Foliations]]
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