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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Reeb stability theorem ： ウィキペディア英語版 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. == Reeb local stability theorem == Theorem:〔 〕 ''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 $\backslash pi:\; U\backslash to\; L$ such that, for every leaf $L\text{'}\backslash subset\; U$, $\backslash pi|\_:L\text{'}\backslash to\; L$ is a covering map with a finite number of sheets and, for each $y\backslash in\; L$, $\backslash pi^(y)$ 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.〔J. Palis, jr., W. de Melo, ''Geometric theory of dinamical systems: an introduction'', — New-York, Springer,1982.〕 This is the case of codimension one, singular foliations $(M^n,F)$, with $n\backslash ge\; 3$, and some center-type singularity in $Sing(F)$. The Reeb local stability theorem also has a version for a noncompact codimension-1 leaf.〔T.Inaba, ''$C^2$ Reeb stability of noncompact leaves of foliations,''— Proc. Japan Acad. Ser. A Math. Sci., 59:158{160, 1983 ()〕〔J. Cantwell and L. Conlon, ''Reeb stability for noncompact leaves in foliated 3-manifolds,'' — Proc. Amer.Math.Soc. 33 (1981), no. 2, 408–410.()〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Reeb stability theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.