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In mathematics, Reeb sphere theorem, named after Georges Reeb, states that : A closed oriented connected manifold ''M'' ''n'' that admits a singular foliation having only centers is homeomorphic to the sphere ''S''''n'' and the foliation has exactly two singularities. ==Morse foliation== A singularity of a foliation ''F'' is of Morse type if in its small neighborhood all leaves of the foliation are levels of a Morse function, being the singularity a critical point of the function. The singularity is a center if it is a local extremum of the function; otherwise, the singularity is a saddle. The number of centers ''c'' and the number of saddles , specifically ''c'' − ''s'', is tightly connected with the manifold topology. We denote ind ''p'' = min(''k'', ''n'' − ''k''), the index of a singularity , where ''k'' is the index of the corresponding critical point of a Morse function. In particular, a center has index 0, index of a saddle is at least 1. A Morse foliation ''F'' on a manifold ''M'' is a singular transversely oriented codimension one foliation of class ''C''2 with isolated singularities such that: * each singularity of ''F'' is of Morse type, * each singular leaf ''L'' contains a unique singularity ''p''; in addition, if ind ''p'' = 1 then is not connected. 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 sphere theorem」の詳細全文を読む スポンサード リンク
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