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In [[mathematics]], '''Reeb sphere theorem''', named after [[Georges Reeb]], states that
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: A closed oriented connected manifold ''M''<sup>&nbsp;''n''</sup> that admits a [[singular foliation]] having only centers  is [[homeomorphism|homeomorphic]] to the [[sphere]] ''S''<sup>''n''</sup> and the foliation has exactly two singularities.
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==Morse foliation==
 
A singularity of a foliation ''F'' is of '''Morse type''' if in its small neighborhood all leaves of the foliation are  [[level set|levels]] of a [[Morse theory|Morse function]], being the singularity a [[Critical point (mathematics)|critical point]] of the function.  The singularity is a '''center''' if it is a [[maxima and minima|local extremum]] of the function; otherwise, the singularity is a [[saddle]].
 
The number of centers ''c'' and the number of saddles <math>s</math>, specifically ''c''&nbsp;&minus;&nbsp;''s'', is tightly connected with the manifold topology.  
 
We denote ind&nbsp;''p''&nbsp;=&nbsp;min(''k'',&nbsp;''n''&nbsp;&minus;&nbsp;''k''), the '''index''' of a singularity <math>p</math>, where ''k'' is the index of the corresponding critical point of a Morse function. In particular, a center has index&nbsp;0, index of a saddle is at least&nbsp;1.
 
A '''Morse foliation''' ''F'' on a manifold ''M'' is a [[singular foliation|singular]] transversely oriented codimension one foliation of class ''C''<sup>2</sup> with isolated singularities such that:
* each singularity of ''F'' is of Morse type,
* each [[singular leaf]] ''L'' contains a unique singularity&nbsp;''p''; in addition, if ind&nbsp;''p''&nbsp;=&nbsp;1 then <math>L\setminus p</math> is not connected.
 
==Reeb sphere theorem==
This is the case ''c''&nbsp;>&nbsp;''s''&nbsp;=&nbsp;0, the case without saddles.
 
Theorem:<ref>{{citation
| last = Reeb | first = Georges | author-link = Georges Reeb
| journal = [[Comptes rendus de l'Académie des sciences|C. R. Acad. Sci. Paris]]
| language = French
| mr = 0015613
| pages = 847–849
| title = Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique
| volume = 222
| year = 1946}}.</ref> ''Let <math>M^n</math> be a closed oriented connected manifold of dimension <math>n\ge 2</math>. Assume that <math>M^n</math> admits a <math>C^1</math>-transversely oriented codimension one foliation <math>F</math> with a non empty set of singularities all of them centers. Then the singular set of <math>F</math> consists of two points and <math>M^n</math> is homeomorphic to the sphere <math>S^n</math>''.  
 
It is a consequence of the [[Reeb stability theorem]].
 
==Generalization==
 
More general case is <math>c>s\ge 0.</math>
 
In 1978, E. Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles. He showed that the number of centers cannot be too much as compared with the number of saddles, notably, <math>c\le s+2</math>. So there are exactly two cases when <math>c>s</math>:
 
:(1)  <math>c=s+2, \, </math>
 
:(2)  <math>c=s+1. \, </math>
 
He obtained a description of the manifold admitting a foliation with singularities that satisfy&nbsp;(1).
 
Theorem:<ref>{{citation
| last = Wagneur | first = E.
| issue = 3
| journal = Annales de l'Institut Fourier
| language = French
| mr = 511820
| pages = xi, 165–176
| title = Formes de Pfaff à singularités non dégénérées
| url = http://www.numdam.org/item?id=AIF_1978__28_3_165_0
| volume = 28
| year = 1978}}.</ref> ''Let <math>M^n</math> be a compact connected manifold admitting  a Morse foliation <math>F</math> with <math>c</math> centers and <math>s</math> saddles. Then <math>c\le s+2</math>.''
''In case <math>c=s+2</math>,''
 
* ''<math>M</math> is homeomorphic to <math>S^n</math>,''
 
* ''all saddles have index'' 1,
 
* ''each regular leaf is diffeomorphic to <math>S^{n-1}</math>.''
 
Finally, in 2008, C. Camacho and B. Scardua considered the case (2), <math>c=s+1</math>. Interestingly, this is possible in a small number of low dimensions.
 
Theorem:<ref>{{citation
| last1 = Camacho | first1 = César
| last2 = Scárdua | first2 = Bruno
| arxiv = math/0611395
| doi = 10.1090/S0002-9939-08-09371-4
| issue = 11
| journal = [[Proceedings of the American Mathematical Society]]
| mr = 2425748
| pages = 4065–4073
| title = On foliations with Morse singularities
| volume = 136
| year = 2008}}.</ref> ''Let <math>M^n</math> be a compact connected manifold and <math>F</math> a Morse foliation on <math>M</math>. If <math>s = c + 1</math>, then''
 
* ''<math>n=2,4,8</math> or <math>16</math>,''
 
* ''<math>M^n</math> is an [[Eells–Kuiper manifold]].''
 
==References==
 
<references/>
 
{{DEFAULTSORT:Reeb Sphere Theorem}}
[[Category:Foliations]]
[[Category:Theorems in topology]]

Latest revision as of 16:30, 13 March 2014

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