Reeb foliation

In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993).

It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf.

By Novikov's compact leaf theorem, every smooth foliation of the 3-sphere includes a compact torus leaf, bounding a solid torus foliated in the same way.

Illustrations

2-dimensional section of Reeb foliation
3-dimensional model of Reeb foliation
gollark: Have you heard of Greg Egan?
gollark: Magic systems generally care about higher-level objects and what humans do and whatever, instead of describing universal physical laws.
gollark: *Our* universe has cold uncaring physics, which life, particularly intelligent life, can exploit like everything else if it researches them enough.
gollark: Thus, my probably horribly flawed way to categorize it is that magic is where the universe/setting is weirdly interested in sentient beings/life/humans/etc, and generally more comprehensible to them.
gollark: I was thinking about this a lot a while ago, and determined that magic wasn't really an aesthetic since there are a few stories which have basically everything be "magic" which does identical things to technology.

References

  • Reeb, Georges (1952). "Sur certaines propriétés topologiques des variétés feuillétées" [On certain topological properties of foliation varieties]. Actualités Sci. Indust. (in French). Paris: Hermann. 1183.
  • Candel, Alberto; Conlon, Lawrence (2000). Foliations. American Mathematical Society. p. 93. ISBN 0-8218-0809-5.
  • Moerdijk, Ieke; Mrčun, J. (2003). Introduction to Foliations and Lie Groupoids. Cambridge Studies in Advanced Mathematics. 91. Cambridge University Press. p. 8. ISBN 0-521-83197-0.
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