Reeb foliage
In mathematics , Reeb scrolling is a special scrolling of the full torus , named after Georges Reeb .
construction
Define a submersion
- by
where is the 2- dimensional disc. The level sets of this submersion form a foliation of . This is invariant under the through
- For
given effect, because with that of independent constants . The induced foliation of the full torus is called Reeb foliation . The bordering torus
is a leaf of this foliation (the level set ).
Reeb components
A foliage of a 3-manifold is said to have a Reeb component if it is an embedded full torus
there, so that the restriction from on is homeomorphic to Reeb foliation.
Example: Reeb foliation of the 3-sphere
The 3-dimensional sphere is obtained by gluing two full gates, see standard Heegaard decomposition of the 3-sphere . The Reeb foliation of the 3-sphere is obtained from the Reeb foliation of the two full torches.
Existence of foliations on 3-manifolds
According to Lickorish's theorem, every closed, orientable 3-manifold is obtained by stretching surgery on a loop in the 3-sphere. One can use this theorem to construct leaves with Reeb components on every closed, orientable 3-manifold.
In contrast, not all closed, orientable 3-manifolds have foliage without Reeb components.
So-called tight foliations ( taut foliations ) have no Reeb components.
properties
The reeb foliation is analytical, but not.
Your leaf space is not Hausdorffsch .
literature
Rosenberg, H .; Roussarie, R. Reeb foliations. Ann. of Math. (2) 91 1970 1-24. pdf