Reeb foliation

In mathematics, the Reeb foliation - is a special foliation of Volltorus, named after Georges Reeb.

Construction

Define a submersion

Wherein the cylindrical coordinates on the two -dimensional circular disc is. The level sets of this submersion form a foliation of which is invariant under the by

Given effect is. The induced foliation of the Reeb foliation is called Volltorus. The bounding torus

Is a leaf of this foliation.

Reeb components

They say that a foliation of a 3-manifold has a Reeb component, if there is an embedded Volltorus

Are such that the restriction of on homeomorphic to the Reeb foliation - is.

Example: Reeb foliation of the 3 - sphere

The 3- dimensional sphere is obtained by bonding two Volltori, see standard Heegaard decomposition of the 3- sphere. The Reeb foliation of the 3 - sphere obtained by the Reeb foliations of the two Volltori.

Existence of foliations on 3-manifolds

By a theorem of Lickorish obtained every closed, orientable 3-manifold by Dehn surgery on a tangle in the 3- sphere. You can use this set to construct on every closed, orientable 3-manifold foliations with Reeb components.

In contrast, not all closed, orientable 3-manifolds without Reeb foliations own components.

So-called taut foliations (English: taut foliations ) have no Reeb components.

Properties

The Reeb foliation - is, but not analytically.

Your hand space is not Hausdorff.