Reeb foliage

In mathematics , Reeb scrolling is a special scrolling of the full torus , named after Georges Reeb .

construction

Cross section through a reeb foliage.

Define a submersion

${\displaystyle f:D^{2}\times {\mathbb {R} }\rightarrow {\mathbb {R} }}$ by

${\displaystyle f\left(x,t\right):=\left(|x|^{2}-1\right)e^{t},}$

where is the 2- dimensional disc. The level sets of this submersion form a foliation of . This is invariant under the through ${\displaystyle D^{2}}$ ${\displaystyle D^{2}\times {\mathbb {R} }}$

${\displaystyle \left(x,t\right)\mapsto \left(x,t+n\right)}$ For ${\displaystyle \left(x,t\right)\in D^{2}\times {\mathbb {R} },n\in {\mathbb {Z} }}$

given effect, because with that of independent constants . The induced foliation of the full torus is called Reeb foliation . The bordering torus ${\displaystyle \mathbb {Z} }$ ${\displaystyle f(x,t+n)=c_{n}f(x,t)}$ ${\displaystyle x,t}$ ${\displaystyle c_{n}=e^{n}}$ ${\displaystyle D^{2}\times S^{1}\cong \left(D^{2}\times {\mathbb {R} }\right)/{\mathbb {Z} }}$

${\displaystyle T^{2}\cong \partial (D^{2}\times S^{1})}$

is a leaf of this foliation (the level set ). ${\displaystyle f=0}$

Reeb components

A foliage of a 3-manifold is said to have a Reeb component if it is an embedded full torus ${\displaystyle {\mathcal {F}}}$ ${\displaystyle M}$

${\displaystyle D^{2}\times S^{1}\subset M}$

there, so that the restriction from on is homeomorphic to Reeb foliation. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle D^{2}\times S^{1}}$

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. ${\displaystyle C^{\infty }}$

Your leaf space is not Hausdorffsch .

literature

Rosenberg, H .; Roussarie, R. Reeb foliations. Ann. of Math. (2) 91 1970 1-24. pdf

Web links

Manifold Atlas