

Line 75: 
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 A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$.   A codimension one foliation $\mathcal{F}$ of $M$ is taut if for every leaf $\lambda$ of $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ which intersects $\lambda$. 
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−  {{beginthmTheorem(Rummler, Sullivan)}}
 
−  The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:
 
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−  a) $\mathcal{F}$ is taut;
 
− 
 
−  b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;
 
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−  c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}}
 
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−  </wikitex>
 
− 
 
−  == Constructing new foliations from old ones ==
 
− 
 
− 
 
−  ==== Pullbacks ====
 
−  <wikitex>;
 
−  {{beginthmTheorem }} If $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ and $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$, then $N$ is foliated by connected components of $f^{1}\left(L\right)$ as $L$ ranges over the leaves of $\mathcal{F}$. {{endthm}}
 
−  {{citeCandel&Conlon2000}}, Theorem 3.2.2
 
−  </wikitex>
 
− 
 
−  ==== Glueing ====
 
−  <wikitex>;
 
−  Let $\left(M_1,{\mathcal{F}}_1\right)$ and $\left(M_2,{\mathcal{F}}_2\right)$ be $n$manifolds with foliations of the same codimension. Assume there is a homeomorphism $f:\partial M_1\rightarrow \partial M_2$.
 
−  If either both foliations are tangent or both foliations are transverse to the boundaries of $M_1$ and $M_2$, then they can be glued to a foliation on $M_1\cup_f M_2$. This is called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.
 
−  </wikitex>
 
− 
 
−  ==== Turbulization ====
 
−  <wikitex>;
 
−  Let $\left(M,{\mathcal{F}}\right)$ be a transversely orientable codimension 1 foliation, and let $\gamma:S^1\rightarrow M$ be an embedding transverse to $\mathcal{F}$.
 
− 
 
−  Define a foliation ${\mathcal{F}}_0$ on a small neighborhood $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n1}$ by
 
−  $$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$
 
−  where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n2}\rightarrow S^1\times D^{n1}$, and $\lambda:\left[0,1\right]\rightarrow\left[\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with $$\lambda\left(0\right)=\frac{\pi}{2}, \lambda\mid_{\left[1\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$$
 
−  The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhood of the boundary of $N\left(\gamma\left(S^1\right)\right)$. The result of glueing these foliations is called the turbulization ${\mathcal{F}}^\prime$ of $\mathcal{F}$.
 
−  </wikitex>
 
− 
 
   
 == References ==   == References == 
Revision as of 11:26, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

1 Foliations
Let
be an
manifold, possibly with boundary, and let
be a decomposition of
into connected, topologically immersed submanifolds of dimension
.
is said to be a codimension
(smooth) foliation of
if
admits an (smooth) atlas
of foliated charts, that is (diffeomorphisms) homeomorphisms
such that for each
the intersection
is a union of plaques
.
2 Defining differential form
If is a smooth, transversely orientable codimension foliation and its tangential plane field, then there is a nonsingular form such that, for each ,
This implies that for some .
3 Leaves
The leaves of are the immersed submanifolds . Each belongs to a unique leaf. The foliation determines its tangential plane field by if .
The space of leaves is with the quotient topology, where if and only if and belong to the same leaf of .
4 Holonomy Cocycle
The holonomy cocycle
of the atlas is given by
A smooth foliation is said to be transversely orientable if everywhere.
5 Special classes of foliations
5.1 Bundles

The most trivial examples of foliations are products , foliated by the leaves . (Another foliation of is given by .)
A more general class are flat bundles with or for a (smooth or topological) manifold . Given a representation , the flat bundle with monodromy is given as , where acts on the universal cober by deck transformations and on by means of the representation . ( is a flat bundle if .)
Flat bundles fit into the frame work of fiber bundles. A (smooth) map
between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each
has an open neighborhood
such that there is a homeomorphism (diffeomorphism)
making the following diagram (with
projection to the first factor) commutative:
The fiber bundle yields a foliation by fibers
. Its space of leaves
is (diffeomeorphic) homeomorphic to
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, in particular
is a Hausdorff manifold.
1.1 Suspensions

A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are
where is the canonical projection. This foliation is called the suspension of the representation .
In particular, if and is a homeomorphism of , then the suspension foliation of is defined to be the suspension foliation of the representation given by . Its space of leaves is , where if for some .
The simplest examples of suspensions are the Kronecker foliations of the 2torus, that is the suspension foliation of the rotation by angle .
If is a rational multiple of , then the leaves of are compact. If is an irrational multiple of , then the leaves of are dense on the 2torus.
1.2 Submersions

Let
be a submersion. Then
is foliated by the preimages
. This includes the case of fiber bundles.
2dimensional Reeb foliation
An example of a submersion, which is not a fiber bundle, is given by
This submersion yields a foliation of
which is invariant under the
actions given by
resp.
for . The induced foliations of are called the 2dimensional Reeb foliation (of the annulus) resp. the 2dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff.
3dimensional Reeb foliation
1.3 Reeb foliations

Define a submersion
by
where
are cylindrical coordinates on
. This submersion yields a foliation of
which is invariant under the
actions given by
for . The induced foliation of is called the ndimensional Reeb foliation. Its leaf space is not Hausdorff.
1.4 Taut foliations

A codimension one foliation of is taut if for every leaf of there is a circle transverse to which intersects .
References