Foliation

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1 Introduction

1 Foliations

Let

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${{Stub}} == Introduction == <wikitex>; === Foliations === Let $M$ be an $n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$ into connected, topologically immersed submanifolds of dimension $n-q$. ${\mathcal{F}}$ is said to be a codimension $q$ (smooth) foliation of $M$ if $M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms $$\phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ such that for each $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$. === Defining differential form === If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$-form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$, $$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$ This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. === Leaves === The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$. Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$. The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$. === Holonomy Cocycle === The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$ A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere. == Special classes of foliations == === Bundles === <wikitex>; The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.) A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.) Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative: $$\begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\ U\ar[r]^{id}&U} \end{xy}$$ The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold. </wikitex> === Suspensions === <wikitex>; A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$ where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$. In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$. The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$. If $\alpha$ is a rational multiple of \pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of \pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus. </wikitex> === Submersions === <wikitex>; Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles. [[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]] An example of a submersion, which is not a fiber bundle, is given by $$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$ $$f\left(x,y\right)=\left(x^2-1\right)e^y.$$ This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp. $$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$ for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff. [[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]] </wikitex> === Reeb foliations === <wikitex>; Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by $$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$ where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$. This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff. </wikitex> === Taut foliations === <wikitex>; 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 $\lambda$. {{beginthm|Theorem|(Rummler, Sullivan)}} The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$: a) $\mathcal{F}$ is taut; b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$; c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}} </wikitex> == Constructing new foliations from old ones == ==== Pullbacks ==== <wikitex>; {{beginthm|Theorem |}} 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}} {{cite|Candel&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 $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by $$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$ where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, 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 == {{#RefList:}} [[Category:Definitions]]M$ be an $n$ $n$ -manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of

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$M$ into connected, topologically immersed submanifolds of dimension $n-q$ $n-q$ . ${\mathcal{F}}$ ${\mathcal{F}}$ is said to be a codimension $q$ $q$ (smooth) foliation of

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$M$ if

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$M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms

$\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$ $\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$

such that for each $\alpha\in{\mathcal{A}}, b\in B$ $\alpha\in{\mathcal{A}}, b\in B$ the intersection $F_b\cap U_\alpha$ $F_b\cap U_\alpha$ is a union of plaques $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$ $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$ .

2 Defining differential form

If $\mathcal{F}$ is a smooth, transversely orientable codimension $q$ foliation and $E$ its tangential plane field, then there is a nonsingular $q$ -form $\omega\in\Omega^q\left(M\right)$ such that, for each $x\in M$ ,

$\displaystyle \omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$ $\displaystyle \omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$

This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$ .

3 Leaves

The leaves of $\mathcal{F}$ are the immersed submanifolds $F_b$ . Each $x\in M$ belongs to a unique leaf. The foliation $\mathcal{F}$ determines its tangential plane field $E\subset TM$ by $E_x:=T_xF_b\subset T_xM$ if $x\in F_b$ .

The space of leaves is $L=M/\sim$ with the quotient topology, where $x\sim y$ if and only if $x$ and $y$ belong to the same leaf of $\mathcal{F}$ .

4 Holonomy Cocycle

The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by

$\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$ $\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$

A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere.

5 Special classes of foliations

5.1 Bundles

The most trivial examples of foliations are products $M=B\times F$ $M=B\times F$ , foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$ $F_b:=\left\{b\right\}\times F, b\in B$ . (Another foliation of

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$M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$ $B_f:=\left\{f\right\}\times B, f\in F$ .) A more general class are flat $G$ $G$ -bundles with $G=\Diff\left(F\right)$ $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$ $F$ . Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$ $\pi_1B\rightarrow \Homeo\left(F\right)$ , the flat $\Homeo\left(F\right)$ $\Homeo\left(F\right)$ -bundle with monodromy $\rho$ $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$ $M=\left(\widetilde{B}\times F\right)/\pi_1B$ , where $\pi_1B$ $\pi_1B$ acts on the universal cober $\widetilde{B}$ $\widetilde{B}$ by deck transformations and on $F$ $F$ by means of the representation $\rho$ $\rho$ . (

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$M$ is a flat $\Diff\left(F\right)$ $\Diff\left(F\right)$ -bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$ $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$ .) Flat bundles fit into the frame work of fiber bundles. A (smooth) map

$\displaystyle \pi:M\rightarrow B$ $\displaystyle \pi:M\rightarrow B$

between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ $b\in B$ has an open neighborhood $U$ $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ $p_1$ projection to the first factor) commutative:

$\displaystyle \begin{xy} \xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\ U\ar[r]^{id}&U} \end{xy}$

The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$ . Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$ , in particular $L$ 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

$\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$ $\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$

where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$ .

In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$ , then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$ . Its space of leaves is $L=F/\sim$ , where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$ .

The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$ . If $\alpha$ is a rational multiple of $2\pi$ , then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$ , then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus.

1.2 Submersions

Let

$\displaystyle f:M\rightarrow B$ $\displaystyle f:M\rightarrow B$

be a submersion. Then

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$M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$ $\pi^{-1}\left(b\right), b\in B$ . This includes the case of fiber bundles.

2-dimensional Reeb foliation

An example of a submersion, which is not a fiber bundle, is given by

$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$ $\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$

$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$ $\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$

This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$ ${\mathbb Z}$ -actions given by

$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$ $\displaystyle z\left(x,y\right)=\left(x,y+z\right)$

resp.

$\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$ $\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$

for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$ . The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff.

3-dimensional Reeb foliation

1.3 Reeb foliations

Define a submersion

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

by

$\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$ $\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$

where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$ $D^n$ . This submersion yields a foliation of $D^n\times{\mathbb R}$ $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$ ${\mathbb Z}$ -actions given by

$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$ $\displaystyle z\left(x,y\right)=\left(x,y+z\right)$

for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$ . The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.

1.4 Taut foliations

A codimension one foliation $\mathcal{F}$ $\mathcal{F}$ of

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$M$ is taut if for every leaf $\lambda$ $\lambda$ of $\mathcal{F}$ $\mathcal{F}$ there is a circle transverse to $\mathcal{F}$ $\mathcal{F}$ which intersects $\lambda$ $\lambda$ .

Theorem 3.1 (Rummler, Sullivan).

The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds

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$M$ :

a) $\mathcal{F}$ is taut;

b) there is a flow transverse to $\mathcal{F}$ $\mathcal{F}$ which preserves some volume form on

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$M$ ; c) there is a Riemannian metric on

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$M$ for which the leaves of $\mathcal{F}$ $\mathcal{F}$ are least area surfaces.

2 Constructing new foliations from old ones

2.1 Pullbacks

Theorem 4.1. If $\left(M,{\mathcal{F}}\right)$ $\left(M,{\mathcal{F}}\right)$ is a foliated manifold of codimension $q$ $q$ and $f:N\rightarrow M$ $f:N\rightarrow M$ is a smooth manifold transverse to $\mathcal{F}$ $\mathcal{F}$ , then $N$ $N$ is foliated by connected components of $f^{-1}\left(L\right)$ $f^{-1}\left(L\right)$ as $L$ $L$ ranges over the leaves of $\mathcal{F}$ $\mathcal{F}$ .

[Candel&Conlon2000], Theorem 3.2.2

2.2 Glueing

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 ${\mathcal{F}}_2$ .

2.3 Turbulization

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 $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by

$\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$ $\displaystyle cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$

where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$ $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$ , and $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ $\lambda:\left[0,1\right]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ is a smooth function with

$\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$ $\displaystyle \lambda\left(0\right)=-\frac{\pi}{2}, \lambda\mid_{\left[1-\epsilon,1\right]}\equiv \frac{\pi}{2}\mbox{\ for\ some\ }\epsilon>0, \lambda^\prime\left(t\right)>0\mbox{\ for\ all\ }t\in\left(0,1-\epsilon\right), \lambda^{\left(k\right)}\left(0\right)=0\mbox{\ for\ all\ }k\ge 1.$

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 $\mathcal{F}$ .

3 References

[Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001

Foliation

Revision as of 12:23, 27 March 2013

Contents

1 Introduction

1 Foliations

2 Defining differential form

3 Leaves

4 Holonomy Cocycle

5 Special classes of foliations

5.1 Bundles

1.1 Suspensions

1.2 Submersions

1.3 Reeb foliations

1.4 Taut foliations

2 Constructing new foliations from old ones

2.1 Pullbacks

2.2 Glueing

2.3 Turbulization

3 References

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@@ Line 1: / Line 1: @@
 {{Stub}}
-== Introduction ==
-<wikitex>;
 == Introduction ==
 <wikitex>;
@@ Line 23: / Line 21: @@
+== Special classes of foliations ==
+=== Bundles ===
+<wikitex>;
+The most trivial examples of foliations are products $M=B\times F$, foliated by the leaves $F_b:=\left\{b\right\}\times F, b\in B$. (Another foliation of $M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$.)
+A more general class are flat $G$-bundles with $G=\Diff\left(F\right)$ or $G=\Homeo\left(F\right)$ for a (smooth or topological) manifold $F$. Given a representation $\pi_1B\rightarrow \Homeo\left(F\right)$, the flat $\Homeo\left(F\right)$-bundle with monodromy $\rho$ is given as $M=\left(\widetilde{B}\times F\right)/\pi_1B$, where $\pi_1B$ acts on the universal cober $\widetilde{B}$ by deck transformations and on $F$ by means of the representation $\rho$. ($M$ is a flat $\Diff\left(F\right)$-bundle if $\rho\left(\pi_1B\right)\subset \Diff\left(F\right)$.)
+Flat bundles fit into the frame work of fiber bundles. A (smooth) map $$\pi:M\rightarrow B$$ between (smooth) manifolds is a (smooth) fiber bundle if there is a (smooth) manifold F such that each $b\in B$ has an open neighborhood $U$ such that there is a homeomorphism (diffeomorphism) $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:
+$$\begin{xy}
+\xymatrix{ \pi^{-1}\left(U\right)\ar[d]^\pi\ar[r]^\phi &U\times F\ar[d]^{p_1}\\
+U\ar[r]^{id}&U}
+\end{xy}$$
+The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$ is (diffeomeorphic) homeomorphic to $B$, in particular $L$ is a Hausdorff manifold.
 </wikitex>
+=== Suspensions ===
+<wikitex>;
+A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are $$L_f:=
+\left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$$
+where $p:\widetilde{B}\times F\rightarrow M$ is the canonical projection. This foliation is called the suspension of the representation $\rho:\pi_1B\rightarrow \Homeo\left(F\right)$.
+In particular, if $B=S^1$ and $\phi:F\rightarrow F$ is a homeomorphism of $F$, then the suspension foliation of $\phi$ is defined to be the suspension foliation of the representation $\rho:{\mathbb Z}\rightarrow \Homeo\left(F\right)$ given by $\rho\left(z\right)=\Phi^z$. Its space of leaves is $L=F/\sim$, where $x\sim y$ if $y=\Phi^n\left(x\right)$ for some $n\in{\mathbb Z}$.
+The simplest examples of suspensions are the Kronecker foliations ${\mathcal{F}}_\alpha$ of the 2-torus, that is the suspension foliation of the rotation $R_\alpha:S^1\rightarrow S^1$ by angle $\alpha\in\left[0,2\pi\right)$.
+If $\alpha$ is a rational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are compact. If $\alpha$ is an irrational multiple of $2\pi$, then the leaves of ${\mathcal{F}}_\alpha$ are dense on the 2-torus.
 </wikitex>
-== Definition ==
+=== Submersions ===
 <wikitex>;
-...
+Let $$f:M\rightarrow B$$ be a submersion. Then $M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$. This includes the case of fiber bundles.
+[[Image:200px-Reebfoliation-ring-2d-2.svg.png|thumb|200px|2-dimensional Reeb foliation]]
+An example of a submersion, which is not a fiber bundle, is given by
+$$f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$$
+$$f\left(x,y\right)=\left(x^2-1\right)e^y.$$
+This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$ resp.
+$$z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$$
+for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliaton (of the Möbius band). Their leaf spaces are not Hausdorff.
+[[Image:Reeb_foliation_half-torus_POV-Ray.png|thumb|300px|3-dimensional Reeb foliation]]
 </wikitex>
-== Examples ==
+=== Reeb foliations ===
 <wikitex>;
-...
+Define a submersion $$f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$$ by
+$$f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$$
+where $\left(r,\theta\right)\in \left[0,1\right]\times S^{n-1}$ are cylindrical coordinates on $D^n$.  This submersion yields a foliation of $D^n\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-actions given by $$z\left(x,y\right)=\left(x,y+z\right)$$
+for $\left(x,y\right)\in D^n\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(D^n\times{\mathbb R}\right)$ is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
 </wikitex>
+=== Taut foliations ===
+<wikitex>;
+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 $\lambda$.
+{{beginthm|Theorem|(Rummler, Sullivan)}}
+The following conditions are equivalent for transversely orientable codimension one foliations $\left(M,{\mathcal{F}}\right)$ of closed, orientable, smooth manifolds $M$:
+a) $\mathcal{F}$ is taut;
+b) there is a flow transverse to $\mathcal{F}$ which preserves some volume form on $M$;
+c) there is a Riemannian metric on $M$ for which the leaves of $\mathcal{F}$ are least area surfaces.{{endthm}}
+</wikitex>
+== Constructing new foliations from old ones ==
+==== Pullbacks ====
+<wikitex>;
+{{beginthm|Theorem |}} 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}}
+{{cite|Candel&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 $N\left(\gamma\left(S^1\right)\right)\simeq S^1\times D^{n-1}$ by
+$$cos\left(\lambda\left(r\right)\right)dr+sin\left(\lambda\left(r\right)\right)dt=0,$$
+where $\left(t,r,\theta\right)\in S^1\times \left[0,1\right]\times S^{n-2}\rightarrow S^1\times D^{n-1}$, 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 ==