Foliation

1 Introduction

This page gives the definition of the term foliation. For further information, see the page Foliations.

1.1 Foliations

Let $M$${{Stub}} == Introduction == This page gives the definition of the term ''foliation''. For further information, see the page [[Foliations]]. === 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). {{cite|Godbillon1991}} === 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 === ; 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. === Suspensions === ; 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. === Submersions === ; 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]] === Reeb foliations === ; 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. === Taut foliations === ; 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. == References == {{#RefList:}} == External links == * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Foliation foliations] * The Wikipedia page about [[Wikipedia:Foliation|foliations]] [[Category:Definitions]]M$ be an $n$$n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$$M$ into connected, topologically immersed submanifolds of dimension $n-q$$n-q$. ${\mathcal{F}}$${\mathcal{F}}$ is said to be a codimension $q$$q$ (smooth) foliation of $M$$M$ if $M$$M$ admits an (smooth) atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$$\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is (diffeomorphisms) homeomorphisms
$\displaystyle \phi_\alpha=\left(x_\alpha,y_\alpha\right):U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$
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)$.

1.2 Defining differential form

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

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

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

1.3 Leaves

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

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

1.4 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
$\displaystyle \gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$

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

2 Special classes of foliations

2.1 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 $M$$M$ is given by $B_f:=\left\{f\right\}\times B, f\in F$$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$. ($M$$M$ is a flat $\Diff\left(F\right)$$\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
$\displaystyle \pi:M\rightarrow B$
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$$F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its space of leaves $L$$L$ is (diffeomeorphic) homeomorphic to $B$$B$, in particular $L$$L$ is a Hausdorff manifold.

2.2 Suspensions

A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers, whose leaves are
$\displaystyle L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}\ \mbox{ for }\ f\in F,$

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

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

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

2.3 Submersions

Let
$\displaystyle f:M\rightarrow B$
be a submersion. Then $M$$M$ is foliated by the preimages $\pi^{-1}\left(b\right), b\in B$$\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

$\displaystyle f:\left[-1,1\right]\times {\mathbb R}\rightarrow{\mathbb R}$
$\displaystyle f\left(x,y\right)=\left(x^2-1\right)e^y.$
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
$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$
resp.
$\displaystyle z\left(x,y\right)=\left(\left(-1\right)^zx,y\right)$

for $\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$$\left(x,y\right)\in\left[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliations of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)$${\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

2.4 Reeb foliations

Define a submersion
$\displaystyle f:D^{n}\times {\mathbb R}\rightarrow{\mathbb R}$
by
$\displaystyle f\left(r,\theta,t\right):=\left(r^2-1\right)e^t,$
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
$\displaystyle z\left(x,y\right)=\left(x,y+z\right)$

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

2.5 Taut foliations

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