# Foliation

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## Introduction

### 1 Foliations

Let $M$${{Stub}} == Introduction == ; == Introduction == ; === 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. == Definition == ; ... == Examples == ; ... == References == {{#RefList:}} [[Category:Definitions]]M$ be an $n$$n$-manifold, possibly with boundary, and let ${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$${\mathcal{F}}=\left\{F_b\right\}_{b\in B}$ be a decomposition of $M$$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)$.

### 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)$.

### 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}$.

### 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.

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