Foliation

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This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

Introduction

1 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
\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 the intersection F_b\cap U_\alpha is a union of plaques \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.

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\} 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}} is said to be transversely orientable if det\left(D\gamma_{\alpha\beta}\right)>0 everywhere.




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2 Definition

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3 Examples

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4 References

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