Foliation
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== Introduction == | == Introduction == | ||
+ | This page gives the definition of the term ''foliation''. For further information, see the page [[Foliations]] and | ||
+ | {{cite|Godbillon1991}}. | ||
+ | === Foliations === | ||
<wikitex>; | <wikitex>; | ||
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− | |||
− | |||
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 $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$. | 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 $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$. | ||
+ | </wikitex> | ||
=== Defining differential form === | === Defining differential form === | ||
+ | <wikitex>; | ||
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 $x\in M$, | 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 $x\in M$, | ||
$$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$ | $$\omega_x\left(v_1\wedge\ldots\wedge v_q\right)=0\Longleftrightarrow \mbox{\ at\ least\ one\ }v_i\in E_x.$$ | ||
This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. | This implies that $d\omega=\omega\wedge\eta$ for some $\eta\in\Omega^1\left(M\right)$. | ||
− | + | </wikitex> | |
=== Leaves === | === Leaves === | ||
+ | <wikitex>; | ||
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 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 $\mathcal{F}$. | 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 $\mathcal{F}$. | ||
− | + | </wikitex> | |
=== Holonomy Cocycle === | === Holonomy Cocycle === | ||
+ | <wikitex>; | ||
The holonomy cocycle $\left\{\gamma_{\alpha\beta}: \alpha,\beta\in{\mathcal{A}}\right\}$ of the atlas is given by $$\gamma_{\alpha\beta}:=y_\alpha y_\beta^{-1}:y_\beta\left(U_\alpha\cap U_\beta\right)\rightarrow y_\alpha\left(U_\alpha\cap U_\beta\right).$$ | 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. | A smooth foliation ${\mathcal{F}}$ is said to be transversely orientable if $det\left(D\gamma_{\alpha\beta}\right)>0$ everywhere. | ||
+ | </wikitex> | ||
+ | == 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> | </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> | </wikitex> | ||
− | == | + | |
+ | === Submersions === | ||
<wikitex>; | <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> | </wikitex> | ||
− | == | + | |
+ | === Reeb foliations === | ||
<wikitex>; | <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> | </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 $\lambda$. | ||
+ | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#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]] | [[Category:Definitions]] |
Latest revision as of 10:35, 28 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
This page gives the definition of the term foliation. For further information, see the page Foliations and [Godbillon1991].
1.1 Foliations
1.2 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 .
1.3 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 .
1.4 Holonomy Cocycle
A smooth foliation is said to be transversely orientable if everywhere.
2 Special classes of foliations
2.1 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) mapThe fiber bundle yields a foliation by fibers . Its space of leaves is (diffeomeorphic) homeomorphic to , in particular is a Hausdorff manifold.
2.2 Suspensions
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 2-torus, 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 2-torus.
2.3 Submersions
An example of a submersion, which is not a fiber bundle, is given by
for . The induced foliations of 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.
2.4 Reeb foliations
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
2.5 Taut foliations
A codimension one foliation of is taut if for every leaf of there is a circle transverse to which intersects .
3 References
- [Godbillon1991] C. Godbillon, Feuilletages, Birkhäuser Verlag, 1991. MR1120547 (93i:57038) Zbl 0724.58002
4 External links
- The Encylopedia of Mathematics article on foliations
- The Wikipedia page about foliations