Foliation
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== Introduction == | == Introduction == | ||
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+ | == 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 $\lambda$. | ||
+ | |||
+ | {{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 == | == References == |
Revision as of 12:23, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
1 Foliations
LetTex syntax errorbe an -manifold, possibly with boundary, and let be a decomposition of
Tex syntax errorinto connected, topologically immersed submanifolds of dimension . is said to be a codimension (smooth) foliation of
Tex syntax errorif
Tex syntax erroradmits an (smooth) atlas of foliated charts, that is (diffeomorphisms) homeomorphisms
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 .
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 .
4 Holonomy Cocycle
The holonomy cocycle of the atlas is given byA smooth foliation is said to be transversely orientable if everywhere.
5 Special classes of foliations
5.1 Bundles
Tex syntax erroris 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 . (
Tex syntax erroris a flat -bundle if .) Flat bundles fit into the frame work of fiber bundles. A (smooth) map
The fiber bundle yields a foliation by fibers . Its space of leaves is (diffeomeorphic) homeomorphic to , in particular is a Hausdorff manifold.
1.1 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.
1.2 Submersions
Tex syntax erroris foliated by the preimages . This includes the case of fiber bundles.
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.
1.3 Reeb foliations
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
1.4 Taut foliations
Tex syntax erroris taut if for every leaf of there is a circle transverse to which intersects .
Theorem 3.1 (Rummler, Sullivan).
The following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifoldsTex syntax error:
a) is taut;
b) there is a flow transverse to which preserves some volume form onTex syntax error; c) there is a Riemannian metric on
Tex syntax errorfor which the leaves of are least area surfaces.
2 Constructing new foliations from old ones
2.1 Pullbacks
[Candel&Conlon2000], Theorem 3.2.2
2.2 Glueing
Let and be -manifolds with foliations of the same codimension. Assume there is a homeomorphism . If either both foliations are tangent or both foliations are transverse to the boundaries of and , then they can be glued to a foliation on . This is called the tangential resp. the transversal glueing of and .
2.3 Turbulization
Let be a transversely orientable codimension 1 foliation, and let be an embedding transverse to .
Define a foliation on a small neighborhood by
The foliations and agree on a neighborhood of the boundary of . The result of glueing these foliations is called the turbulization of .
3 References
- [Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001