Foliations

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	==== Foliations of surfaces ====		==== Foliations of surfaces ====
	<wikitex>;		<wikitex>;
−	...	+	Codimension one foliations exist only on surfaces $S$ with $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the M\"obius band.
		+
		+	{{beginthm|Theorem}}
		+	a) Let $\left(S,{\mathcal{F}}\right) be a foliated torus or Klein bottle. Then we have one of the two exclusive situations:
		+	(1) $\mathcal{F}$ is the foliation by fibres of a flat $Homeo\left(S^1\right)$-bundle or
		+	(2) $\mathcal{F}$ contains a Reeb component (orientable or not).
		+	b) Every foliation of the annulus $S^1\times I$ tangent to the boundary is obtained by glueing together a finite number of Reeb components and a finite number of flat $Homeo\left(I\right)$-bundles
		+	c) Every foliation of the Möbius band tangent to the boundary is one of the fllowing three possibly glued together with a foliation on $S^1\times I$:
		+	(1) the non-orientable Reeb component
		+	(2) the orientable Reeb component identified on one boundary circle by means of a fixed point free involution
		+	(3) a flat $Homeo\left(I\right)$-bundle with monodromy $\rho:{\mathbb Z}\rightarrow Homeo\left(I\right)$ such that $\rho\left(1\right)$ is orientation-reversing.{{endthm}}
		+
	</wikitex>		</wikitex>

---

Revision as of 15:52, 7 June 2010

Contents 1 Introduction 2 Construction and examples 2.1 Bundles 2.2 Submersions 2.3 Constructing new foliations from old ones 2.3.1 Pullbacks 2.3.2 Glueing 2.3.3 Turbulization 3 Classification/Characterization 3.1 Codimension one foliations 3.1.1 Foliations of surfaces 4 Further discussion 5 References

1 Introduction

Let $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; 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 $q$ foliation of $M$ if $M$ admits an atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is charts (diffeomorphisms) $$\phi_\alpha:U_\alpha\rightarrow B_{\alpha,\tau}\times B_{\alpha,\pitchfork}\subset {\mathbb R}^{n-q}\times{\mathbb R}^q$$ 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> == Construction and examples == === 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 $Diff\left(F\right)$-bundles. Given a representation $\pi_1B\rightarrow Diff\left(F\right)$, the flat $Diff\left(F\right)$-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 $\rho$. A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers. 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 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 leaf space is (diffeomeorphic) homeomorphic to $B$. </wikitex> === Submersions === <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. 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 ${\mathbb Z}$-action given by $z\left(x,y\right)=\left(x,y+z\right)$ for $\left(x,y\right)\in\left[[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)=\left[-1,1\right]\times S^1$ is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff. </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}} </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 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 neighborhod 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 $\mathcal{F}$. == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == === Codimension one foliations === <wikitex>; A necessary condition for the existence of a codimension one foliation on a manifold $M$ is the vanishing of the Euler characteristic $\chi\left(M\right)=0$. </wikitex> ==== Foliations of surfaces ==== <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}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 $q$ foliation of $M$ if $M$ admits an atlas $\left\{U_\alpha,\phi_\alpha\right\}_{\alpha\in {\mathcal{A}}}$ of foliated charts, that is charts (diffeomorphisms) 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 Construction and examples

2.1 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 $Diff\left(F\right)$-bundles. Given a representation $\pi_1B\rightarrow Diff\left(F\right)$, the flat $Diff\left(F\right)$-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 $\rho$. A flat bundle has a foliation by fibres and it also has a foliation transverse to the fibers.

Flat bundles fit into the frame work of fiber bundles. A (smooth) map 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 diffeomorphism $\phi:\pi^{-1}\left(U\right)\rightarrow U\times F$ making the following diagram (with $p_1$ projection to the first factor) commutative:

The fiber bundle yields a foliation by fibers $F_b:=\pi^{-1}\left(\left\{b\right\}\right), b\in B$. Its leaf space is (diffeomeorphic) homeomorphic to $B$.

2.2 Submersions

Let 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. An example of a submersion, which is not a fiber bundle, is given by

This submersion yields a foliation of $\left[-1,1\right]\times{\mathbb R}$ which is invariant under the ${\mathbb Z}$-action given by $z\left(x,y\right)=\left(x,y+z\right)$ for $\left(x,y\right)\in\left[[-1,1\right]\times{\mathbb R}, z\in{\mathbb Z}$. The induced foliation of ${\mathbb Z}\backslash \left(\left[-1,1\right]\times{\mathbb R}\right)=\left[-1,1\right]\times S^1$ is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff.

2.3 Constructing new foliations from old ones

2.3.1 Pullbacks

[Candel&Conlon2000]

2.3.2 Glueing

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 called the tangential resp. the transversal glueing of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$.

2.3.3 Turbulization

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

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

The foliations ${\mathcal{F}}\mid_{M\setminus N\left(\gamma\left(S^1\right)\right)}$ and $\mathcal{F}_0$ agree on a neighborhod 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 $\mathcal{F}$.

Invariants

...

3 Classification/Characterization

3.1 Codimension one foliations

A necessary condition for the existence of a codimension one foliation on a manifold $M$ is the vanishing of the Euler characteristic $\chi\left(M\right)=0$.

3.1.1 Foliations of surfaces

Codimension one foliations exist only on surfaces $S$ with $\chi\left(S\right)=0$, that is on the Torus, the Klein bottle, the annulus and the M\"obius band.

4 Further discussion

...

5 References

• [Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001

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