Foliations
(^^) |
|||
Line 48: | Line 48: | ||
{{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}} | {{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|Conlon I}} | {{cite|Conlon I}} | ||
+ | |||
+ | ==== 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 $\mathcal{F}$. | ||
Revision as of 14:54, 7 June 2010
Contents |
1 Introduction
2 Construction and examples
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. 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 . 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) mapThe fiber bundle yields a foliation by fibers . Its leaf space is (diffeomeorphic) homeomorphic to .
2.2 Submersions
This submersion yields a foliation of which is invariant under the -action given by for . The induced foliation of is called the 2-dimensional Reeb foliation. Its leaf space is not Hausdorff.
2.3 Constructions
2.3.1 Pullbacks
[Conlon I]
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 neighborhod of the boundary of . The result of glueing these foliations is called the turbulization of .
Invariants
...
3 Classification/Characterization
...
4 Further discussion
...
5 References
This page has not been refereed. The information given here might be incomplete or provisional. |