Foliations
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{{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}} | ||
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Revision as of 15:37, 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 Constructing new foliations from old ones
2.3.1 Pullbacks
2.3.2 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 called the tangential resp. the transversal glueing of and .
2.3.3 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 neighborhod of the boundary of . The result of glueing these foliations is called the turbulization of .
Invariants
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3 Classification/Characterization
3.1 Codimension one foliations
A necessary condition for the existence of a codimension one foliation on a manifold is the vanishing of the Euler characteristic .
3.1.1 Foliations of surfaces
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4 Further discussion
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5 References
- [Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001
This page has not been refereed. The information given here might be incomplete or provisional. |