Foliations
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== Introduction == | == Introduction == | ||
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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 $q$ 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 $\phi_\alpha^{-1}\left(B_{\alpha,\tau}\times\left\{y\right\}\right)$. |
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Revision as of 14:12, 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]
Invariants
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3 Classification/Characterization
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4 Further discussion
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5 References
This page has not been refereed. The information given here might be incomplete or provisional. |