Foliation
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Contents |
[edit] 1 Introduction
This page gives the definition of the term foliation. For further information, see the page Foliations and [Godbillon1991].
[edit] 1.1 Foliations
[edit] 1.2 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 .
[edit] 1.3 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 .
[edit] 1.4 Holonomy Cocycle
A smooth foliation is said to be transversely orientable if everywhere.
[edit] 2 Special classes of foliations
[edit] 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 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 . ( is a flat -bundle if .)
Flat bundles fit into the frame work of fiber bundles. A (smooth) mapThe fiber bundle yields a foliation by fibers . Its space of leaves is (diffeomeorphic) homeomorphic to , in particular is a Hausdorff manifold.
[edit] 2.2 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.
[edit] 2.3 Submersions
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.
[edit] 2.4 Reeb foliations
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
[edit] 2.5 Taut foliations
A codimension one foliation of is taut if for every leaf of there is a circle transverse to which intersects .
[edit] 3 References
- [Godbillon1991] C. Godbillon, Feuilletages, Birkhäuser Verlag, 1991. MR1120547 (93i:57038) Zbl 0724.58002
[edit] 4 External links
- The Encylopedia of Mathematics article on foliations
- The Wikipedia page about foliations