Foliation
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Contents |
1 Introduction
1 Foliations
LetTex syntax errorbe an -manifold, possibly with boundary, and let be a decomposition of
Tex syntax errorinto connected, topologically immersed submanifolds of dimension . is said to be a codimension (smooth) foliation of
Tex syntax errorif
Tex syntax erroradmits an (smooth) atlas of foliated charts, that is (diffeomorphisms) homeomorphisms
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 .
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 .
4 Holonomy Cocycle
The holonomy cocycle of the atlas is given byA smooth foliation is said to be transversely orientable if everywhere.
5 Special classes of foliations
5.1 Bundles
Tex syntax erroris 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 . (
Tex syntax erroris a flat -bundle if .) Flat bundles fit into the frame work of fiber bundles. A (smooth) map
Tex syntax error, in particular is a Hausdorff manifold.
1.1 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.
1.2 Submersions
Tex syntax erroris foliated by the preimages . This includes the case of fiber bundles.
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.
1.3 Reeb foliations
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
1.4 Taut foliations
Tex syntax erroris taut if for every leaf of there is a circle transverse to which intersects .