Foliation
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1 Introduction
1 Foliations
Let be an -manifold, possibly with boundary, and let be a decomposition of into connected, topologically immersed submanifolds of dimension . is said to be a codimension (smooth) foliation of if admits an (smooth) atlas of foliated charts, that is (diffeomorphisms) homeomorphisms2 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
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) mapTex 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
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
A codimension one foliation of is taut if for every leaf of there is a circle transverse to which intersects .
Theorem 3.1 (Rummler, Sullivan). The following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifolds :
a) is taut;
b) there is a flow transverse to which preserves some volume form on ;
c) there is a Riemannian metric on for which the leaves of are least area surfaces.
2 Constructing new foliations from old ones
2.1 Pullbacks
[Candel&Conlon2000], Theorem 3.2.2
2.2 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 is called the tangential resp. the transversal glueing of and .
2.3 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 neighborhood of the boundary of . The result of glueing these foliations is called the turbulization of .
3 References
- [Candel&Conlon2000] A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000. MR1732868 (2002f:57058) Zbl 0936.57001