Foliation

The foliation (French Feuilletage closely. Foliation ) of a manifold is a concept from the mathematical branch of differential topology. The topological theory of foliations was founded mainly by Georges Reeb.

One -dimensional foliation of a manifold is a decomposition of into disjoint, path-connected quantities at any point to look locally as a layering parallel -dimensional submanifolds. The elements are called the leaves of; the leaves are not necessarily complete or even compact.

Definition

Let be a smooth manifold. A partition of path-connected into disjoint sets is called foliation of when an atlas exists (ie is an open cover and which are diffeomorphisms ), so that the image of every non-empty connected component is mapped from taking in a plane. The elements are called the leaves of

Examples

• Be a non-vanishing vector field on, then form the flux lines of a one-dimensional foliation.
• In general, leaves form a global no submanifold. On the torus, consider the constant vector field. Each flow line winds tightly around the torus. Thus, the topology of such a sheet does not match the topology of match (This is also an example that not every subgroup of a Lie group is a Lie subgroup ).
• Be a fiber bundle, then a foliation.
• More generally, is a submersion, then a foliation. An example of a submersion, which is not a fiber bundle. This provides a foliation of, invariant under translation, the induced foliation on the 2-dimensional orientable Reeb - foliation. Furthermore, the foliation is invariant under also, in this case, the induced foliation on the Möbius band, the 2-dimensional non- orientable Reeb - foliation.
• Be a homeomorphism of a manifold M, then the Abbildungstorus of f has a foliation transverse to the fibers, the so-called suspension foliation.
• The Hopf fibration is a foliation in circles. It follows from the theorem of Vogt that also has a foliation into circles.

Integrability

In the above examples, a partition has not been directly determined, but instead only one direction was specified at each point, and it raised the question whether there is a foliation such that each leaf is tangent to the given direction at each point. Often found in practice in similar situations: on a manifold one -dimensional distribution is given. This is one-dimensional sub- bundle of the tangent space. Whether there is a distribution to this foliation, which is tangential to can often be answered by using the Frobenius theorem.

The Lie bracket of two vector fields, which are defined on a manifold produces again a vector field on the manifold. Since each leaf of a foliation locally has the form of a submanifold, then, that for any two vector fields that are tangent to ( and which must be defined only on this sheet ) also is tangent again follows. The Frobenius theorem, however, also implies the reverse direction.

Frobenius theorem (after Ferdinand Georg Frobenius ): exists to a -dimensional distribution if and only tangential to one -dimensional foliation if for any vector fields that lie in whose Lie bracket again forms a section in.

A topological obstruction to integrability of distributions provides the set of Bott.

Set of Bott ( by Raoul Bott ): If one -dimensional distribution has a tangential -dimensional foliation, then disappears from the Pontryagin classes of ring generated in dimensions.

Existence theorem

Set of Thurston (after William Thurston ): A closed smooth n- dimensional manifold has exactly then a smooth (n-1 )-dimensional foliation if its Euler characteristic is zero. If the Euler characteristic is zero, then each (n-1 )-dimensional hyperplane field is homotopic to a smooth foliation Tangentialebenenfeld.

Taut foliations

An elaborate structure theory there is in codimension one, especially for tight foliations. These do not contain any Reeb foliations and there is a Riemannian metric so that all leaves are minimal surfaces.

Foliations of surfaces

If F is a closed surface leafed, then F is either a torus or a Klein bottle and the foliation is either the suspension foliation of a homeomorphism or it consists of several ( orientable or non- orientable ) Reeb foliations.

Foliations of 3-manifolds

Codimension 1

Set of Novikov - Zieschang ( according to Sergei Novikov and Heiner Zieschang ): If there is on a closed, orientable 3-manifold is a 2- dimensional foliation without Reeb components, then, and all the leaves are incompressible.

Set of Palmeira: If there is on a closed, orientable 3-manifold is a 2- dimensional foliation without Reeb components, then the universal covering is diffeomorphic to the highly sophisticated and foliation is a foliation of diffeomorphic through leaves to.

Set of Gabai ( by David Gabai ): with Let M be a closed, irreducible 3-manifold, then there is M on a 2-dimensional foliation without Reeb components.

Codimension 2

Set of Epstein ( by David Epstein ): Any foliation of a compact 3-manifold by circles is a Seifert fibration.

Set of Vogt ( by Elmar Vogt ): If a 3-manifold has a foliation by circles, then each contributes by removing finitely many points from a resulting manifold (not necessarily differentiable ) foliation by circles.

Invariants of foliations

• Godbillon - Vey invariant
• Base -like cohomology