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In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension.
such that for overlapping pairs the transition functions defined by
take the form
where denotes the first coordinates, and denotes the last co-ordinates. That is,
In the chart , the stripes constant match up with the stripes on other charts . Technically, these stripes are called plaques of the foliation. In each chart, the plaques are dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.
The notion of leaves allows for a more intuitive way of thinking about a foliation. A -dimensional foliation of an -manifold may be thought of as simply a collection of pairwise-disjoint, connected, immersed -dimensional submanifolds (the leaves of the foliation) of , such that for every point in , there is a chart with homeomorphic to containing such that for every leaf , meets in either the empty set or a countable collection of subspaces whose images under in are -dimensional affine subspaces whose first coordinates are constant.
If we shrink the chart it can be written in the form , where
- and ,
and is homeomorphic to the plaques and the points of parametrize the plaques in . If we pick a
is a submanifold of that intersects every plaque exactly once. This is called a local transversal section of the foliation. Note that due to monodromy there might not exist global transversal sections of the foliation.
Consider an -dimensional space, foliated as a product by subspaces consisting of points whose first co-ordinates are constant. This can be covered with a single chart. The statement is essentially that
with the leaves or plaques being enumerated by . The analogy is seen directly in three dimensions, by taking and : the two-dimensional leaves of a book are enumerated by a (one-dimensional) page number.
A rather trivial example of foliations are products , foliated by the leaves . (Another foliation of M is given by .)
A more general class are flat G-bundles with for a manifold F. Given a representation , the flat -bundle with monodromy is given by , where acts on the universal cover by deck transformations and on F by means of the representation .
Flat bundles fit into the frame work of fiber bundles. A map between manifolds is a fiber bundle if there is a manifold F such that each has an open neighborhood U such that there is a homeomorphism with , with projection to the first factor. The fiber bundle yields a foliation by fibers . Its space of leaves L is homeomorphic to B, in particular L is a Hausdorff manifold.
If is a covering between manifolds, and is a foliation on , then it pulls back to a foliation on . More generally, if the map is merely a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back.
If (where ) is a submersion of manifolds, it follows from the inverse function theorem that the connected components of the fibers of the submersion define a codimension foliation of . Fiber bundles are an example of this type.
An example of a submersion, which is not a fiber bundle, is given by
This submersion yields a foliation of which is invariant under the -actions given by
for . The induced foliations of are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliation (of the Möbius band). Their leaf spaces are not Hausdorff.
Define a submersion
where are cylindrical coordinates on the n-dimensional disk . This submersion yields a foliation of which is invariant under the -actions given by
for . The induced foliation of is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.
For n = 2, this gives a foliation of the solid torus which can be used to define the Reeb foliation of the 3-sphere by gluing two solod tori along their boundary. Foliations of odd-dimensional spheres are also explicitly known.
Lie group actions
The set of lines on the torus T = R2/Z2 with the same slope θ forms a foliation. The leaves are obtained by projecting straight lines of slope θ in the plane R2 onto the torus. If the slope is rational then all leaves are closed curves homeomorphic to the circle, while if it is irrational, the leaves are noncompact, homeomorphic to the real line, and dense in the torus (cf Irrational rotation). The irrational case is known as the Kronecker foliation, after Leopold Kronecker. A similar construction using a foliation of Rn by parallel lines yields a one-dimensional foliation of the n-torus Rn/Zn associated with the linear flow on the torus.
A flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are
where is the canonical projection. This foliation is called the suspension of the representation .
In particular, if and is a homeomorphism of F, then the suspension foliation of is defined to be the suspension foliation of the representation given by . Its space of leaves is , where whenever for some .
The Kronecker foliations of the 2-torus are the suspension foliations of the rotations by angle .
Foliations and integrability
There is a close relationship, assuming everything is smooth, with vector fields: given a vector field on that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension foliation).
This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution (i.e. an dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under Lie bracket. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from to a reducible subgroup.
The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist. For example, in the codimension 1 case, we can define the tangent bundle of the foliation as , for some (non-canonical) (i.e. a non-zero co-vector field). A given is integrable iff everywhere.
There is a global foliation theory, because topological constraints exist. For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0. There are many deep connections with contact topology, which is the "opposite" concept.
Existence of foliations
Haefliger (1970) gave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution. Thurston (1974, 1976) showed that any compact manifold with a distribution has a foliation of the same dimension.
- Classifying space for foliations
- Haefliger structure, a generalization of foliations closed under taking pullbacks.
- Reeb foliation of the 3-sphere.
- Taut foliation
- Durfee: Foliations of Odd-Dimensional Spheres. Annals of Mathematics, Second Series, Vol. 96, No. 2 (Sep., 1972), pp. 407-411.
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- Foliations at the Manifold Atlas