Foliation
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In mathematics, a foliation is a geometric device used to study manifolds. Informally speaking, a foliation is a kind of "clothing" worn on a manifold, cut from a stripy fabric. On each sufficiently small piece of the manifold, these stripes give the manifold a local product structure. This product structure does not have to be consistent outside local patches (i. e. well-defined globally): a stripe followed around long enough might return to a different, nearby stripe.
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[edit] Definition
More formally, a dimension p foliation F of an n-dimensional manifold M is a covering by charts Ui together with maps
such that on the overlaps the transition functions defined by
take the form
where x denotes the first n − p co-ordinates, and y denotes the last p co-ordinates. That is,
and
- .
In the chart Ui, the stripes x = constant match up with the stripes on other charts Uj. Technically, these stripes are called plaques of the foliation. In each chart, the plaques are n − p dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.
[edit] Examples
[edit] Flat space
Consider an n-dimensional space, foliated as a product by subspaces consisting of points whose first n − p 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 n = 3 and p = 1: the two-dimensional leaves of a book are enumerated by a (one-dimensional) page number.
[edit] Covers
If is a covering between manifolds, and F is a foliation on N, then it pulls back to a foliation on M. 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.
[edit] Lie groups
If G is a Lie group, and H is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of G, then G is foliated by cosets of H.
[edit] Foliations and integrability
There is a close relationship, assuming everything is smooth, with vector fields: given a vector field X on M that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension n − 1 foliation).
This observation generalises to a theorem of Ferdinand Georg Frobenius (the Frobenius theorem), saying that the necessary and sufficient conditions for a distribution (i.e. an n − p dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, are 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 GL(n) 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.
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.
[edit] See also
- G-structure
- Classifying space for foliations
- Reeb foliation
- Taut foliation