- Foliation
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 striped fabric. On each sufficiently small piece of the manifold, thesestripe s 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.Definition
More formally, a
dimension foliation of an -dimensional manifold is a covering by charts together with maps:
such that on the overlaps the
transition function s defined by:
take the form
:
where denotes the first co-ordinates, and denotes the last "p" co-ordinates. That is,: and :. 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
submanifold s. These submanifolds piece together from chart to chart to form maximal connected injectivelyimmersed submanifold s called the leaves of the foliation.If we shrink the chart it can be written in the form where and and is isomorphic 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.Examples
Flat space
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.
Covers
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.
ubmersions
If (where ) is a
submersion of manifolds, it follows from theinverse 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.Lie groups
If is a
Lie group , and is asubgroup obtained by exponentiating a closedsubalgebra of theLie algebra of , then is foliated bycoset s of .Lie group actions
Let be a Lie group acting smoothly on a manifold . If the action is a
locally free action orfree action , then the orbits of define a foliation of .Foliations and integrability
There is a close relationship, assuming everything is smooth, with
vector field s: given a vector field on that is never zero, itsintegral curve s will give a 1-dimensional foliation. (i.e. a codimension foliation).This observation generalises to a theorem of
Ferdinand Georg Frobenius (the Frobenius theorem), saying that thenecessary and sufficient conditions for a distribution (i.e. an dimensionalsubbundle of thetangent 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 underLie bracket . One can also phrase this differently, as a question ofreduction of the structure group of thetangent 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 requiredblock 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 anorientable compact surface only for thetorus . This is a consequence of thePoincaré-Hopf index theorem , which shows theEuler characteristic will have to be 0.ee also
*
G-structure
*Classifying space for foliations
*Reeb foliation
*Taut foliation References
*Lawson, H. Blaine, [http://www.ams.org/bull/1974-80-03/S0002-9904-1974-13432-4/S0002-9904-1974-13432-4.pdf "Foliations"]
*I.Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs)
Wikimedia Foundation. 2010.