- 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 p foliation F of an n-dimensional manifold M is a covering by charts U_i together with maps:phi_i:U_i o R^n
such that on the overlaps U_i cap U_j the
transition function s varphi_{ij}:mathbb{R}^n omathbb{R}^n defined by:varphi_{ij} =phi_j phi_i^{-1}
take the form
:varphi_{ij}(x,y) = (varphi_{ij}^1(x),varphi_{ij}^2(x,y))
where x denotes the first n-p co-ordinates, and y denotes the last "p" co-ordinates. That is,:varphi_{ij}^1:mathbb{R}^{n-p} omathbb{R}^{n-p} and :varphi_{ij}^2:mathbb{R}^n omathbb{R}^{p}. In the chart U_i, the stripes x= constant match up with the stripes on other charts U_j. Technically, these stripes are called plaques of the foliation. In each chart, the plaques are n-p 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 U_i it can be written in the form U_{ix} imes U_{iy} where U_{ix}subsetmathbb{R}^{n-p} and U_{iy}subsetmathbb{R}^p and U_{iy} is isomorphic to the plaques and the points of U_{ix} parametrize the plaques in U_i. If we pick a y_0in U_{iy}, U_{ix} imes{y_0} is a submanifold of U_i 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 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
:mathbb{R}^n=mathbb{R}^{n-p} imes mathbb{R}^{p}
with the leaves or plaques mathbb{R}^{n-p} being enumerated by mathbb{R}^{p}. 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.
Covers
If M o N 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.
ubmersions
If M^n o N^q (where q leq n ) is a
submersion of manifolds, it follows from theinverse function theorem that the connected components of the fibers of the submersion define a codimension q foliation of M .Fiber bundles are an example of this type.Lie groups
If G is a
Lie group , and H is asubgroup obtained by exponentiating a closedsubalgebra of theLie algebra of G, then G is foliated bycoset s of H.Lie group actions
Let G be a Lie group acting smoothly on a manifold M . If the action is a
locally free action orfree action , then the orbits of G define a foliation of M .Foliations and integrability
There is a close relationship, assuming everything is smooth, with
vector field s: given a vector field Xon M that is never zero, itsintegral curve s 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 thenecessary and sufficient conditions for a distribution (i.e. an n-p 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 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 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)
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