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Foliation
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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.
formally, a dimension foliation of an -dimensional manifold
is a covering by charts together with maps
such that on the overlaps the transition functions defined by
take the form
where denotes the first co-ordinates, and denotes the last p co-ordinates.
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Encyclopedia
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.
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 functions 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 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 a -manifold may be thought of as simply a collection of pairwise-disjoint, connected -dimensional sub-manifolds (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 preimages in are -dimensional affine subspaces whose last coordinates are constant.
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.
Submersions
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.
Lie groups
If is a Lie group, and is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of , then is foliated by cosets of .
Lie group actions
Let be a Lie group acting smoothly on a manifold . If the action is a locally free action or free action, then the orbits of define a foliation of .
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, 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 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.
Existence of foliations
gave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution. showed that any compact manifold with a distribution has a foliation of the same dimension.
See also
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