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Learn more about "Foliation"

Foliation

In mathematics, a '''foliation''' is a geometric device used to study manifold (mathematics)|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, 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.

Definition

More formally, a dimension

p

foliation

F

of an

n

-dimensional manifold

M

is a covering by chart (topology)|charts

U_i

together with maps
-

\phi_i:U_i \to \R^n

such that on the overlaps

U_i \cap U_j

the transition functions

\varphi_{ij}:\mathbb{R}^n\to\mathbb{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}\to\mathbb{R}^{n-p}

and
-

\varphi_{ij}^2:\mathbb{R}^n\to\mathbb{R}^{p}

. In the chart

U_i

, the '''stripes'''

x=

mathematical constant|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 submanifolds. These submanifolds piece together from chart to chart to form maximal connected space|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

p

-dimensional foliation of a

n

-manifold

M

may be thought of as simply a collection

M_{a}

of pairwise-disjoint, connected

p

-dimensional sub-manifolds (the leaves of the foliation) of

M

, such that for every point

x

M

, there is a chart

(U,\phi)

with

U

homeomorphic to

\mathbb{R}^{n}

containing

x

such that for every leaf

M_{a}

M_{a}

meets

U

in either the empty set or a countable collection of subspaces whose preimages in

U

are

p

-dimensional affine subspaces whose last

n-p

coordinates are constant. If we shrink the chart

U_i

it can be written in the form

U_{ix}\times U_{iy}

where

U_{ix}\subset\mathbb{R}^{n-p}

and

U_{iy}\subset\mathbb{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_0\in U_{iy}

U_{ix}\times\{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#monodromy groupoid|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}\times \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

M \to 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 (mathematics)|locus is transverse to the foliation, then the foliation can be pulled back.

Submersions

M^n \to N^q

(where

q \leq n

) 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

q

foliation of

M

. Fiber bundles are an example of this type.

Lie groups

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

Lie group actions

Let

G

be a Lie group acting smoothly on a manifold

M

. If the action is a locally free action or free action, then the orbits of

G

define a foliation of

M

Foliations and integrability

There is a close relationship, assuming everything is smooth function|smooth, with vector fields: given a vector field

X

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 (differential topology)|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.

References

- Lawson, H. Blaine, "Foliations"
- I.Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs) Category:Foliations| Category:Structures on manifolds

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