foliation

Other Definitions
foliation (dict)

In mathematics, 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. More formally, a codimension

p

foliation

F

of an

n

-dimensional manifold

M

is a covering by 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}

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. 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 submanifolds. These submanifolds piece together from chart to chart to form maximal connected submanifolds called the leaves of the foliation. Example:

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. Example: If

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 is transverse to the foliation, then the foliation can be pulled back. Example: 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

. There is a close relationship, assuming everything is 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), 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 Poincar-Hopf index theorem, which shows the Euler characteristic will have to be 0.

Foliation

See also