Foliation

For other senses of the term (such as in geology), see foliation (disambiguation).

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 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.

Definition

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 $\varphi_{ij}:\mathbb{R}^n\to\mathbb{R}^n$ 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=

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 injectively immersed submanifolds called the leaves 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 is transverse to the foliation, then the foliation can be pulled back.

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

Foliations and integrability

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 Poincaré-Hopf index theorem, which shows the Euler characteristic will have to be 0.