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, 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.

Contents

Definition

Examples

Flat space

Covers

Lie groups

Foliations and integrability

See also

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 functions defined by
:
take the form
:
where $x$ denotes the first $n-p$ co-ordinates, and $y$ denotes the last ''p'' co-ordinates. That is,
:
and
:.
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
:$mathbb\{R\}^n=mathbb\{R\}^\{n-p\}\; imes\; mathbb\{R\}^\{p\}$
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.

Lie groups

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 $H$.

Foliations and integrability

There is a close relationship, assuming everything is smooth, with vector fields: given a vector field $X$
on $M$ 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.

See also

★ G-structure

★ Classifying space for foliations

★ Reeb foliation

★ Taut foliation