# Dictionary Definition

foliation

### Noun

1 (botany) the process of forming leaves [syn:
leafing]

2 (geology) the arrangement of leaflike layers in
a rock

3 (architecture) leaf-like architectural ornament
[syn: foliage]

4 the production of foil by cutting or beating
metal into thin leaves

5 the work of coating glass with metal foil

# User Contributed Dictionary

### Noun

- The process of forming into a leaf or leaves.
- The manner in which the young leaves are disposed within the bud.
- The act of beating a metal into a thin plate, leaf, foil, or lamina.
- The act of coating with an amalgam of tin foil and quicksilver, as in making looking-glasses.
- The enrichment of an opening by means of foils, arranged in trefoils, quatrefoils, etc.; also, one of the ornaments.
- The property, possessed by some crystalline rocks, of dividing into plates or slabs, which is due to the cleavage structure of one of the constituents, as mica or hornblende. It may sometimes include slaty structure or cleavage, though the latter is usually independent of any mineral constituent, and transverse to the bedding, it having been produced by pressure.
- A set of submanifolds of a given manifold, each of which is of lower dimension than it, but which, taken together, are coextensive with it.

#### Translations

topology: a set of subspaces coextensive with a
manifold

- Chinese: 叶层结构 (yè céng jiégòu)
- French: foliation, feuilletage
- German: Blätterung
- Portuguese: folheação, folheações
- Russian: листоватость
- Spanish: foliación

### See also

# Extensive Definition

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.

## 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 \to \R^n

such that on the overlaps U_i \cap U_j the
transition
functions \varphi_:\mathbb^n\to\mathbb^n defined by

- \varphi_ =\phi_j \phi_i^

take the form

- \varphi_(x,y) = (\varphi_^1(x),\varphi_^2(x,y))

where x denotes the first n-p co-ordinates, and y
denotes the last p co-ordinates. That is,

- \varphi_^1:\mathbb^\to\mathbb^

- \varphi_^2:\mathbb^p\to\mathbb^.

If we shrink the chart U_i it can be written in
the form U_\times U_ where U_\subset\mathbb^ and U_\subset\mathbb^p
and U_ is isomorphic to the plaques and the points of U_
parametrize the plaques in U_i. If we pick a y_0\in U_, U_\times\
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
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^n=\mathbb^\times \mathbb^

with the leaves or plaques \mathbb^ being
enumerated by \mathbb^. 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 \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.### Submersions

If 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

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.### 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, 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.

## References

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

foliation in German: Blätterung

foliation in Spanish: Foliación

foliation in Chinese: 叶层结构

# Synonyms, Antonyms and Related Words

accounting, census, counting, dactylonomy, delamination, desquamation, enumeration, exfoliation, flakiness, foliage, frondage, furfuration, inventorying, lamellation, lamination, leafage, leafiness, measurement, numbering, numeration, pagination, quantification, quantization, scaliness, stratification, tallying, telling, umbrage