# Foliation

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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.

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 $varphi_\left\{ij\right\}:mathbb\left\{R\right\}^n omathbb\left\{R\right\}^n$ defined by

:$varphi_\left\{ij\right\} =phi_j phi_i^\left\{-1\right\}$

take the form

:$varphi_\left\{ij\right\}\left(x,y\right) = \left(varphi_\left\{ij\right\}^1\left(x\right),varphi_\left\{ij\right\}^2\left(x,y\right)\right)$

where $x$ denotes the first $n-p$ co-ordinates, and $y$ denotes the last "p" co-ordinates. That is,:$varphi_\left\{ij\right\}^1:mathbb\left\{R\right\}^\left\{n-p\right\} omathbb\left\{R\right\}^\left\{n-p\right\}$ and :$varphi_\left\{ij\right\}^2:mathbb\left\{R\right\}^n omathbb\left\{R\right\}^\left\{p\right\}$. 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.

If we shrink the chart $U_i$ it can be written in the form $U_\left\{ix\right\} imes U_\left\{iy\right\}$ where $U_\left\{ix\right\}subsetmathbb\left\{R\right\}^\left\{n-p\right\}$ and $U_\left\{iy\right\}subsetmathbb\left\{R\right\}^p$ and $U_\left\{iy\right\}$ is isomorphic to the plaques and the points of $U_\left\{ix\right\}$ parametrize the plaques in $U_i$. If we pick a $y_0in U_\left\{iy\right\}$, $U_\left\{ix\right\} imes\left\{y_0\right\}$ 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

:$mathbb\left\{R\right\}^n=mathbb\left\{R\right\}^\left\{n-p\right\} imes mathbb\left\{R\right\}^\left\{p\right\}$

with the leaves or plaques $mathbb\left\{R\right\}^\left\{n-p\right\}$ being enumerated by $mathbb\left\{R\right\}^\left\{p\right\}$. 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.

ubmersions

If $M^n o N^q$ (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 $GL\left(n\right)$ 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.

ee also

*G-structure
*Classifying space for foliations
*Reeb foliation
*Taut foliation

References

*Lawson, H. Blaine, [http://www.ams.org/bull/1974-80-03/S0002-9904-1974-13432-4/S0002-9904-1974-13432-4.pdf "Foliations"]
*I.Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs)

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### Look at other dictionaries:

• foliation — [ fɔljasjɔ̃ ] n. f. • 1757; du lat. folium « feuille » 1 ♦ Bot. Disposition des feuilles sur la tige. Développement des feuilles. ⇒ feuillaison. 2 ♦ Géol. Structure en feuillets observée dans certaines roches. ● foliation nom féminin (latin… …   Encyclopédie Universelle

• Foliation — Fo li*a tion, n. [Cf. F. foliation.] 1. The process of forming into a leaf or leaves. [1913 Webster] 2. The manner in which the young leaves are dispo?ed within the bud. [1913 Webster] The . . . foliation must be in relation to the stem. De… …   The Collaborative International Dictionary of English

• foliation — (n.) 1620s, from Fr. foliation or directly from L. foliatus (see FOLIATE (Cf. foliate)) …   Etymology dictionary

• Foliation — (v. lat.), die Belaubung, der Act, wodurch eine Pflanze Blätter erhält, od. auch die Gesammtheit der Blätter. Foliatus, beblättert …   Pierer's Universal-Lexikon

• Foliation — Foliation, lat. deutsch, das Ausschlagen der Bäume …   Herders Conversations-Lexikon

• foliation — [fō΄lē ā′shən] n. [ML foliatio: see FOLIATE] 1. a growing of or developing into a leaf or leaves; leaf formation 2. the state of being in leaf 3. the way leaves are arranged in the bud; vernation 4. the act or process of beating metal into layers …   English World dictionary

• foliation — /foh lee ay sheuhn/, n. 1. the act or process of putting forth leaves. 2. the state of being in leaf. 3. Bot. a. the arrangement of leaves within a bud. b. the arrangement of leaves on a plant. 4. leaves or foliage. 5. Print. the consecutive… …   Universalium

• Foliation — La foliation (du latin folium, feuille) est une structuration en plans distincts des roches métamorphiques. La structure est marquée par l orientation préférentielle de minéraux visibles à l œil nu le plus souvent les micas et aussi en… …   Wikipédia en Français

• FOLIATION — n. f. T. de Botanique Disposition des feuilles autour de la tige. Il se dit aussi du Moment où les bourgeons commencent à développer leurs feuilles. L’époque de la foliation. Dans ce sens il est synonyme de FEUILLAISON …   Dictionnaire de l'Academie Francaise, 8eme edition (1935)

• foliation — sluoksniavimasis statusas T sritis chemija apibrėžtis Išsiskirstymas sluoksniais. atitikmenys: angl. delamination; exfoliation; foliation; lamination; settling; splitting; stratification rus. расслаивание; расслоение …   Chemijos terminų aiškinamasis žodynas

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