- 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, thesestripe s 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 function s $varphi\_\{ij\}:mathbb\{R\}^n\; omathbb\{R\}^n$ 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,:$varphi\_\{ij\}^1:mathbb\{R\}^\{n-p\}\; omathbb\{R\}^\{n-p\}$ and :$varphi\_\{ij\}^2:mathbb\{R\}^n\; omathbb\{R\}^\{p\}$. 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 $n-p$ dimensionalsubmanifold s. These submanifolds piece together from chart to chart to form maximal connected injectivelyimmersed submanifold s called the**leaves**of the foliation.If we shrink the chart $U\_i$ it can be written in the form $U\_\{ix\}\; imes\; U\_\{iy\}$ where $U\_\{ix\}subsetmathbb\{R\}^\{n-p\}$ and $U\_\{iy\}subsetmathbb\{R\}^p$ and $U\_\{iy\}$ is isomorphic to the plaques and the points of $U\_\{ix\}$ parametrize the plaques in $U\_i$. If we pick a $y\_0in\; U\_\{iy\}$, $U\_\{ix\}\; imes\{y\_0\}$ 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

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

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.

**ubmersions**If $M^n\; o\; N^q$ (where $q\; leq\; n$) is a

submersion of manifolds, it follows from theinverse 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 asubgroup obtained by exponentiating a closedsubalgebra of theLie algebra of $G$, then $G$ is foliated bycoset s 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 orfree 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 field s: given a vector field $X$on $M$ that is never zero, itsintegral curve s 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 thenecessary and sufficient conditions for a distribution (i.e. an $n-p$ dimensionalsubbundle of thetangent 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 underLie bracket . One can also phrase this differently, as a question ofreduction of the structure group of thetangent 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 requiredblock 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 anorientable compact surface only for thetorus . This is a consequence of thePoincaré-Hopf index theorem , which shows theEuler 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