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-definedglobally): a stripe followed around long enough might return to a different, nearby stripe.
More formally, a
dimensionfoliation of an -dimensional manifold is a covering by charts together with maps
such that on the overlaps the
transition functions defined by
take the form
where denotes the first co-ordinates, and denotes the last "p" co-ordinates. That is,: and :. In the chart , the stripes constant match up with the stripes on other charts . Technically, these stripes are called plaques of the foliation. In each chart, the plaques are 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 it can be written in the form where and and is isomorphic to the plaques and the points of parametrize the plaques in . If we pick a , is a submanifold of 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.
Consider an -dimensional space, foliated as a product by subspaces consisting of points whose first co-ordinates are constant. This can be covered with a single chart. The statement is essentially that
with the leaves or plaques being enumerated by . The analogy is seen directly in three dimensions, by taking and : the two-dimensional leaves of a book are enumerated by a (one-dimensional) page number.
If is a covering between manifolds, and is a foliation on , then it pulls back to a foliation on . 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.
If (where ) is a
submersionof manifolds, it follows from the inverse function theoremthat the connected components of the fibers of the submersion define a codimension foliation of . Fiber bundlesare an example of this type.
If is a
Lie group, and is a subgroupobtained by exponentiating a closed subalgebraof the Lie algebraof , then is foliated by cosets of .
Lie group actions
Let be a Lie group acting smoothly on a manifold . If the action is a
locally free actionor free action, then the orbits of define a foliation of .
Foliations and integrability
There is a close relationship, assuming everything is smooth, with
vector fields: given a vector field on that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension foliation).
This observation generalises to a theorem of
Ferdinand Georg Frobenius(the Frobenius theorem), saying that the necessary and sufficient conditionsfor a distribution (i.e. an dimensional subbundleof the tangent bundleof 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 groupof the tangent bundlefrom 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 structureexist.
There is a global foliation theory, because topological constraints exist. For example in the
surfacecase, an everywhere non-zero vector field can exist on an orientable compactsurface only for the torus. This is a consequence of the Poincaré-Hopf index theorem, which shows the Euler characteristicwill have to be 0.
Classifying space for foliations
*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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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