Foliation

For other uses, see Foliation (disambiguation).

In mathematics, a foliation is a geometric device used to study manifolds. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc

1 Definition
2 Examples
3 Foliations and integrability
4 See also
5 References

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 $\varphi_{ij}:\mathbb{R}^n\to\mathbb{R}^n$ 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. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how In Mathematics, a transition function has several different meanings In Topology, a transition function is a Homeomorphism That is,

$\varphi_{ij}^1:\mathbb{R}^{n-p}\to\mathbb{R}^{n-p}$

and

$\varphi_{ij}^2:\mathbb{R}^p\to\mathbb{R}^{p}$ .

In the chart $U i$ , the stripes $x =$ constant match up with the stripes on other charts $U j$ . A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. Technically, these stripes are called plaques of the foliation. In each chart, the plaques are $n - p$ dimensional submanifolds. In Mathematics, a submanifold of a Manifold M is a Subset S which itself has the structure of a manifold and for which the Inclusion These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, a submanifold of a Manifold M is a Subset S which itself has the structure of a manifold and for which the Inclusion

If we shrink the chart $U i$ it can be written in the form $U_{ix}\times U_{iy}$ where $U_{ix}\subset\mathbb{R}^{n-p}$ and $U_{iy}\subset\mathbb{R}^p$ and $U i y$ is isomorphic to the plaques and the points of $U i x$ parametrize the plaques in $U i$ . If we pick a $y_0\in U_{iy}$ , $U_{ix}\times\{y_0\}$ 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. In Mathematics, monodromy is the study of how objects from Mathematical analysis, Algebraic topology and algebraic and Differential geometry

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

$\mathbb{R}^n=\mathbb{R}^{n-p}\times \mathbb{R}^{p}$

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 $M \to N$ 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. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property

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 $M$ . In Mathematics, the inverse function theorem gives sufficient conditions for a Vector-valued function to be Invertible on an Open region containing Fiber bundles are an example of this type. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space.

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 $H$ . In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

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 $M$ . In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

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. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, an integral curve for a Vector field defined on a Manifold is a curve in the manifold whose tangent vector (i (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. Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions In Mathematics, Frobenius' theorem gives Necessary and sufficient conditions for finding a maximal set of independent solutions of an Overdetermined system 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. In Mathematics, a subbundle U of a Vector bundle V on a Topological space X is a collection of Linear subspaces In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the Lie bracket can refer to Lie algebra Lie bracket of vector fields One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from $G L (n)$ to a reducible subgroup. In Mathematics, in particular the theory of Principal bundles one can ask if a G-bundle "comes from" a subgroup H. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the

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. In Mathematics, certain systems of Partial differential equations are usefully formulated from the point of view of their underlying geometric and algebraic structure in terms In the mathematical discipline of Matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices

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. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar This is a consequence of the Poincaré-Hopf index theorem, which shows the Euler characteristic will have to be 0. In Mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

References

Lawson, H. In Differential geometry, a G -structure on an n - Manifold M, for a given Structure group G, is a G In Mathematics, the Reeb foliation is a particular Foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920-1992 In Mathematics, a taut foliation is a Codimension 1 Foliation of a 3-manifold with the property there is a single transverse circle intersecting Blaine, "Foliations"
I. Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs)

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.
(topology) 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.

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