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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 stripy 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
A stripe may be one of a pattern of areas created by a family of parallel lines, as on the flag of the United States, also known as the stars and stripes in a candy-stripe pattern, on a diagonal and twisted round a cylinder, as for a candy cane...
In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...
Definition
More formally, a codimension p foliation F of an n-dimensional manifold M is a covering by charts Ui together with maps In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. ...
In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...
 such that on the overlaps  the transition functions  defined by In mathematics, a transition function has several different meanings: In topology, a transition function is a homeomorphism from one coordinate chart to another. ...
 take the form  where x denotes the first n − p co-ordinates, and y denotes the last p co-ordinates. That is,  and  . In the chart Ui, the stripes x = constant match up with the stripes on other charts Uj. 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. A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...
This is a glossary of terms specific to differential geometry and differential topology. ...
Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...
In mathematics, in differential geometry, the image, say , of a differentiable manifold to another differentiable manifold with respect to a differentiable mapping may not be a submanifold. ...
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  with the leaves or plaques  being enumerated by  . 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  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 common property. ...
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 is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the...
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...
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). In mathematics, a smooth function is one that is infinitely differentiable, i. ...
Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...
Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...
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. Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ...
In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ...
In logic, the words necessary and sufficient describe problems that consist between propositions or states of affairs, if one is accidental on the other. ...
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ...
In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ...
In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
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 of a system of differential forms. ...
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...
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. An open surface with X-, Y-, and Z-contours shown. ...
This article or section should be merged with Orientable manifold. ...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
A torus. ...
In mathematics, the Poincaré-Hopf Theorem (also known as the Poincaré-Hopf index formula, Poincaré-Hopf index theorem, or Hopf index theorem) states: Let M be a compact differentiable manifold. ...
It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...
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