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Encyclopedia > Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties. Distributions are used to build up notions of integrability, and specifically of a foliation of a manifold. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... 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 âˆˆ M and v âˆˆ Tx(M), the tangent space... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems. ... In mathematics, a foliation is a geometric device used to study manifolds. ...

Let M be a $C^infty$ manifold of dimension m, and let $n leq m$. Suppose that for each $x in M$, we assign an n-dimensional subspace $Delta_x subset T_x(M)$ of the tangent space in such a way that for a neighbourhood $N_x subset M$ of x there exist n linearly independent smooth vector fields $X_1,ldots,X_n$ such that for any point $y in N_x$, $X_1(y),ldots,X_n(y)$ span Δy. We let Δ refer to the collection of all the Δx for all $x in M$ and we then call Δ a distribution of dimension n on M, or sometimes a $C^infty$ n-plane distribution on M. The set of smooth vector fields ${ X_1,ldots,X_n }$ is called a local basis of Δ. The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. ... Vector field given by vectors of the form (âˆ’y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... This article is about sets in mathematics. ...

The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis. However the naming is in wide use. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

## Involutive distributions

We say that a distribution Δ on M is involutive if for every point $x in M$ there exists a local basis ${ X_1,ldots,X_n }$ of the distribution in a neighbourhood of x such that for all $1 leq i, j leq n$, [Xi,Xj] (the Lie bracket of two vector fields) is in the span of ${ X_1,ldots,X_n }.$ That is, if [Xi,Xj] is a linear combination of ${ X_1,ldots,X_n }.$ Normally this is written as $[ Delta , Delta ] subset Delta.$ 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, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

Involutive distributions are the tangent spaces to foliations. Involutive distributions are important in that they satisfy the conditions of the Frobenius theorem, and thus lead to integrable systems. In mathematics, a foliation is a geometric device used to study manifolds. ... 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 mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems. ...

A related idea occurs in Hamiltonian mechanics: two functions f and g on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...

## References

• William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.

Results from FactBites:

 Category:Differential geometry - Wikipedia, the free encyclopedia (120 words) In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using calculus.
 Information geometry - Wikipedia, the free encyclopedia (809 words) In mathematics and especially in statistical inference, information geometry is the study of probability and information by way of differential geometry. The main tenet of information geometry is that many important structures in probability theory, information theory and statistics can be treated as structures in differential geometry by regarding a space of probabilities as a differential manifold endowed with a Riemannian metric and a family of affine connections. For example, a family of probability distributions, such as Gaussian distributions, may be transformed into another family of distributions, such as log-normal distributions, by a change of variables.
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