In the following we will C when we say smooth.


Let M be a smooth manifold of dimensionMathworldPlanetmathPlanetmath m. Let nm and for each xM, we assign an n-dimensional subspaceMathworldPlanetmathPlanetmath ΔxTx(M) of the tangent spaceMathworldPlanetmath in such a way that for a neighbourhood NxM of x there exist n linearly independentMathworldPlanetmath smooth vector fields X1,,Xn such that for any point yNx, X1(y),,Xn(y) span Δy. We let Δ refer to the collectionMathworldPlanetmath of all the Δx for all xM and we then call Δ a distribution of dimension n on M, or sometimes a C n-plane distribution on M. The set of smooth vector fields {X1,,Xn} is called a local basis of Δ.

Note: The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis (http://planetmath.org/Distribution). However the naming is in wide use.


We say that a distribution Δ on M is involutive if for every point xM there exists a local basis {X1,,Xn} in a neighbourhood of x such that for all 1i,jn, [Xi,Xj] (the commutator of two vector fields) is in the span of {X1,,Xn}. That is, if [Xi,Xj] is a linear combinationMathworldPlanetmath of {X1,,Xn}. Normally this is written as [Δ,Δ]Δ.


  • 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
Title distribution
Canonical name Distribution1
Date of creation 2013-03-22 14:51:57
Last modified on 2013-03-22 14:51:57
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 53-00
Synonym C^∞n-plane distribution
Related topic FrobeniussTheorem
Defines involutive
Defines involutive distribution
Defines local basis