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| {{for|other meanings|Distribution (disambiguation)}}
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| In [[differential geometry]], a discipline within [[mathematics]], a '''distribution''' is a subset of the [[tangent bundle]] of a [[differentiable manifold|manifold]] satisfying certain properties. Distributions are used to build up notions of [[Integrable system|integrability]], and specifically of a [[foliation]] of a manifold.
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| Even though they share the same name, distributions we discuss in this article have nothing to do with [[distribution (mathematics)|distribution]]s in the sense of analysis.
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| ==Definition==
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| Let <math>M</math> be a <math>C^\infty</math> manifold of dimension <math>m</math>, and let <math>n \leq m</math>. Suppose that for each <math>x \in M</math>, we assign an <math>n</math>-dimensional [[linear subspace|subspace]] <math>\Delta_x \subset T_x(M)</math> of the [[tangent space]] in such a way that for a [[neighbourhood (mathematics)|neighbourhood]] <math>N_x \subset M</math> of <math>x</math> there exist <math>n</math> [[linear independence|linearly independent]] smooth [[vector field]]s <math>X_1,\ldots,X_n</math> such that for any point <math>y \in N_x</math>, <math>X_1(y),\ldots,X_n(y)</math> [[linear span|span]] <math>\Delta_y.</math> We let <math>\Delta</math> refer to the [[set (mathematics)|collection]] of all the <math>\Delta_x</math> for all <math>x \in M</math> and we then call <math>\Delta</math> a ''distribution'' of dimension <math>n</math> on <math>M</math>, or sometimes a ''<math>C^\infty</math> <math>n</math>-plane distribution'' on <math>M.</math> The set of smooth vector fields <math>\{ X_1,\ldots,X_n \}</math> is called a ''local basis'' of <math>\Delta.</math>
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| ==Involutive distributions==
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| We say that a distribution <math>\Delta</math> on <math>M</math> is ''involutive'' if for every point <math>x \in M</math> there exists a local basis <math>\{ X_1,\ldots,X_n \}</math> of the distribution in a neighbourhood of <math>x</math> such that for all <math>1 \leq i, j \leq n</math>, <math>[X_i,X_j]</math> (the [[Lie bracket of vector fields|Lie bracket]] of two vector fields) is in the span of <math>\{ X_1,\ldots,X_n \}.</math> That is, if <math>[X_i,X_j]</math> is a [[linear combination]] of <math>\{ X_1,\ldots,X_n \}.</math> Normally this is written as <math>[ \Delta , \Delta ] \subset \Delta.</math>
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| Involutive distributions are the tangent spaces to [[foliation]]s. Involutive distributions are important in that they satisfy the conditions of the [[Frobenius theorem (differential topology)|Frobenius theorem]], and thus lead to [[integrable system]]s.
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| 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.
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| ==Generalized distributions==
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| A '''generalized distribution''', or '''Stefan-Sussmann distribution''', is similar to a distribution, but the subspaces <math>\Delta_x \subset T_xM</math> are not required to all be of the same dimension. The definition requires that the <math>\Delta_x</math> are determined locally by a set of vector fields, but these will no longer be linearly independent everywhere. It is not hard to see that the dimension of <math>\Delta_x</math> is [[lower semicontinuous]], so that at special points the dimension is lower than at nearby points.
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| One class of examples is furnished by a non-free action of a [[Lie group]] on a manifold, the vector fields in question being the infinitesimal generators of the [[group action]] (a free action gives rise to a genuine distribution). Another arises in [[dynamical systems]], where the set of vector fields in the definition is the set of vector fields that commute with a given one. There are also examples and applications in [[Control theory]], where the generalized distribution represents infinitesimal constraints of the system.
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| == References ==
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| * William M. Boothby. Section IV. 8. Frobenius's Theorem in ''An Introduction to Differentiable Manifolds and Riemannian Geometry'', Academic Press, San Diego, California, 2003.
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| * P. Stefan, Accessible sets, orbits and foliations with singularities. ''Proc. London Math. Soc.'' '''29''' (1974), 699-713.
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| * H.J. Sussmann, Orbits of families of vector fields and integrability of distributions. ''Trans. Amer. Math. Soc.'' '''180''' (1973), 171-188.
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| {{PlanetMath attribution|id=6541|title=Distribution}}
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| [[Category:Differential geometry]]
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| [[Category:Foliations]]
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