Distribution (differential geometry)

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For other meanings, see Distribution (disambiguation).

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.

Even though they share the same name, distributions we discuss in this article have nothing to do with distributions in the sense of analysis.

Definition[edit]

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 \Delta_y. We let \Delta refer to the collection of all the \Delta_x for all x \in M and we then call \Delta 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 \Delta.

Involutive distributions[edit]

We say that a distribution \Delta 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, [X_i,X_j] (the Lie bracket of two vector fields) is in the span of \{ X_1,\ldots,X_n \}. That is, if [X_i,X_j] is a linear combination of \{ X_1,\ldots,X_n \}. Normally this is written as [ \Delta , \Delta ] \subset \Delta.

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.

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.

Generalized distributions[edit]

A generalized distribution, or Stefan-Sussmann distribution, is similar to a distribution, but the subspaces \Delta_x \subset T_xM are not required to all be of the same dimension. The definition requires that the \Delta_x 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 \Delta_x is lower semicontinuous, so that at special points the dimension is lower than at nearby points.

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.

References[edit]

  • William M. Boothby. Section IV. 8. Frobenius's Theorem in An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
  • P. Stefan, Accessible sets, orbits and foliations with singularities. Proc. London Math. Soc. 29 (1974), 699-713.
  • H.J. Sussmann, Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973), 171-188.

External links[edit]

This article incorporates material from Distribution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.