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.

[edit] Definition

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.

[edit] 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.

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.

[edit] References

  • William M. Boothby. Section IV. 8. Frobenius's Theorem in An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.

This article incorporates material from Distribution on PlanetMath, which is licensed under the GFDL.