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.

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

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

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

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 upper 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

• 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.

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