Distribution (differential geometry)

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

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 Δ_y. We let Δ refer to the collection of all the Δ_x 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.

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 $1 \leq i, j \leq n$, [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 Δ_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 Δ_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

• 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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