Distribution (differential geometry)

Definition

Let M be a C ∞ manifold of dimension m , and let n ≤ m . Suppose that for each x ∈ M , we assign an n -dimensional subspace Δ x ⊂ T x ( M ) of the tangent space in such a way that for a neighbourhood N x ⊂ M of x there exist n linearly independent smooth vector fields X 1 , … , X n such that for any point y ∈ N x , span { X 1 ( y ) , … , X n ( y ) } = Δ y . We let Δ refer to the collection of all the Δ x for all x ∈ M and we then call Δ a distribution of dimension n on M , or sometimes a C ∞ n -plane distribution on M . The set of smooth vector fields { X 1 , … , 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 ∈ M there exists a local basis { X 1 , … , X n } of the distribution in a neighbourhood of x such that for all 1 ≤ i , j ≤ n , [ X i , X j ] (the Lie bracket of two vector fields) is in the span of { X 1 , … , X n } . That is, if [ X i , X j ] is a linear combination of { X 1 , … , X n } . Normally this is written as [ Δ , Δ ] ⊂ Δ .

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 Δ x ⊂ T x M 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.