Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle
Distributions satisfying a further integrability condition give rise to foliations, i. e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, e. g. integrable systems, Poisson geometry, non commutative geometry, sub Riemannian geometry, differential topology, etc. Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis.

Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists in a collection of vector subspaces with the following property. Around any there exist a neighbourhood and a collection of vector fields such that, for any point , span

The set of smooth vector fields is also called a local basis of Note that the number may be different for different neighbourhoods. The notation is used to denote both the assignment and the subset .

Given an integer , a smooth distribution on is called regular of rank if all the subspaces have the same dimension. Locally, this amounts to ask that every local basis is given by linearly independent vector fields. More compactly, a regular distribution is a vector subbundle of rank (this is actually the most commonly used definition). A rank distribution is sometimes called an plane distribution, and when , one talks about hyperplane distributions.

Unless stated otherwise, by "distribution" we mean a smooth regular distribution (in the sense explained above).

Given a distribution , its sections consist of the vector fields which are tangent to , and they form a vector subspace of the space of all vector fields on A distribution is called involutive if is also a Lie subalgebra: in other words, for any two vector fields , the Lie bracket belongs to
Locally, this condition means that for every point there exists a local basis of the distribution in a neighbourhood of such that, for all , the Lie bracket is in the span of , i. e. is a linear combination of

Involutive distributions are a fundamental ingredient in the study of integrable systems. A related idea occurs in Hamiltonian mechanics: two functions and on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.

An integral manifold for a rank distribution is a submanifold of dimension such that for every A distribution is called integrable if through any point there is an integral manifold. The base spaces of the bundle are thus disjoint, maximal, connected integral manifolds, also called leaves; that is, defines an n dimensional foliation of
Locally, integrability means that for every point there exists a local chart such that, for every , the space is spanned by the coordinate vectors In other words, every point admits a foliation chart, i. e. the distribution is tangent to the leaves of a foliation. Moreover, this local characterisation coincides with the definition of integrability for a structures, when is the group of real invertible upper triangular block matrices (with and blocks). It is easy to see that any integrable distribution is automatically involutive. The converse is less trivial but holds by Frobenius theorem.

Any vector field on defines a rank 1 distribution, by setting , which is automatically integrable: the image of any integral curve is an integral manifold. The trivial distribution of rank on is generated by the first coordinate vector fields It is automatically integrable, and the integral manifolds are defined by the equations , for any constants In general, any involutive/integrable distribution is weakly regular (with for every ), but it is never bracket generating.

The Martinet distribution on is given by , for ; equivalently, it is generated by the vector fields and It is bracket generating since , but it is not weakly regular: has rank 3 everywhere except on the surface The contact distribution on is given by , for ; equivalently, it is generated by the vector fields and , for It is weakly regular, with grow vector , and bracket generating, with One can also define an abstract contact structures on a manifold as a hyperplane distribution which is maximally non integrable, i. e. it is as far from being involutive as possible. An analogue of the Darboux theorem shows that such structure has the unique local model described above. The Engel distribution on is given by , for and ; equivalently, it is generated by the vector fields and It is weakly regular, with grow vector , and bracket generating. One can also define an abstract Engel structure on a manifold as a weakly regular rank 2 distribution such that has rank 3 and has rank 4; Engel proved that such structure has the unique local model described above. In general, a Goursat structure on a manifold is a rank 2 distribution which is weakly regular and bracket generating, with grow vector For and one recovers, respectively, contact distributions on 3 dimensional manifolds and Engel distributions. Goursat structures are locally diffeomorphic to the Cartan distribution of the jet bundles .

A singular distribution, generalised distribution, or Stefan Sussmann distribution, is a smooth distribution which is not regular. This means that the subspaces may have different dimensions, and therefore the subset is no longer a smooth subbundle. In particular, the number of elements in a local basis spanning will change with , and those vector fields will no longer be linearly independent everywhere. It is not hard to see that the dimension of is lower semicontinuous, so that at special points the dimension is lower than at nearby points.

Given a Lie group action of a Lie group on a manifold , its infinitesimal generators span a singular distribution which is always integrable; the leaves of the associated singular foliation are precisely the orbits of the group action. The distribution/foliation is regular if and only if the action is free. Given a Poisson manifold , the image of is a singular distribution which is always integrable; the leaves of the associated singular foliation are precisely the symplectic leaves of The distribution/foliation is regular If and only if the Poisson manifold is regular. More generally, the image of the anchor map of any Lie algebroid defines a singular distribution which is automatically integrable, and the leaves of the associated singular foliation are precisely the leaves of the Lie algebroid. The distribution/foliation is regular if and only if has constant rank, i. e. the Lie algebroid is regular. Considering, respectively, the action Lie algebroid and the cotangent Lie algebroid , one recovers the two examples above. In dynamical systems, a singular distribution arise from the set of vector fields that commute with a given one. There are also examples and applications in control theory, where the generalised distribution represents infinitesimal constraints of the system.

William M. Boothby. Section IV. 8 in An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003. John M. Lee, Chapter 19 in Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer Verlag, 2003. Richard Montgomery, Chapters 2, 4 and 6 in A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. Amer. Math. Soc., Providence, RI, 2002. Álvaro del Pino, Topological aspects in the study of tangent distributions. Textos de Matemática. Série B, 48. Universidade de Coimbra, 2019.