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Wikipedia / Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution on a manifold of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle. If the length of the vector is not needed, i.e. if one is concerned only with the subspaces themselves, projectivized, then it can also be constructed as a subbundle of the contact 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, including integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology.

Even though they share the same name, distributions presented in this article have nothing to do with distributionss in the sense of analysis.

Definition

be a smooth manifold; a (smooth) distribution there exist a neighbourhood and a collection of vector fields, span

is also called a local basis. These need not be linearly independent at every point, and so aren't formally a vector space basis at every point; thus, the term local generating set.

Regular distributions

is called regular linearly independent vector fields.

More compactly, a regular distribution is a vector subbundle, one talks about hyperplane distributions.

Special classes of distributions

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

Involutive distributions

forming a vector subspace is the space of sections is called involutive is also a Lie subalgebra, the Lie bracket.

is a linear combination

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.

Integrable distributions and foliations

An integral manifold is a submanifold. A distribution is called integrable are thus disjoint, maximal, connected integral manifolds, also called leaves defines an n-dimensional foliation.

is tangent to the leaves of a foliation. Moreover, this local characterisation coincides with the definition of integrability for a G-structures, when G is the group of real invertible upper-triangular block matrices-blocks).

It is easy to see that any integrable distribution is automatically involutive. The converse is less trivial but holds by Frobenius theorem.

Weakly regular distributions

, the associated Lie flag is a grading, defined as

. Some authors use a negative decreasing grading for the definition.

is called weakly regular (or just regular by some authors) if there exists a sequence \{T^i M \subseteq TM \}_i of nested vector subbundles such that \Gamma(T^iM) = \Delta^{(i)})., since the ranks of T^iM are bounded from above by \mathrm{rank}(TM) = \mathrm{dim}(M) is then called the grow vector.

, called the -curvature. In particular, the -curvature vanishes identically if and only if the distribution is involutive.

, also called the Levi bracket is also called the nilpotentisation.

is also called regular (or strongly regular by some authors). Note that the names (strongly, weakly) regular being constant.

Bracket-generating distributions

is called bracket-generating (or non-holonomic, or it is said to satisfy the Hörmander condition steps, or has depth.

from the two definitions is the same.

Thanks to the Chow-Rashevskii theorem can be joined by a path tangent to the distribution.

Stable distributions

is stable. There exists only 3 cases:

  • Rank 1: a fibration, essentially a vector field without length.

  • Corank 1: a hyperplane distribution. When the ambient dimension is odd, it is the contact form. When the ambient dimension is even, it is the even contact form.

Equivalently, Engel distributions are characterized by

Examples of regular distributions

Integrable distributions

  • , which is automatically integrable: the image of any integral curve is an integral manifold.

  • .

  • ), but it is never bracket-generating.

Non-integrable distributions

  • The Martinet distribution.

  • The contact distribution. One can also define an abstract contact structures 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, and bracket-generating. One can also define an abstract Engel structurehas rank 4; Engel proved that such structure has the unique local model described above.

  • In general, a Goursat structure 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.

Singular distributions

A singular distribution, generalised distribution, or Stefan-Sussmann distribution is no longer a smooth subbundle.

is lower semicontinuous, so that at special points the dimension is lower than at nearby points.

Integrability and singular foliations

The definitions of integral manifolds and of integrability given above applies also to the singular case (removing the requirement of the fixed dimension). However, Frobenius theorem does not hold in this context, and involutivity is in general not sufficient for integrability (counterexamples in low dimensions exist).

After several partial results, the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan and Sussmann. is integrable if and only if the following two properties hold:

  • of vector fields;

  • is the flow.

Similarly to the regular case, an integrable singular distribution defines a singular foliation) of different dimensions.

The definition of singular foliation can be made precise in several equivalent ways. Actually, in the literature there is a plethora of variations, reformulations and generalisations of the Stefan-Sussman theorem, using different notion of singular foliations according to which applications one has in mind, e.g. Poisson geometry or non-commutative geometry.

Examples

  • Given a Lie group action of a Lie group, 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 distribution/foliation is regular If and only if the Poisson manifold is regular.

  • of any 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.

References