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, 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 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 . 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* can be more appropriate. 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 vector fields on forming a vector subspace of the space of all vector fields on . (Notation: is the space of sections of ) 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.

Given any distribution , the **associated Lie flag** is a grading, defined as

where , and . In other words, denotes the set of vector fields spanned by the -iterated Lie brackets of elements in . Some authors use a negative decreasing grading for the definition.

Then is called **weakly regular** (or just regular by some authors) if there exists a sequence of nested vector subbundles such that (hence ).^{[1]} Note that, in such case, the associated Lie flag stabilises at a certain point , since the ranks of are bounded from above by . The string of integers is then called the **grow vector** of .

Any weakly regular distribution has an associated graded vector bundle

Moreover, the Lie bracket of vector fields descends, for any , to a -linear bundle morphism , called the

Patching together the curvatures, one obtains a morphism , also called the **Levi bracket**, which makes into a bundle of nilpotent Lie algebras; for this reason, is also called the **nilpotentisation** of .^{[1]}

The bundle , however, is in general not locally trivial, since the Lie algebras are not isomorphic when varying the point . If this happens, the weakly regular distribution is also called **regular** (or strongly regular by some authors).^{[clarification needed]} Note that the names *(strongly, weakly) regular* used here are completely unrelated with the notion of regularity discussed above (which is always assumed), i.e. the dimension of the spaces being constant.

A distribution is called **bracket-generating** (or **non-holonomic**, or it is said to satisfy the **Hörmander condition**) if taking a finite number of Lie brackets of elements in is enough to generate the entire space of vector fields on . With the notation introduced above, such condition can be written as for certain ; then one says also that is bracket-generating in *steps*, or has *depth* .

Clearly, the associated Lie flag of a bracket-generating distribution stabilises at the point . Even though being weakly regular and being bracket-generating are two independent properties (see the examples below), when a distribution satisfies both of them, the integer from the two definitions is the same.

Thanks to the Chow-Rashevskii theorem, given a bracket-generating distribution on a connected manifold, any two points in can be joined by a path tangent to the distribution.^{[2]}^{[3]}

- 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.^{[4]} - 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 .

Not to be confused with Singular distribution. |

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.

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,^{[5]} the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan^{[6]}^{[7]} and Sussmann.^{[8]}^{[9]} It states that a singular distribution is integrable if and only if the following two properties hold:

- is generated by a family of vector fields;
- is invariant with respect to every , i.e. , where is the flow of , and .

Similarly to the regular case, an integrable singular distribution defines a **singular foliation**, which intuitively consists in a partition of into submanifolds (the maximal integral manifolds of ) 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^{[10]}^{[11]} or non-commutative geometry.^{[12]}^{[13]}

- 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.