In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a subbundle U of a vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

 V on a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X is a collection of linear subspace

Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

s U_x of the fibers V_x of V at x in X, that make up a vector bundle in their own right.

In connection with foliation

Foliation

In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....

 theory, a subbundle of the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 of a smooth manifold may be called a distribution

Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties...

 (of tangent vector

Tangent vector

A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

s).

If a set of vector fields Y_k span

Linear span

In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

 the vector space U, and all Lie commutators [Y_i,Y_j] are linear combinations of the Y_k, then one says that U is an involutive distribution.