nLab integrable distribution of subspaces

This entry is about the concept of subspaces of vector bundles and Lie algebroids. For the concept in functional analysis see at distribution.

Idea

Let $p:V\to M$ be a smooth vector bundle. Any smooth family of $k$-dimensional subspaces $W_m\subset p^{-1}(m)$ where $m\in M$ is called a distribution of k-dimensional subspaces in $V\to M$. If $V = T M$ is the tangent bundle of $M$ then we talk about distributions of tangent vectors.

A distribution of tangent vectors is called integrable if the Lie bracket of its sections is involutive, i.e. if $X,Y: M\to W\subset T M$ are two sections (vector fields belonging to the distribution) of $W\to M$ then the bracket $[X,Y]$ of these vector fields is also a section of $W$: $[X,Y]\in W$.

More generally, if a vector bundle is equipped with the structure of a Lie algebroid, then a distribution of subspaces is integrable if its sections are closed under the given Lie bracket. This reduces to the previous case for the tangent Lie algebroid. Hence integrable distributions are sub-Lie algebroids.

Properties

A basic result on integrability is the Frobenius theorem (Wikipedia) which relates involutivity to integrability in the sense of partial differential equations. Examples include complex analytic manifolds which correspond exactly to complex manifolds with an integrable almost complex structure. Courant algebroids are a quite general tool to express the integrability of geometric structure that include these as special cases.

Related concepts

• foliation

  • foliation of a Lie algebroid

  • foliation of a Lie groupoid

• polarization

• exterior differential system