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.

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

Let p:VMp:V\to M be a smooth vector bundle. Any smooth family of kk-dimensional subspaces W mp 1(m)W_m\subset p^{-1}(m) where mMm\in M is called a distribution of k-dimensional subspaces in VMV\to M. If V=TMV = TM is the tangent bundle of MM 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:MWTMX,Y: M\to W\subset TM are two sections (vector fields belonging to the distribution) of WMW\to M then the bracket [X,Y][X,Y] of these vector fields is also a section of WW: [X,Y]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 the integrability in the sense of partial differential equations. An examples are complex analytic manifolds which correspond exactly to complex manifolds with integrable almost complex structure. Courant algebroids are a quite general tool to express the integrability of geometric structure including these as special cases.

Revised on August 14, 2017 02:35:57 by Urs Schreiber (94.220.75.2)