integrable distribution

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 subpaces in $V\to M$. If $V = TM$ 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 $X,Y: M\to W\subset TM$ 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 $W$: $[X,Y]\in W$.

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 April 22, 2010 13:16:23
by David Corfield
(86.168.161.231)