# distribution

In the following we will ${C}^{\mathrm{\infty}}$ when we say smooth.

###### Definition.

Let $M$ be a smooth manifold of dimension^{} $m$. Let $n\le m$ and for each $x\in M$, we assign an $n$-dimensional subspace^{}
${\mathrm{\Delta}}_{x}\subset {T}_{x}(M)$ of the tangent space^{} in such a way that for a
neighbourhood ${N}_{x}\subset M$ of $x$ there exist $n$ linearly independent^{}
smooth vector fields ${X}_{1},\mathrm{\dots},{X}_{n}$ such that for any point $y\in {N}_{x}$,
${X}_{1}(y),\mathrm{\dots},{X}_{n}(y)$ span ${\mathrm{\Delta}}_{y}$. We let $\mathrm{\Delta}$ refer to the
collection^{} of all the ${\mathrm{\Delta}}_{x}$ for all $x\in M$ and we then call $\mathrm{\Delta}$ a
distribution of dimension $n$ on $M$, or sometimes a
${C}^{\mathrm{\infty}}$ $n$-plane distribution on $M$. The set of smooth
vector fields $\{{X}_{1},\mathrm{\dots},{X}_{n}\}$ is called a local basis of $\mathrm{\Delta}$.

Note: The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis (http://planetmath.org/Distribution). However the naming is in wide use.

###### Definition.

We say that a distribution $\mathrm{\Delta}$ on $M$
is involutive if for every point $x\in M$ there exists a local basis
$\{{X}_{1},\mathrm{\dots},{X}_{n}\}$
in a neighbourhood of $x$ such that for all $1\le i,j\le n$, $[{X}_{i},{X}_{j}]$
(the commutator of two vector fields) is in the span of
$\{{X}_{1},\mathrm{\dots},{X}_{n}\}$. That is, if
$[{X}_{i},{X}_{j}]$ is a linear combination^{} of $\{{X}_{1},\mathrm{\dots},{X}_{n}\}$.
Normally this is written as $[\mathrm{\Delta},\mathrm{\Delta}]\subset \mathrm{\Delta}$.

## References

- 1 William M. Boothby. , Academic Press, San Diego, California, 2003.

Title | distribution |
---|---|

Canonical name | Distribution1 |

Date of creation | 2013-03-22 14:51:57 |

Last modified on | 2013-03-22 14:51:57 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 53-00 |

Synonym | C^∞n-plane distribution |

Related topic | FrobeniussTheorem |

Defines | involutive |

Defines | involutive distribution |

Defines | local basis |