# integral manifold

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

###### Definition.

Let $M$ be a smooth manifold^{} of dimension $m$ and let $\mathrm{\Delta}$ be a
distribution of dimension $n$ on $M$. Suppose that $N$ is a connected
submanifold^{} of $M$ such that for every $x\in N$ we have that
${T}_{x}(N)$ (the tangent space of $N$ at $x$) is contained in ${\mathrm{\Delta}}_{x}$
(the distribution at $x$). We can abbreviate this by saying that
$T(N)\subset \mathrm{\Delta}$. We then say that $N$ is an integral manifold
of $\mathrm{\Delta}$.

Do note that $N$ could be of lower dimension then $\mathrm{\Delta}$ and is not required to be a regular submanifold of $M$.

###### Definition.

We say that a distribution $\mathrm{\Delta}$ of dimension $n$ on $M$ is completely integrable if for each point $x\in M$ there exists an integral manifold $N$ of $\mathrm{\Delta}$ passing through $x$ such that the dimension of $N$ is equal to the dimension of $\mathrm{\Delta}$.

An example of an integral manifold is the integral curve of a non-vanishing vector field and then of course the span of the vector field is a completely integrable distribution.

## References

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

Title | integral manifold |
---|---|

Canonical name | IntegralManifold |

Date of creation | 2013-03-22 14:52:00 |

Last modified on | 2013-03-22 14:52:00 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 53B25 |

Classification | msc 52-00 |

Classification | msc 37C10 |

Related topic | FrobeniussTheorem |

Defines | completely integrable |

Defines | completely integrable distribution |