# local Nagano theorem

###### Theorem (Local Nagano Theorem).

Let $\mathrm{\Omega}\mathrm{\subset}{\mathrm{R}}^{n}$ be an open neighbourhood of a point
${x}^{\mathrm{0}}$. Further let $\mathrm{g}$ be a Lie subalgebra of the Lie algebra^{} of
real analytic real vector fields on $\mathrm{\Omega}$ which is also a
${C}^{\omega}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{;}\mathrm{R}\mathrm{)}$-module. Then there exists a real analytic
submanifold $M\mathrm{\subset}\mathrm{\Omega}$ with ${x}^{\mathrm{0}}\mathrm{\in}M$, such that for all $x\mathrm{\in}M$
we have

$${T}_{x}(M)=\U0001d524(x).$$ |

Furthermore the germ of $M$ at $x$ is the unique germ of a submanifold with this property.

Here note that ${T}_{x}(M)$ is the tangent space of $M$ at $x$,
${C}^{\omega}(\mathrm{\Omega};\mathbb{R})$ are the real analytic real valued functions
on $\mathrm{\Omega}$. Also real analytic real vector fields on $\mathrm{\Omega}$ are the
real analytic sections^{} of $T(\mathrm{\Omega})$, the real tangent bundle^{} of $\mathrm{\Omega}$.

###### Definition.

The germ of the manifold^{} $M$ is called the local Nagano leaf of
$\U0001d524$ at ${x}_{0}$.

###### Definition.

The union of all connected real analytic embedded submanifolds of $\mathrm{\Omega}$ whose germ at ${x}_{0}$ coincides with the germ of $M$ at ${x}_{0}$ is called the global Nagano leaf.

The global Nagano leaf turns out to be a connected immersed real analytic submanifold which may however not be an embedded submanifold of $\mathrm{\Omega}$.

## References

- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title | local Nagano theorem |
---|---|

Canonical name | LocalNaganoTheorem |

Date of creation | 2013-03-22 14:48:27 |

Last modified on | 2013-03-22 14:48:27 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 17B99 |

Classification | msc 53B25 |

Synonym | Nagano’s theorem |

Defines | local Nagano leaf |

Defines | Nagano leaf |

Defines | global Nagano leaf |