# algebraic manifold

###### Definition.

Let $k$ be a field and let $M\subset {k}^{n}$ be a submanifold^{}. $M$ is said to
be an *algebraic manifold ^{}* (or $k$-algebraic

^{}) if there exists an irreducible

^{}algebraic variety $V\subset {k}^{n}$ such that $dimV=dimM$ and $M\subset V$. If $k=\mathbb{R}$, then $M$ is called a

*Nash manifold*.

It can be proved that such a manifold^{} is defined as the zero set^{} of a finite collection of analytic algebraic functions.

## References

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

Title | algebraic manifold |
---|---|

Canonical name | AlgebraicManifold |

Date of creation | 2013-03-22 15:36:08 |

Last modified on | 2013-03-22 15:36:08 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 14P20 |

Classification | msc 14-00 |

Classification | msc 58A07 |

Synonym | algebraic submanifold |

Synonym | $k$-algebraic manifold |

Synonym | $k$-algebraic submanifold |

Defines | Nash manifold |

Defines | Nash submanifold |