# germ

GermFernando Sanz GÃÂ¡miz

###### Definition 1 (Germ).

Let $M$ and $N$ be manifolds^{} and $x\in M$. We consider all smooth
mappings $f:{U}_{f}\to N$, where ${U}_{f}$ is some open neighborhood of
$x$ in $M$. We define an equivalence relation^{} on the set of mappings
considered, and we put $f\underset{\mathit{x}}{\sim}g$ if there is some
open neighborhood $V$ of $x$ with ${f|}_{V}={g|}_{V}$. The equivalence
class^{} of a mapping $f$ is called the *germ of f at x*, denoted
by $\overline{f}$ or, sometimes, $ger{m}_{x}f$, and we write

$$\overline{f}:(M,x)\to (N,f(x))$$ |

###### Remark 1.

Germs arise naturally in differential^{} topolgy. It is very convenient
when dealing with derivatives^{} at the point $x$, as every mapping in
a germ will have the same derivative values and properties in $x$,
and hence can be identified for such purposes: every mapping in a
germ gives rise to the same *tangent vector* of $M$ at $x$.

Title | germ |
---|---|

Canonical name | Germ |

Date of creation | 2013-03-22 17:25:36 |

Last modified on | 2013-03-22 17:25:36 |

Owner | fernsanz (8869) |

Last modified by | fernsanz (8869) |

Numerical id | 5 |

Author | fernsanz (8869) |

Entry type | Definition |

Classification | msc 53B99 |

Related topic | TangentSpace |

Defines | Germ |

Defines | function germ. |