germ

GermFernando Sanz Gámiz

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 $f\underset{𝑥}{\sim }g$ 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.