germ

GermFernando Sanz Gámiz

Definition 1 (Germ).

Let $M$ and $N$ be manifolds and $x\in M$. We consider all smoothmappings $f:U_f\to N$, where $U_f$ is some open neighborhood of$x$ in $M$. We define an equivalence relation on the set of mappingsconsidered, and we put $f\underset{𝑥}{\sim }g$ if there is someopen neighborhood $V$ of $x$ with ${f|}_V={g|}_V$. The equivalenceclass of a mapping $f$ is called the germ of f at x, denotedby $\overline{f}$ 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 convenientwhen dealing with derivatives at the point $x$, as every mapping ina germ will have the same derivative values and properties in $x$,and hence can be identified for such purposes: every mapping in agerm gives rise to the same tangent vector of $M$ at $x$.