germ

GermFernando Sanz GÃÂ¡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 $f\underset{x}{\sim}g$ 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, $germ_{x}f$, 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 Germ 2013-03-22 17:25:36 2013-03-22 17:25:36 fernsanz (8869) fernsanz (8869) 5 fernsanz (8869) Definition msc 53B99 TangentSpace Germ function germ.