germGerm2007-07-27 06:03:282007-07-27 11:21:11DefinitionGermfunction germ.% this is the default PlanetMath preamble. as your knowledge
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\numberwithin{equation}{section}\title{Germ}%
\author{Fernando Sanz Gámiz}%
\begin{defn}[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 \emph{germ of f at x}, denoted
by $\overline{f}$ or, sometimes, $germ_x f$, and we write
$$\overline{f}:(M,x) \to (N,f(x))$$
\end{defn}
\bigskip
\begin{rem}
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 \emph{tangent vector} of $M$ at $x$.
\end{rem}