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germ (Definition)
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 $x$ .




"germ" is owned by fernsanz.
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See Also: tangent space

Also defines:  Germ, function germ.
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Cross-references: tangent vector, properties, point, derivatives, equivalence class, mappings, equivalence relation, neighborhood, open, smooth mappings, manifolds
There are 21 references to this entry.

This is version 2 of germ, born on 2007-07-27, modified 2007-07-27.
Object id is 9801, canonical name is Germ.
Accessed 2487 times total.

Classification:
AMS MSC53B99 (Differential geometry :: Local differential geometry :: Miscellaneous)

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