GermFernando Sanz Gámiz

Definition 1 (Germ).

Let M and N be manifoldsMathworldPlanetmath and xM. We consider all smooth mappings f:UfN, where Uf is some open neighborhood of x in M. We define an equivalence relationMathworldPlanetmath on the set of mappings considered, and we put f𝑥g if there is some open neighborhood V of x with f|V=g|V. The equivalence classMathworldPlanetmath of a mapping f is called the germ of f at x, denoted by f¯ or, sometimes, germxf, and we write


Remark 1.

Germs arise naturally in differentialMathworldPlanetmath topolgy. It is very convenient when dealing with derivativesMathworldPlanetmathPlanetmath 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.