germ of smooth functions

If $x$ is a point on a smooth manifold $M$ , then a germ of smooth functions near $x$ is represented by a pair $(U,f)$ where $U\subseteq M$ is an open neighbourhood of $x$ , and $f$ is a smooth function $U\rightarrow\mathbb{R}$ . Two such pairs $(U,f)$ and $(V,g)$ are considered equivalent if there is a third open neighbourhood $W$ of $x$ , contained in both $U$ and $V$ , such that $f|_{W}=g|_{W}$ . To be precise, a germ of smooth functions near $x$ is an equivalence class of such pairs.

In more fancy language: the set $\mathcal{O}_{x}$ of germs at $x$ is the stalk at $x$ of the sheaf $\mathcal{O}$ of smooth functions on $M$ . It is clearly an $\mathbb{R}$ -algebra.

Germs are useful for defining the tangent space $T_{x}M$ in a coordinate-free manner: it is simply the space of all $\mathbb{R}$ -linear maps $X:\mathcal{O}_{x}\rightarrow\mathbb{R}$ satisfying Leibniz’ rule $X(fg)=X(f)g+fX(g)$ . (Such a map is called an $\mathbb{R}$ -linear derivation of $\mathcal{O}_{x}$ with values in $\mathbb{R}$ .)

Title	germ of smooth functions
Canonical name	GermOfSmoothFunctions
Date of creation	2013-03-22 13:05:08
Last modified on	2013-03-22 13:05:08
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 53B99