# 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 GermOfSmoothFunctions 2013-03-22 13:05:08 2013-03-22 13:05:08 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Definition msc 53B99