GermOfSmoothFunctions 2002-10-01 12:20:59 2002-10-01 12:20:59 Definition vector fields local functions % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here If $x$ is a point on a smooth manifold $M$, then a {\em 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 $\mathbb{R}$.)