Embedding3 2004-12-11 01:00:42 2007-04-16 15:25:32 Definition Whitney's theorem % 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,amscd} \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 \PMlinkescapeword{characterization} \PMlinkescapeword{states} Let $M$ and $N$ be manifolds and $f\colon M\to N$ a smooth map. Then $f$ is an \emph{embedding} if \begin{enumerate} \item $f(M)$ is a submanifold of $N$, and \item $f\colon M\to f(M)$ is a diffeomorphism. (There's an abuse of notation here. This should really be restated as the map $g\colon M\to f(M)$ defined by $g(p)=f(p)$ is a diffeomorphism.) \end{enumerate} The above characterization can be equivalently stated: $f\colon M\to N$ is an embedding if \begin{enumerate} \item $f$ is an immersion, and \item by abuse of notation, $f\colon M\to f(M)$ is a homeomorphism. \end{enumerate} \textbf{Remark}. A celebrated theorem of Whitney states that every $n$ dimensional manifold admits an embedding into $\mathbb{R}^{2n+1}$.