NashIsometricEmbeddingTheorem 2006-01-21 13:11:58 2007-10-06 13:43:36 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} \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 Every compact $n$-dimensional Riemannian manifold $M$ of class $C^k$ ($3\le k\le\infty$) can be $C^k$-isometrically imbedded in any small portion of a Euclidean space $\mathbb R^N$, where $$ N=\frac 12 n(3n+11). $$ Every non-compact $n$-dimensional Riemannian manifold $M$ of class $C^k$ ($3\le k\le\infty$) can be $C^k$-isometrically imbedded in any small portion of a Euclidean space $\mathbb R^N$, where $$ N=(n+1)\frac 12 n(3n+11). $$ The original proof due to Nash relying on an iteration scheme has been considerably simplified. For an overview, see \cite{nashsim}. \begin{thebibliography}{9} \bibitem{nash} Nash, J. F., \emph{The imbedding problem for Riemannian manifold}, Ann. of Math. 63 (1956), 20--63 (MR 17, 782) \bibitem{nashsim} D. Yang, \emph{Gunther's proof of Nash's isometric embedding theorem}, \PMlinkexternal{online}{http://www.math.poly.edu/~yang/papers/gunther.pdf} \end{thebibliography}