AlgebraicManifold 2005-12-05 21:07:39 2007-09-30 09:57:03 Definition Nash manifold Nash submanifold % 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 \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \newtheorem*{example}{Example} \newtheorem*{prop}{Proposition} \theoremstyle{definition} \newtheorem*{defn}{Definition} \theoremstyle{remark} \newtheorem*{rmk}{Remark} \begin{defn} Let $k$ be a field and let $M \subset k^n$ be a submanifold. $M$ is said to be an \emph{algebraic manifold} (or $k$-algebraic) if there exists an irreducible algebraic variety $V \subset k^n$ such that $\dim V = \dim M$ and $M \subset V$. If $k = \mathbb{R}$, then $M$ is called a \emph{Nash manifold}. \end{defn} It can be proved that such a manifold is defined as the zero set of a finite collection of analytic algebraic functions. \begin{thebibliography}{9} \bibitem{ber:submanifold} M.\@ Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. {\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, Princeton University Press, Princeton, New Jersey, 1999. \end{thebibliography}