Manifold2 2004-02-22 04:23:33 2007-12-12 12:51:55 Topic % 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 \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} A {\em manifold} is a space that is locally like $\mathbb{R}^n$, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible \PMlinkname{loops}{Curve}, that distinguish it from the topologically trivial $\mathbb{R}^n$. By imposing different restrictions on the transition functions of a manifold, one obtain different types of manifolds: \begin{itemize} \item topological manifolds \item $C^k$ manifolds, smooth manifolds \item real analytic manifold \item complex analytic manifold \item symplectic manifolds, where transition functions are symplectomorphisms. On such manifolds, one can formulate the Hamilton equations. \end{itemize} Special types of manifolds \begin{itemize} \item orientable manifolds \item manifolds with boundary \item compact manifolds \end{itemize} On manifolds, one can introduce more \PMlinkescapetext{structure}. Some examples are: \begin{itemize} \item Riemannian manifolds \item contact manifolds \item CR manifolds \item fiber bundles and sheaves \end{itemize} \subsubsection*{Examples} \begin{itemize} \item space-time manifold in general relativity \item phase space in mechanics \item de Rham cohomology in algebraic topology \end{itemize} \subsubsection*{See also} For the formal definition click \PMlinkname{here}{Manifold}\\ \PMlinkexternal{Manifold entry at Wikipedia}{http://en.wikipedia.org/wiki/Manifold}