|
|
|
Viewing Version
10
of
'topics in manifold theory'
|
[ view 'topics in manifold theory'
|
back to history
]
| Title of object: |
topics in manifold theory |
| Canonical Name: |
Manifold2 |
| Type: |
Topic |
| Created on: |
2004-02-22 04:23:33 |
| Modified on: |
2006-07-22 21:03:38 |
| Classification: |
msc:53-00 |
Preamble:
% 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 |} |
Content:
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 structure. Some examples are:
\begin{itemize}
\item Riemann 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} |
|
|
|
|
|
