| Version 7 |
Version 6 |
| A {\em manifold} is a space that is |
A {\em manifold} is a space that is |
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locally like $\mathbb{R}^n$, however lacking a preferred system of
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locally like $\R^n$, however lacking a preferred system of
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| coordinates. Furthermore, a manifold can have global topological |
coordinates. Furthermore, a manifold can have global topological |
| properties, such as non-contractible \PMlinkname{loops}{Curve}, that distinguish it from |
properties, such as non-contractible \PMlinkname{loops}{Curve}, that distinguish it from |
| the topologically trivial $\mathbb{R}^n$. |
the topologically trivial $\R^n$. |
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| By imposing different restrictions on the transition functions of a manifold, one |
By imposing different restrictions on the transition functions of a manifold, one |
| obtain different types of manifolds: |
obtain different types of manifolds: |
| \begin{itemize} |
\begin{itemize} |
| \item topological manifolds |
\item topological manifolds |
| \item $C^k$ manifolds, smooth manifolds |
\item $C^k$ manifolds, smooth manifolds |
| \item real analytic manifold |
\item real analytic manifold |
| \item symplectic manifolds, where transition functions |
\item symplectic manifolds, where transition functions |
| are symplectomorphisms. On such manifolds, one can formulate the |
are symplectomorphisms. On such manifolds, one can formulate the |
| Hamilton equations. |
Hamilton equations. |
| \end{itemize} |
\end{itemize} |
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| Special types of manifolds |
Special types of manifolds |
| \begin{itemize} |
\begin{itemize} |
| \item orientable manifolds |
\item orientable manifolds |
| \item manifolds with boundary |
\item manifolds with boundary |
| \item compact manifolds |
\item compact manifolds |
| \end{itemize} |
\end{itemize} |
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| On manifolds, one can introduce more structure. Some examples are: |
On manifolds, one can introduce more structure. Some examples are: |
| \begin{itemize} |
\begin{itemize} |
| \item Riemann manifolds |
\item Riemann manifolds |
| \item contact manifolds |
\item contact manifolds |
| \end{itemize} |
\end{itemize} |
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| \subsubsection*{Examples} |
\subsubsection*{Examples} |
| \begin{itemize} |
\begin{itemize} |
| \item space-time manifold in general relativity |
\item space-time manifold in general relativity |
| \item phase space in mechanics |
\item phase space in mechanics |
| \item de Rham cohomology in algebraic topology |
\item de Rham cohomology in algebraic topology |
| \end{itemize} |
\end{itemize} |
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| \subsubsection*{See also} |
\subsubsection*{See also} |
| For the formal definition click \PMlinkname{here}{Manifold}\\ |
For the formal definition click \PMlinkname{here}{Manifold}\\ |
| \PMlinkexternal{Manifold entry at Wikipedia}{http://en.wikipedia.org/wiki/Manifold} |
\PMlinkexternal{Manifold entry at Wikipedia}{http://en.wikipedia.org/wiki/Manifold} |
