| Version 13 |
Version 12 |
| \begin{definition} |
\begin{definition} |
| A category $\mathcal{R}_M$ whose objects are all Riemannian manifolds and whose morphisms are mappings between Riemannian manifolds is defined as the category of Riemannian manifolds. |
A category $\mathcal{R}_M$ whose objects are all Riemannian manifolds and whose morphisms are mappings between Riemannian manifolds is defined as the category of Riemannian manifolds. |
| \end{definition} |
\end{definition} |
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| \subsection{Applications of Riemannian manifolds in mathematical physics} |
\subsection{Applications of Riemannian manifolds in mathematical physics} |
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| \begin{enumerate} |
\begin{enumerate} |
| \item The {\em conformal Riemannian subcategory} $\mathcal{R}_C$ of $\mathcal{R}_M$, whose objects are Riemannian manifolds, and whose morphisms are conformal mappings of Riemannian manifolds, is an important category for mathematical physics, in conformal theories. |
\item The {\em conformal Riemannian subcategory} $\mathcal{R}_C$ of $\mathcal{R}_M$, whose objects are Riemannian manifolds, and whose morphisms are conformal mappings of Riemannian manifolds, is an important category for mathematical physics, in conformal theories. |
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| \item It can be shown that, if $(R_1,g)$ and $(R_2,h)$ are Riemannian manifolds, then |
\item It can be shown that, if $(R_1,g)$ and $(R_2,h)$ are Riemannian manifolds, then |
| a map $f \colon R_1 \to R_2$ is \PMlinkname{conformal}{ConformalMapping} iff $f^* h = s.g$ for some scalar field $s$ |
a map $f \colon R_1 \to R_2$ is \PMlinkname{conformal}{ConformalMapping} iff $f^* h = s.g$ for some scalar field $s$ |
| (on $R_1$), where $f^*$ is the complex conjugate of $f$. |
(on $R_1$), where $f^*$ is the complex conjugate of $f$. |
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| \end{enumerate} |
\end{enumerate} |
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| \subsubsection{Category of pseudo-Riemannian manifolds} |
\subsubsection{Category of pseudo-Riemannian manifolds} |
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| The category of \PMlinkname{pseudo-Riemannian manifolds}{PseudoRiemannianManifold} that generalize Minkowski spaces is similarly defined by replacing ``Riemanian manifolds'' in the above definition with ``pseudo-Riemannian manifolds''; the latter has been claimed to have applications in Einstein's theory of general relativity ($GR$). |
The category of \PMlinkname{pseudo-Riemannian manifolds}{PseudoRiemannianManifold} that generalize Minkowski spaces is similarly defined by replacing ``Riemanian manifolds'' in the above definition with ``pseudo-Riemannian manifolds''; the latter has been claimed to have applications in Einstein's theory of general relativity ($GR$). |
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