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'category of Riemannian manifolds'
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| Title of object: |
category of Riemannian manifolds |
| Canonical Name: |
CategoryOfRiemannianManifolds |
| Type: |
Definition |
| Created on: |
2008-09-22 11:53:21 |
| Modified on: |
2009-01-27 11:38:04 |
| Classification: |
msc:53B21, msc:53B20, msc:18-00, msc:30E20 |
| Keywords: |
Riemannian manifolds, Riemannian metric, conformal mapping of Riemannian manifolds |
| Defines: |
category of pseudo-Riemannian manifolds, conformal Riemannian subcategory, conformal Riemannian manifolds, conformal mappings |
Revision comment (for changes between this and next version):
{\em category of Riemannian manifolds}.
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Content:
\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.
\end{definition}
\subsection{Applications of Riemannian manifolds in mathematical physics}
\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 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 $s$
(on $R_1$), where $f^*$ is the complex conjugate of $f$.
\end{enumerate}
\subsubsection{Category of pseudo-Riemannian manifolds}
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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