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%%% Primary Title: category of Riemannian manifolds
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%%% Filename: CategoryOfRiemannianManifolds.tex
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\begin{document}
\subsection{Introduction}
The very important roles played by Riemannian \htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html} and Riemannian \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} in Albert \htmladdnormallink{Einstein's}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} \htmladdnormallink{General Relativity}{http://planetphysics.us/encyclopedia/GeneralResultsOfTheTheory.html} (\htmladdnormallink{GR}{http://planetphysics.us/encyclopedia/SR.html}) is well known. The following definition provides the proper mathematical framework for studying different Riemannian manifolds and all possible relationships between different Riemannian metrics defined on different Riemannian manifolds; it also provides one with the more general framework for comparing abstract \htmladdnormallink{spacetimes}{http://planetphysics.us/encyclopedia/SR.html} defined `without any Riemann metric, or metric, in general'. The mappings of such Riemannian spacetimes provide the mathematical \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} representing transformations of such spacetimes that are either expanding or `transforming'
in higher dimensions (as perhaps suggested by some of the \htmladdnormallink{superstring}{http://planetphysics.us/encyclopedia/10DBrane.html} `theories'). Other, possible, conformal theory developments based on Einstein's
special relativity (\htmladdnormallink{SR}{http://planetphysics.us/encyclopedia/SR.html}) theory are also concisely discussed.
\subsubsection{Category of pseudo-Riemannian manifolds}
The \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of \htmladdnormallink{pseudo-Riemannian manifolds}{http://planetmath.org/?op=getobj&from=objects&name=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 ($GR$).
In General Relativity space-time may also be modeled as a 4-pseudo Riemannian manifold with signature $(-,+,+,+)$; over such spacetimes one can then consider the \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} conditions for
\htmladdnormallink{Einstein's field equations}{http://planetmath.org/?op=getobj&from=objects&name=EinsteinFieldEquations} in order to find and study possible solutions that are physically meaningful.
\begin{definition}
A category $\mathcal{R}_M$ whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are all Riemannian manifolds and whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are mappings between Riemannian manifolds is defined as the
{\em category of Riemannian manifolds}.
\end{definition}
The 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 \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html}, in conformal theories. 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 conformal iff $f^* h = s.g$ for some \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} $s$
(on $R_1$), where $f^*$ is the complex conjugate of $f$.
\end{document}
