%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: category of pseudo-Riemannian manifolds
%%% Primary Category Code: 02.
%%% Filename: CategoryOfRiemannianManifolds2.tex
%%% Version: 23
%%% Owner: bci1
%%% Author(s): bci1
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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} (GR) 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 spacetimes 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 (SR) 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} $\mathcal{\R}_p$ has as \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} `pseudo-Riemannian manifolds' $\mathbb{R}_p$ representing generalized Minkowski spaces; the latter have been claimed to have applications in general relativity, $GR$. The \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{\R}_p$ are
mappings between pseudo-Riemannian manifolds,
$$\tau : \mathbb{R}^i_p \to \mathbb{R}^j_p.$$
For a selected pseudo-Riemannian manifold, the endomorphisms
$$\epsilon: \mathbb{R}_p \to \mathbb{R}_p$$
represent \htmladdnormallink{dynamic}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} transformations.
In quantized versions of $\mathbb{R}_p$,
as in `\htmladdnormallink{quantum Riemannian geometry}{http://planetphysics.us/encyclopedia/NonabelianAlgebraicTopology3.html}' (\htmladdnormallink{QRG}{http://planetphysics.us/encyclopedia/GCGR.html}), such dynamic transformations may be defined for example by \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between (quantum) \htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}, or \htmladdnormallink{quantum spin `foams}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html}'.
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; it can be shown however that
such boundary conditions are however insufficient to obtain physical solutions.
\end{document}
