category of Riemannian manifolds CategoryOfRiemannianManifolds2 2009-01-27 10:51:27 2009-01-27 11:34:54 Topic Relativity: The Special and General Theory GR SR general and special theories of relativity Einstein GR SR Einstein's field equations % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % there are many more packages, add them here as you need % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Introduction} The very important roles played by Riemannian metric and Riemannian manifolds in Albert Einstein's \PMlinkname{General Relativity}{GeneralResultsOfTheTheory} (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 concept representing transformations of such spacetimes that are either expanding or `transforming' in higher dimensions (as perhaps suggested by some of the superstring `theories'). Other, possible, conformal theory developments based on Einstein's special relativity (SR) theory are also concisely discussed. \subsection{Riemannian manifolds in Einstein's General Relativity (GR)} \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{Other 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 \PMlinkexternal{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 boundary conditions for \PMlinkexternal{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.