PlanetPhysics/Category of Riemannian Manifolds 2

Introduction

The very important roles played by Riemannian metric and Riemannian manifolds in Albert Einstein's General Relativity (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.

Category of pseudo-Riemannian manifolds

The category of pseudo-Riemannian manifolds ${\mathcal {\mathbb {R} }}_{p}$ has as objects `pseudo-Riemannian manifolds' $\mathbb {R} _{p}$ representing generalized Minkowski spaces; the latter have been claimed to have applications in general relativity, $GR$. The morphisms of ${\mathcal {\mathbb {R} }}_{p}$ are mappings between pseudo-Riemannian manifolds, $\tau :\mathbb {R} _{p}^{i}\to \mathbb {R} _{p}^{j}.$

For a selected pseudo-Riemannian manifold, the endomorphisms $\epsilon :\mathbb {R} _{p}\to \mathbb {R} _{p}$ represent dynamic transformations.

In quantized versions of $\mathbb {R} _{p}$, as in `quantum Riemannian geometry' (QRG), such dynamic transformations may be defined for example by functors between (quantum) spin networks, or quantum spin `foams'. 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 Einstein's field equations 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.