PlanetPhysics/Category of Riemannian Manifolds

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 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 Einstein's field equations in order to find and study possible solutions that are physically meaningful.

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 .

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 mathematical physics, 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 scalar field $s$ (on $R_{1}$), where $f^{*}$ is the complex conjugate of $f$.