PlanetPhysics/Category of Riemannian Manifolds

Introduction

The very important roles played by Riemannian [[../MetricTensor/|metric]] and Riemannian [[../NoncommutativeGeometry4/|manifolds]] in Albert [[../AlbertEinstein/|Einstein's]] [[../GeneralResultsOfTheTheory/|General Relativity]] ([[../SR/|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 [[../SR/|spacetimes]] defined `without any Riemann metric, or metric, in general'. The mappings of such Riemannian spacetimes provide the mathematical [[../PreciseIdea/|concept]] representing transformations of such spacetimes that are either expanding or `transforming' in higher dimensions (as perhaps suggested by some of the [[../10DBrane/|superstring]] `theories'). Other, possible, conformal theory developments based on Einstein's special relativity ([[../SR/|SR]]) theory are also concisely discussed.

Category of pseudo-Riemannian manifolds

The [[../Cod/|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 [[../PiecewiseLinear/|boundary]] conditions for Einstein's field equations in order to find and study possible solutions that are physically meaningful.

A category ${ℛ}_M$ whose [[../TrivialGroupoid/|objects]] are all Riemannian manifolds and whose [[../TrivialGroupoid/|morphisms]] are mappings between Riemannian manifolds is defined as the category of Riemannian manifolds .

The subcategory ${ℛ}_C$ of ${ℛ}_M$, whose objects are Riemannian manifolds, and whose morphisms are conformal mappings of Riemannian manifolds, is an important category for [[../PhysicalMathematics2/|mathematical physics]], in conformal theories. It can be shown that, if $(R_1,g)$ and $(R_2,h)$ are Riemannian manifolds, then a map $f:R_1\to R_2$ is conformal iff $f^{*}h=s.g$ for some [[../Vectors/|scalar]] [[../CosmologicalConstant2/|field]] $s$ (on $R_1$), where $f^{*}$ is the complex conjugate of $f$.