PlanetPhysics/Category of Riemannian Manifolds 2

Introduction

The very important roles played by Riemannian [[../MetricTensor/|metric]] and Riemannian [[../NoncommutativeGeometry4/|manifolds]] in Albert [[../AlbertEinstein/|Einstein's]] [[../GeneralResultsOfTheTheory/|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 [[../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) theory are also concisely discussed.

Category of pseudo-Riemannian manifolds

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

For a selected pseudo-Riemannian manifold, the endomorphisms $ϵ:{ℝ}_p\to {ℝ}_p$ represent [[../MathematicalFoundationsOfQuantumTheories/|dynamic]] transformations.

In quantized versions of ${ℝ}_p$, as in `[[../NonabelianAlgebraicTopology3/|quantum Riemannian geometry]]' ([[../GCGR/|QRG]]), such dynamic transformations may be defined for example by [[../TrivialGroupoid/|functors]] between (quantum) [[../SimplicialCWComplex/|spin networks]], or [[../TriangulationMethodsForQuantizedSpacetimes2/|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 [[../PiecewiseLinear/|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.