Riemannian manifolds category ${R}_{M}$
Definition 0.1.
A category^{} ${\mathcal{R}}_{M}$ whose objects are all Riemannian manifolds^{} $R$ and whose morphisms are mappings between Riemannian manifolds ${m}_{R}$ is defined as the category of Riemannian manifolds.
0.1 Applications of Riemannian manifolds in mathematical physics

1.
The conformal Riemannian subcategory ${\mathcal{R}}_{C}$ of ${\mathcal{R}}_{M}$, whose objects are Riemannian manifolds $R$, and whose morphisms are conformal mappings^{} of Riemannian manifolds ${c}_{R}$, is an important category for mathematical physics, in conformal theories.

2.
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 (http://planetmath.org/ConformalMapping) iff ${f}^{*}h=s.g$ for some scalar field $s$ (on ${R}_{1}$), where ${f}^{*}$ is the complex conjugate^{} of $f$.
0.1.1 Category of pseudoRiemannian manifolds
The category of pseudoRiemannian manifolds (http://planetmath.org/PseudoRiemannianManifold) ${\mathcal{R}}_{P}$ that generalize Minkowski spaces^{} ${M}_{k}$ is similarly defined by replacing the Riemanian manifolds^{} $R$ in the above definition with pseudoRiemannian manifolds^{} ${R}_{P}$. PseudoRiemannian manifolds ${R}_{P}$s were claimed to have applications in Einstein’s theory of general relativity ($GR$), whereas the subcategory^{} $\mathrm{\mathbf{M}\mathbf{i}\mathbf{n}\mathbf{k}}$ of fourdimensional Minkowski spaces in ${\mathcal{R}}_{P}$ plays the central role in special relativity ($SR$) theories.
Title  Riemannian manifolds category ${R}_{M}$ 

Canonical name  RiemannianManifoldsCategoryRM 
Date of creation  20130322 18:25:13 
Last modified on  20130322 18:25:13 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  26 
Author  bci1 (20947) 
Entry type  Definition 
Classification  msc 30E20 
Classification  msc 1800 
Classification  msc 53B20 
Classification  msc 53B21 
Related topic  RiemannianMetric 
Related topic  ConformalMapping 
Related topic  ExampleOfConformalMapping 
Related topic  PseudoRiemannianManifold 
Related topic  IndexOfCategories 
Related topic  EinsteinFieldEquations 
Defines  category of pseudoRiemannian manifolds 
Defines  conformal Riemannian subcategory 
Defines  conformal Riemannian manifold 
Defines  conformal mapping 
Defines  ${c}_{R}$ 