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. 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. 2.

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 (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 pseudo-Riemannian manifolds

The category of pseudo-Riemannian 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 pseudo-Riemannian manifolds $R_{P}$. Pseudo-Riemannian manifolds $R_{P}$s were claimed to have applications in Einstein’s theory of general relativity ($GR$), whereas the subcategory ${\bf Mink}$ of four-dimensional Minkowski spaces in $\mathcal{R}_{P}$ plays the central role in special relativity ($SR$) theories.

Title Riemannian manifolds category $R_{M}$ RiemannianManifoldsCategoryRM 2013-03-22 18:25:13 2013-03-22 18:25:13 bci1 (20947) bci1 (20947) 26 bci1 (20947) Definition msc 30E20 msc 18-00 msc 53B20 msc 53B21 RiemannianMetric ConformalMapping ExampleOfConformalMapping PseudoRiemannianManifold IndexOfCategories EinsteinFieldEquations category of pseudo-Riemannian manifolds conformal Riemannian subcategory conformal Riemannian manifold conformal mapping $c_{R}$