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 $f\colon R_{1}\to R_{2}$ 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 ( $G R$ ), whereas the subcategory ${\bf Mink}$ of four-dimensional Minkowski spaces in $\mathcal{R}_{P}$ plays the central role in special relativity ( $S R$ ) theories.

Title	Riemannian manifolds category $R_{M}$
Canonical name	RiemannianManifoldsCategoryRM
Date of creation	2013-03-22 18:25:13
Last modified on	2013-03-22 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 18-00
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 pseudo-Riemannian manifolds
Defines	conformal Riemannian subcategory
Defines	conformal Riemannian manifold
Defines	conformal mapping
Defines	$c_{R}$

Riemannian manifolds category RM

Definition 0.1.

0.1 Applications of Riemannian manifolds in mathematical physics

0.1.1 Category of pseudo-Riemannian manifolds

Riemannian manifolds category $R_{M}$