Riemannian manifolds category $R_M$

Definition 0.1.

A category ${ℛ}_M$ 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 ${ℛ}_C$ of ${ℛ}_M$, 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 pseudo-Riemannian manifolds

The category of pseudo-Riemannian manifolds (http://planetmath.org/PseudoRiemannianManifold) ${ℛ}_P$ 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 $\mathrm{𝐌𝐢𝐧𝐤}$ of four-dimensional Minkowski spaces in ${ℛ}_P$ plays the central role in special relativity ($SR$) 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	$c_R$