Riemannian manifolds category RM

Definition 0.1.

A categoryMathworldPlanetmath M whose objects are all Riemannian manifoldsMathworldPlanetmath R and whose morphisms are mappings between Riemannian manifolds mR is defined as the category of Riemannian manifolds.

0.1 Applications of Riemannian manifolds in mathematical physics

  1. 1.

    The conformal Riemannian subcategory C of M, whose objects are Riemannian manifolds R, and whose morphisms are conformal mappingsMathworldPlanetmath of Riemannian manifolds cR, is an important category for mathematical physics, in conformal theories.

  2. 2.

    It can be shown that, if (R1,g) and (R2,h) are Riemannian manifolds, then a map f:R1R2 is conformal (http://planetmath.org/ConformalMapping) iff f*h=s.g for some scalar field s (on R1), where f* is the complex conjugateMathworldPlanetmath of f.

0.1.1 Category of pseudo-Riemannian manifolds

The category of pseudo-Riemannian manifolds (http://planetmath.org/PseudoRiemannianManifold) P that generalize Minkowski spacesMathworldPlanetmath Mk is similarly defined by replacing the Riemanian manifoldsMathworldPlanetmath R in the above definition with pseudo-Riemannian manifoldsMathworldPlanetmath RP. Pseudo-Riemannian manifolds RPs were claimed to have applications in Einstein’s theory of general relativity (GR), whereas the subcategoryMathworldPlanetmath 𝐌𝐢𝐧𝐤 of four-dimensional Minkowski spaces in P plays the central role in special relativity (SR) theories.

Title Riemannian manifolds category RM
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 cR