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单词 RiemannianManifoldsCategoryRM
释义

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, thena 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.

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更新时间:2025/5/4 20:12:36