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, thena 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.

单词	RiemannianManifoldsCategoryRM
释义	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, thena 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.
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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 $R_{M}$