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# 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.

# 0.1.1 Category of pseudo-Riemannian manifolds

The category of pseudo-Riemannian manifolds $\mathcal{R}_{P}$ 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 ($GR$), whereas the subcategory ${\bf Mink}$ of four-dimensional Minkowski spaces in $\mathcal{R}_{P}$ plays the central role in special relativity ($SR$) theories.

## Mathematics Subject Classification

30E20*no label found*18-00

*no label found*53B20

*no label found*53B21

*no label found*

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