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, then a map $f \colon R_1 \to R_2$ is conformal iff $f^* h = s.g$ for some scalar field $s$ (on $R_1$ ), where $f^*$ is the complex conjugate of $f$ .

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.