Riemannian manifolds category
Definition 0.1.
A category whose objects are all Riemannian manifolds
and whose morphisms are mappings between Riemannian manifolds is defined as the category of Riemannian manifolds.
0.1 Applications of Riemannian manifolds in mathematical physics
- 1.
The conformal Riemannian subcategory of , whose objects are Riemannian manifolds , and whose morphisms are conformal mappings
of Riemannian manifolds , is an important category for mathematical physics, in conformal theories.
- 2.
It can be shown that, if and are Riemannian manifolds, thena map is conformal (http://planetmath.org/ConformalMapping) iff for some scalar field (on ), where is the complex conjugate
of .
0.1.1 Category of pseudo-Riemannian manifolds
The category of pseudo-Riemannian manifolds (http://planetmath.org/PseudoRiemannianManifold) that generalize Minkowski spaces is similarly defined by replacing the Riemanian manifolds
in the above definition with pseudo-Riemannian manifolds
. Pseudo-Riemannian manifolds s were claimed to have applications in Einstein’s theory of general relativity (), whereas the subcategory
of four-dimensional Minkowski spaces in plays the central role in special relativity () theories.