Hyperbolic geometry
It is the mathematical theory that results from substitute the 5th Euclid's axiom by the existence of an infinite number of parallels to a given line by an exterior point. The name is in part justified by this reflection: Elliptic, hyperbolic and parabolic geometry.
Dimension 2
In the case of dimension 2 it can be "modeled" in several ways:
- The Poincare half plane
- The pseudosphere
- The Poincare disk
Lambert showed, from the axioms, that the angular excess of a triangle is a fixed negative multiple of its area. Beltrami thought that maybe this had to do with local Gauss-Bonnet theorem applied to a surface with constant
Dimension 3
In the case of dimension 3 I only know the model
Remarkably, the three "two-dimensional geometries of constant curvature" live inside
Group of isometries
In the two-dimensional case is a subgroup of Moebius transformations but in the tridimensional case is the full group of Moebius transformations.