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:

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 K=1, and this idea led him to the pseudosphere.

Dimension 3

In the case of dimension 3 I only know the model H3, a generalization of the Poincare half plane.
Remarkably, the three "two-dimensional geometries of constant curvature" live inside H3 (Needham_2021 page 82). Of course the hyperbolic plane is contained in H3, but also the Euclidean plane (for us, looking from outside H3, it takes the shape of a sphere touching the "horizon", and it is called the horosphere; for the insiders I don't know what it is) and the spherical geometry (for us is any other sphere not touching the horizon, and for the "inhabitants" of H3 would be also a sphere with different centre).

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.