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.
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.
In the case of dimension 3 I only know the model $\mathbb{H}^3$, a generalization of the Poincare half plane.
Remarkably, the three "two-dimensional geometries of constant curvature" live inside $\mathbb{H}^3$ (Needham_2021 page 82). Of course the hyperbolic plane is contained in $\mathbb{H}^3$, but also the Euclidean plane (for us, looking from outside $\mathbb{H}^3$, 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 $\mathbb{H}^3$ would be also a sphere with different centre).
In the two-dimensional case is a subgroup of Moebius transformations but in the tridimensional case is the full group of Moebius transformations.
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Author of the notes: Antonio J. Pan-Collantes
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