What does it mean for something to be a model of hyperbolic space?

In the book "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author uses the term "analytic models of hyperbolic space." (p. 19) Some examples are the Klein model and the Hyperboloid model, which are mentioned on Wikipedia as well. However, he does not explain what such a model is. Why do they qualify as models of hyperbolic space, and how can both be valid?

You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.

The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.

In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.

If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.

A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.

For example $\mathbb{Z}$, $\mathbb{C}$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.

• Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates? – Tensor McTensorstein Jul 17 '18 at 10:27
• I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book. – Rad80 Jul 17 '18 at 10:41
• Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$. – Lee Mosher Jul 18 '18 at 2:59

There is another notion of "model" which characterizes the hyperbolic plane and is expressed in the language of differential geometry, namely:

The hyperbolic plane is the unique simply connected, complete Riemannian manifold of dimension 2 and constant curvature $-1$.

The meaning of "uniqueness" in this statement is uniqueness up to isometry. More formally, if $\mathbb H_1$ and $\mathbb H_2$ are two simply connected, complete Riemannian manifolds of dimension 2 and constant curvature $-1$ then there exists an isometry $f : \mathbb H_1 \to \mathbb H_2$.

This notion of uniqueness can be used to prove that all of the models mentioned in your question are isometric to each other, by verifying that the Klein model, the hyperboloid model, and the Poincare model are all complete, simply connected, Riemannian 2-manifolds of constant curvature $-1$.

I would say the word "model" here is used for historical purposes. The history is that people have been wondering whether Euclid's fifth postulate can be proven from the other postulates. This has been shown to be false by constructing a "model" (in the sense of the model theory), i.e., a structure which satisfies the other postulates, but not the fifth postulate. And thus we have models of non-Euclidean geometry.

Comparing with the spherical geometry could be useful. We can view the surface of Earth as a 3D object, or we can make a flat map of Earth using many well-known projections, e.g., stereographic, gnomonic, Mercator. If we apply the stereographic projection to the Minkowski hyperboloid, we get the Poincaré model of the hyperbolic plane. Likewise, gnomonic projection = Klein model, Mercator projection = band model, orthogonal = Gans model. Thus, the word "projection" would be appropriate for some of the models (we have to be very careful though, as the Minkowski hyperboloid lives in the Minkowski space, not the usual $\mathbb{R}^3$) but "model" is still more common. I have written more about this here.