Categoricity of Hyperbolic Geometry

Question

I have been reading George E Martin's classic text on Foundations of Geometry. In this book, the authors states that "axioms for hyperbolic geometry are not categorical" rather they are "similar" ie. the system can be made categorical by adding an additional axiom fixing the distance scale. Recently, I attended a lecture on Axiomatic Geometry and the lecturer said that both Euclidean and Hyperbolic Geometry are categorical. Is it because of the slight difference in meaning of "isomorphic models" by two people?

Answer 1

Hyperbolic geometry, as a Riemann geometry, is modelled by the plane of constant negative curvature $\kappa < 0$. However $\kappa$ can be any negative number. And if $\kappa \neq \kappa^\prime$, then two Riemannian surfaces of constant negative curvature $\kappa$ and $\kappa^\prime$ respectively are not isometric, i.e. there is no distance-preserving diffeomorphism between them. Perhaps this lack of isometry is what Martin refers to as having to "fixing the distance scale" in order to get a unique surface.

As an axiomatic geometry, non-Euclidean geometry is absolute geometry plus the axiom that, given a point $P$ that is not on a given line $\ell$, there are infinitely many line through $P$ that are parallel to $\ell$. Absolute geometry plus the Playfair Axiom, i.e. that there is exactly one parallel line through $P$ gives Euclidean Geometry. The language of absolute geometry involve points, lines, and perhaps circles. It talks about betweeness, congruence, containment etc. Any proposition that can be proven by the axioms of non-Euclidean geometry is satisfied by every constant negative curvature Riemannian surface, regardless of its curvature $\kappa < 0$, where lines are interpreted as geodesics. In this sense, the language of axiomatic geometry is unable to detect the distance scale because we are not comparing between two Riemannian surfaces of different constant negative curvature, but only comparing line segments in the same model when asked whether they are congruent.

The axiomatisation of absolute geometry was made rigourous by Hilbert. Models of absolute geometry are known as Hilbert planes. Other than the hyperbolic model, there are other models of non-Euclidean geometry, which have been classified by Pejas. So non-Euclidean geometry is not categorical. Do see Greenberg's article.