A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem yields spherical, Euclidean and hyperbolic space. This is a uniform analytic construction of these three fundamental spaces.
Q. Now all three of these spaces have a synthetic description: Euclidean geometry by Euclid, Hyperbolic geometry by Lobachevsky and spherical geometry by Riemann.
Q. Is there a uniform description of all three in that there is an absolute geometry'absolute geometry' that underlies them?
ItTo explain what I mean by this, recall it was Bolyai who introduced the term 'absolute geometry' by dropping the parallel axiom from Euclid's plane geometry called the remaining axiomatic system this name. It turns out we can still construct parallel lines here and hence it cannot underlie spherical geometry. However, it underlies both the Euclidean and hyperbolic plane. According to Wikipedia's page on this topic, it is possible to modify the axioms so we also include spherical geometry as well as "elliptic geometry" which I think is the geometry of the real projective plane. They don't give any details but do refer to Ewald 1971, Geometry: An Introduction.
I don't have access to this book, can someone familiar with it please verify Wikipedia's claim and perhaps give a summary of the axioms used if so.