Absolute geometry (as I know the term) is just (Euclidean geometry) $-$ (parallel postulate).
(It is sometimes also called neutral geometry, because it is "neutral" w.r.t. the parallel postulate.)
The only spaces I know of which satisfy its axioms are Euclidean spaces and hyperbolic spaces, both of which are obviously Riemannian manifolds.
Question: Are these the only possible metric spaces which can satisfy the axioms of absolute geometry? If so, is there a way to prove this?
Attempt: By Postulate 4 here, it follows that absolute geometry is a subset of metric geometry. In addition to the existence of a metric, it also requires the existence of lines and angles, in order for the remaining postulates to be defined.
Both lines and angles appear to be a concept of ordered geometry, thus we have to restrict to metric spaces with some notion of intermediacy, i.e. "betweenness".
One way to establish such a notion is by restricting to geodesic spaces, and then to say that the point $P$ lies between the points $A$ and $B$ if and only if it is on a geodesic connecting $A$ and $B$. (The uniqueness of geodesics seems unnecessary, for example elliptic geometry should also be an ordered geometry, I think, but elliptic geometry does not have unique geodesics.)
But is it really necessary, rather than just sufficient, to restrict to geodesic spaces, in order to have ordered geometry? Ordered geometry doesn't even seem like it requires the existence of a metric, so why would it require the existence of geodesics?
Moreover, I see no reason how or why to restrict further from geodesic spaces to Riemannian manifolds in order to satisfy the axioms of absolute geometry.