Robinson [1] proved that given the primitive notion of points, Pieri’s primitive ternary equidistance relation is sufficient to axiomatize Euclidean, elliptic, and hyperbolic geometries. Is Pieri’s relation also sufficient to axiomatize arbitrary Riemannian geometries (including “pseudo-Riemannian” geometries which are not positive-definite)?
[1] Robinson, Raphael M. (1970). Binary Relations as Primitive Notions in Elementary Geometry. Journal of Symbolic Logic 35 (1):148-148.