non-Euclidean Riemannian geometry

Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line." In order to achieve a consistent system, however, the basic axioms of neutral geometry must be partially modified. Most notably, the axioms of betweenness are no longer sufficient (essentially because betweenness on a great circle makes no sense, namely if A and B are on a circle and C is between them, then the relative position of C is not uniquely specified), and so must be replaced with the axioms of subsets.

axiom of subsets | elliptic space | Euclidean geometry | hyperbolic geometry | non-Euclidean geometry