One example: oriented (i.e., reflections not included) elliptic geometry (i.e., the surface of an n -sphere with opposite points identified) and oriented spherical geometry (the same non-Euclidean geometry, but with opposite points not identified) have isomorphic automorphism group, SO(n +1) for even n. These may appear to be distinct. It turns out, however, that the geometries are very closely related, in a way that can be made precise.