Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3 / a / Solution

The curvature constant specifies the intrinsic spatial curvature of homogeneous constant-time slices. Positive curvature has spherical geometry, zero curvature Euclidean geometry, and negative curvature hyperbolic geometry. A conventional coordinate normalization uses $k=+1,0,-1$, with the scale factor carrying the curvature radius; rescaling that factor changes the numerical magnitude assigned to a dimensional curvature coefficient.

The sign alone does not determine the future expansion once a cosmological constant is admitted. Nor does negative or zero local curvature by itself determine the global topology: quotient spaces can share the same local curvature.