= 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.
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