 Having separated our "d" dimensional space into the product of two subspaces, Einstein's field equations allow us to solve for the curvature of these two sub-manifolds, and we find 
We find that the Ricci curvatures of the "s"- and "(d-s)"-dimensional sub-manifolds are necessarily opposite in sign.One must have positive curvature, and the other must have negative curvature, and so one of these manifolds must be compact.Consequently, at scales significantly larger than that of the compact manifold, the universe appears to have either "s" or "(d-s)" dimensions, as opposed to the underlying "d".