Hi, as we all know, the FLRW metric has 3 types of spatial curvatures, spherical, flat, and hyperbolic. I understand that of course due to homogeneity, this curvature must be spatially everywhere the same and so can not depend on the spatial coordinates. However, I can't recall what is the reason the curvature can't go from flat to spherical or spherical to hyperbolic during the dynamic evolution of the universe, as is assumed. Obviously changing from a compact manifold to a non-compact one seems like the manifold must "tear" or something, so it seems to be a good assumption that the universe can't switch between the 3 cases, but what is the actual mathematical or physical argument for why? It is escaping me at the moment.