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Isotropy and homogeneity of space leads to the spacetime metric of the form $$ ds^2=-dt^2+d\sigma_k^2, $$ where $d\sigma_k^2$ is the metric on one of the standard manifolds (the 3-sphere, Euclidean 3-space, and the hyperbolic 3-space) depending on the curvature $k$. Here $k$ may depend on $t$, and there seem to be nothing to prevent $k$ from changing its sign.

However, the Robertson-Walker model assumes $$ ds^2=-dt^2+a(t)^2d\sigma_k^2, $$ where now $k$ is fixed, and the time evolution of the geometry is delegated to the scale factor $a(t)$. This means that the curvature sign-change is forbidden in the Robertson-Walker models.

My question is: Are there cosmological models where the space is allowed to change the sign of its curvature, or is there an argument that it simply cannot happen?

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RW assumes that matter is uniformly distributed over the spatial slices, and elliptic, flat, and hyperbolic geometries have very different distributions, in terms of the amount of matter within a given distance of any given point. There's no way you could move the matter around to be homogeneous in a different geometry without violating homogeneity in the process. So homogeneity is enough to force $k$ to be constant, it isn't a separate assumption.

The real world is obviously somewhat inhomogeneous, and if the size of the universe (or the approximately RW part of it) is small compared to the radius of curvature, then the value of $k$ could be somewhat ill-defined. This isn't really a change of geometry, though, it's just that different RW models with different values of $k$ would all approximate the reality about equally well.

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