Cosmological spacetimes with spatially constant sign-changing curvature

Globally hyperbolic spacetimes endowed with a time function $t$ whose spacelike slices $t=t_0$ have constant curvature $k(t_0)$ and where the sign of $k(t_0)$ (as well as the topology of the slice) varies with $t_0$, can be constructed despite some common claims about the implications of the classical Cosmological Principle. Here, we stress the possibilities of these cosmologies and announce the development of new models obtained in collaboration with G. Garc\'{\i}a-Moreno, B. Janssen, A. Jim\'enez-Cano, M. Mars and R. Vera