In the relativistic point of view, the geometry of space should evolve with time, in a manner directed by Einstein equations. I will briefly summarize two interesting aspects with open questions:Bianchi cosmologies: these are 3+1 dimensional lorentzian manifolds satisfying Einstein equation (for this talk, in the void) and admitting a locally free isometric spacelike action by a 3-dimensional Lie group. The space of Bianchi cosmologies, as a whole, admits a very rich and interesting dynamical feature which has not yet been fully investigated.
Constant curvature case: interesting and paradigmatic cases of solutions of Einstein equations (even if physically questionable) are space-times with constant curvature (i.e locally modeled on Minkowski, de Sitter or anti-de Sitter space). In the 2+1 dimensional case, G. Mess gave a very nice description of these space-times and a close connection with Teichmüller space. In the higher dimensional case, they give rise to a proof of the following:
Theorem:Let Gamma be a cocompact lattice in $SO(1,n) (n\geq 2)$. Then, in the space $Rep(\Gamma, SO(2,n))$ of representations of $\Gamma$ into $SO(2,n)$, every representation contained in the connected component containing the inclusion $\Gamma \subset SO(1,n) \subset SO(2,n)$ is faithful and discrete.