# Can we have positively curved space within negatively curved spacetime?

Thinking about the universe as a whole. One could imagine that the three spatial dimensions each have the same, say positive, curvature, making space spherical, while time is negatively curved, making spacetime hyperbolic (saddle shaped). Is this possible? If it were possible, would $$k$$, the curvature term in the FWLR model, be referring to the curvature of space or of spacetime?

• De Sitter spacetime is of constant positive curvature. In such a spacetime one can have flat spatial sections, or sections with positive or negative 3d curvature. I don't know, if this is the same for anti de Sitter, but I would think so. Commented Apr 10, 2021 at 12:39

The idea of constant spatial curvature comes from the idea, i.e. observation, that the spatial part of the universe at a certain instance in time (!) should be homogenous and isotropic. This leaves the three possibilities of $$k=1,0,-1$$, if all spatial coordinates are normalised correspondingly, because these three spatial metrics fulfill the conditions.
You could think of a universe where time loops, being a circle, I think this would correspond to your idea of time having $$k=1$$ (which, again, it has not, since mathematically a 1D circle still has no curvature). I think this would make it harder to solve the Friedmann equations compared to Standard cosmology, since you would have boundary conditions for $$a(t)$$ at the beginning and everytime the time component loops, which for example forbids exponential growth as in a dark energy dominated universe.