bapowell said:
The global geometry of the universe is determined by its energy density -- the universe can be closed, flat, or open, depending on whether the energy density is greater than, equal to, or less than the critical energy density, respectively. If the the universe is closed, the spatial topology/geometry is that of a sphere, and an observer could indeed 'wrap around' the universe and return to their starting point. A flat universe can be planar or have the topology of a torus. An open universe has hyperbolic geometry. In each of these cases, the time coordinate is non-compact.
This isn't entirely true. This is a simplification that results from the assumption that the energy density of the universe is a constant everywhere, which it isn't (obviously: the Earth is a bit more dense than the space between the Earth and the Moon, for instance).
When you take into account the fact that the density of the universe varies from place to place, this assumption breaks down. For instance, if we end up measuring our region of the universe to be closed, that could just be due to our observable region being slightly overdense. It doesn't mean the universe as a whole is closed, because the rest of it could, on average, be less dense.
Instead, we are forced to separate between the global geometry (whether or not the universe wraps back on itself) and our local geometry. Whether we measure a local universe that is closed, open, or flat, the universe as a whole might well wrap back on itself or not depending upon whether or not the locally-measured curvature is representative of the global curvature.
And even then, it is also possible for a flat or open universe to wrap back on itself. This is most easily understood in the context of a flat universe, with the simple arcade game of asteroids: in the game of asteroids, the play field is flat (and yes, it does have perfectly flat geometry in the proper General Relativistic sense). And yet, the game field wraps back on itself: when you go off the right side of the screen, you reappear on the left side. This is known as a toroidal topology, because if you took a sheet and wrapped it up into a tube so that its edges touched, then wrapped that tube so that the ends of the tube met, you'd have a torus (a donut shape).
So unfortunately, measuring the local curvature will tell us very little about whether or not the universe wraps back on itself. It may, it may not. We don't know.
