A mathematical note: Tori can be flat.

If the universe (and here we are talking about the large spacetime structure, not any of the weird tiny compactified extra string dimension stuff) is globally flat, it can still have the structure of a torus.

The torus when viewed as a 2 dimensional space in 3 dimensions, is not flat-- it has some positively curved parts (think the outer edge of the donut) and some negatively curved portions (think the saddle like regions on the inner ring of the torus.) However, the total curvature (when I sum up all contributing curvatures) on the torus is zero. This is related to a mathematical fact that the total curvature of any surface is given by a topological quantity called the genus. In simpler terms, no matter how I deform the torus, the sum of the curvature will be zero. This is very different from the sphere, whose total curvature is always 2\pi.

So, a flat universe would imply that we cannot live on a 4 sphere, because such objects must always have at least some positive curvature. However, there are examples of tori that have no curvature.

In the 2 dimensional case, it is best to see this from the ``Pac-Man'' perspective. The pacman game is played on a flat surface, and whenever you head off the top of the screen, you arrive at the bottom, and whenever you go off the left side of the screen you wind up on the right hand side. This describes a possible shape for the universe, and this shape is the torus! To see this, imagine that you took the playing field, and glue the top and bottom sides together. That would give you a cylinder. Taking the left and right sides and gluing them together would give you a torus. Now that we believe the pacman game is played on a torus, notice that the original interpretation was a flat surface. So , there is a flat representation of the torus.

To avoid some confusion and people trying to draw flat tori in 3 dimensional space, it can't be done. Every surface viewed in 3 dimensional space will necessarily have some positive curvature around its maximal value. Sorry folks!

In fact, of all the 2 dimensional (compact) surfaces, the only one that has a flat representation is the torus. So, if the universe is compact (and 2 dimensional, which seems unlikely,) there is hope for a Pacman world.