Okay, so I saw on wikipedia that you can have a 'flat torus' --- at least a 2 dimensional torus inbedded in 4-space. So you can also have a 3-torus, that's flat?
And would time just be added on as an additional, orthogonal dimension?

I'm a little confused about how/why this is considered flat. In the same wikipedia article, it says that a (2)cylinder is also flat---this is news to me. The analogy they make is that bending a flat piece of paper into a cylinder doesn't require any stretching/deformation of the paper. Okay. And I also realize that a 2cylinder would still have triangles whose angles add to 180 degrees... etc etc.

These things definitely aren't true for the standard 2-torus in 3D; I would have assumed the 3-torus was the same.

My understanding of differential geometry is rudimentary--only what I've gleamed from attempts at GR. I have no experience with 'topology' per se. None-the-less, equations would be welcome.

The metric on this 2-torus in 3D-space would not be the regular one you have in our three-dimensional space. To see this, look at the lines in images like this one. They span a grid of equidistant lines - but in our 3D-space, they are not equidistant.

You can consider the 3-torus as three-dimensional cube, where the faces are connected via "magic".

It's not really magic. It's just counter-intuitive. General Relativity simply doesn't require that our 3+1 dimensional space-time be embedded in a higher-dimensional space-time for it to be connected.

When people think of a connected space-time, they usually think of a space-time that wraps back on itself in higher dimensions. For example, the surface of a sphere is a two-dimensional curved surface that wraps back on itself embedded in three dimensions. Within General Relativity, we can refer to such a surface without ever referencing any more than those two dimensions of the surface. And so, for example, if we find that our universe has the spatial topology of a 3-sphere, then there wouldn't be any reason to believe that it was embedded in higher dimensions in order to wrap back on itself.

And then there are projections of moving submanifolds within a flat torus. Makes me wonder: in a cosmological context, what if it's not spacetime that's actually curved? The old idea of spacetime curvature got a boost from measuring astronomical lensing of light, but what if some mass-induced polarization of the medium of quantum transmission is found to be the cause? In recent explorations of "nothing", the zero points of each quantum field allow for much to happen between the level of an absolutely ultimate space, and the level of lepto-quark phenomena.

What would be fun is to find a relationship that emulates what has been treated as "spacetime curvature". The way time appears always to move forward, and how gravity appears always to suck, may point to something about the motion of our massive manifold when projected within a toroidal topology. In such a case, what would it take to make it appear that space is performing a period expansion, or the reverse?

With LHC warming up, and JWST soon to launch... great time to be in the field.