A user on a physics forum wanted to construct a metric of a hypothetical closed universe expanding with acceleration. When I proposed a simple example of a 3-sphere $S^3$, he asked if this space could be flat instead like "infinity mirrors". His requirements were, "if you fly straight, you come back to the same point from behind", and, "a flat space with no curvature". I am not a mathematician, so I've decided to ask the professionals here.
Intuitively, I can imagine this along one dimention, something like $x=x+D$ where $D$ is the size of the universe in the direction of $x$. As you cross $x=D$, you simply reappear at $x=0$. The same in 3 dimensions defines a cube whose opposite sides are magically joined: getting out on one side puts you back inside on the other.
If there is any rational seed in this idea, does this space have a known name or description? Would it have any unique properties? For example, would it support General Relativity (e.g. locally metric space). Would this space have detectable edges (like the sides of the cube above) or not? Are there any other ways to fulfill these requirements?
Thanks for your help!
Yes, a flat 3-space could be a $3$-torus $\mathbb T^3$.