I found a very interesting question on another forum:
If one starts going always in the same direction does one then return to the starting point (in our universe)?
I mean we dont know so much about the universe and a recent theory proposes the universe being a dodecahedron with opposite faces identified. So I generalize the question a bit:
Are there straight lines on closed manifolds that are not closed?
But indeed this question can be answered with 'yes', as is hopefully clear from the following pictures (without further explanation). We take as counterexample the 2dim torus:
Perhaps it should be added that the 2-torus is submanifold of the 3-torus and so if our universe would be a 3-torus there would be directions to start with, such that we would never return to our starting point.
What now really is not obvious to me, whether there are closed manifolds, with points, where *every* straight line through this point is not closed. I.e. regardless what initial direction we start our walk, we would never return to our starting point.
Another associated question is: If we start in a direction that does not return to the starting point on a closed manifold, do we then automaticly/generally come arbitrary close to our starting point (as is the case with the 2-torus example).