When thinking about the shape of the universe in my story I have been looking into it being a 3-manifold. As I understand it, space would be a 3-dimensional surface and for a small scale observer space would seem flat but if somehow it could be viewed from outside its shape would be curved.
Something I am not able to visualise or probably that I don't understand is if there is a thickness which would be a much shorter length than moving in other directions? So if you moved across the circumference of the surface/shape you would either return to where you started or move in a seemingly endless path looping around the shape but if you moved at 90 degrees to the circumference, would it be a much shorter distance to travel and if so, what would happen there? Would space curve back on itself?
Two of the easiest topologies to imagine this surface on is a hollow sphere and a torus, in this image the surface of the torus would be 3-dimensional space time and the grey cross section area is the depth or thickness of space which I am wondering if it would be a much shorter distance to reach than taking other paths and what would happen to space when you reach there?
Edits have been made to the question to correct parts which were mentioned in the comments.