How to distinguish between walking on a sphere and walking on a torus? Imagine that you're a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can't see the difference from this point of view.
If you are interested, this question arose while I was watching this video about the shape of space by Jeff Weeks.

I think that the higher rated answers answer the question except that they require a kind of total knowledge and impracticality. 
What if we assume that light travels in the usual manner on the surface, in 'straight' lines?  Then a program of sending out beams from a single, fixed point and measuring the possibly returned beam could yield sufficient information. 
On a sphere the beam will always return and with the same attenuation from 180 degrees from the emitter regardless of the direction chosen. Whereas on a torus there will be a range of values with two minimums. The total flight time required could, I haven't checked, be made arbitrarily large depending on the angle chosen. 
