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.
Get a (two-dimensional) dog and a very long (one-dimensional) leash.  Send your dog out exploring, letting the leash play out.  When the dog returns, try to pull in the leash.  (Meaning, you try to reel in the loop with you and the dog staying put.)  On a sphere, the leash can always be pulled in; on a torus, sometimes it can't be.

(See homotopy.)
Topologically, two geodesics emanating from the same point on a sphere will intersect in an antipodal point, whereas this is not true on a torus.  So with a friend (and enough time), you can do this.