Question 1. How can one distinguish the sphere from a plane (Flatland) based solely on experiences on the surface?
Question 2. How can one distinguish the sphere from a torus based solely on experiences on the surface?
Look at shadows at the same time of day? This is a "local" feature of the surface.
This is also a local property.
Go West -> return from the East, then go North-> return from the south.
On a sphere: There will always be 2 points of intersection of the curves determined by the two routes.
On a torus: There will be only one point of intersection of the two routes.
On a plane: Go West -> you keep going... there seems to be no return???