to the precession of the perihelion of the Mercury). for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces).
Now, from the ants' perspective who aren't aware that they are living in a curved space, this will seem that there is a force between them because their distance will be changing in time non-linearly (think about those meridians again). He'll also produce circle and the two circles will cross at two points (you can imagine those circles as meridians and the crossing points as a north resp. Imagine a second ant and suppose he'll start to walk from the same point as the first ant and at the same speed but into a different direction. If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve). Luboš's answer is of course perfectly correct.
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