Is the surface of a donut distinguishable from pac-man's world? Pac-man's world is topologically like the surface a donut. Pac-man's world is also locally flat. For example, the interior angles of a small triangle will always add up to 180 degrees. Conversely, when I picture a donut in my mind, it's not very flat. Dissimilar to pac-man's world. Yet I'm having trouble coming up with a way to distinguish between the donut's surface and pac-man's world from the inside of the space. (When I say 'donut' I mean a literal donut embedded in 3d space, with a hole through the middle and all that, but from the perspective of ants walking on the surface trying to figure out if they're pac-man or not.) 

Your picture of the surface of a donut is correct: it is curved.

Not all metrics on a topological surface will have the same local properties.

For example, although a donut surface does indeed have flat metrics, it also has nonflat metrics, and the donut surface is one of them. 2d entities living on the surface of a donut could indeed distinguish their donut from pac-man's world, by making little measurements of length and area. For instance, they could measure the radius and area of a small circle on their world, and for almost any circle they chose, they would discover that the formula $A=\pi r^2$ fails, so their world is not flat.

More specifically, if they take a small circle centered at a point on the inside of the torus, $A > \pi r^2$ (the surface is negatively curved in the inside region). On the other hand, if they take a small circle centered at a point on the outside of the torus, $A < \pi r^2$ (the surface is positively curved in the outside region).