I was thinking of the Riemann curvature tensor which is just some measure of curvature at each point. If the bug were at some point of S^2 he would notice the curvature.

Of course, this is assuming S^2 is getting its geometry from a certain embedding into R^3 that comes to mind. You could define different Riemann curvature tensors over S^2 that may have zero curvature in certain places, like squishing a balloon against a flat table for example.

I guess my point was that topology and curvature are different things (like you pointed out with your comment about RP^2), and saying that "a sphere looks flat locally" is missing the point!

Also, it didn't occur to me that this notion of curvature doesn't make sense in 1d, i.e. for S^1 like you said.