[alcolade]

If the bug were truly mathematical, he would notice the curvature of S^1 and S^2 without walking around (since curvature is defined at each point), but that is obviously very pedantic of me and shows why I wouldn't write an article nearly as interesting as this one.

[improbable22]

But S^1 isn't curved. At least if we mean intrinsic curvature, which is the kind that belongs to the manifold itself.

S^2 is curved, but locally you can't tell it's not (say) RP^2.

[alcolade]

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.

[improbable22]

Not only is there no intrinsic curvature in 1D, but the 2D and 3D versions are also simplified special cases.

In 2D there is only scalar curvature, i.e. you don't need the whole Riemann tensor, just one number (at a given point).

In 3D you need the Ricci tensor, but still not the full curvature tensor. This is still much simpler, for instance IIRC this is why you cannot have gravitational waves in 3 (meaning 2+1) dimensional spacetime. In 4D you get the full complication.