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.
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.