14 Nov 2005
A friend of mine asked a question about the possible ways of bending a piece of paper. (For example, you can easily bend the paper into a cylinder, but you can never bend it into a sphere.) He's got a background in math so he guessed that the answer might have something to do with Gaussian curvature, but he wasn't really sure what he was looking for. Here's my reply:
I'll start with the punchline. Roughly speaking, you can only bend a sheet of paper in one direction at a time (but that direction could be different from point to point on the sheet).
More precisely, the criterion you want in general is "any surface whose Riemann curvature tensor components are all zero". That's a unique characterization of an intrinsically flat surface in any dimension, which is what you want. ("Intrinsic" curvature has to do with whether the surface itself is necessarily curved; a sheet of paper necessarily is not. "Extrinsic" curvature has to do with how you've embedded the surface in a larger space.)
That sounds complicated, and in general it is. But the number of independent Riemann tensor components for a surface of dimension d is d2(d2 - 1)/12. So for d = 2, that's just one component to deal with. Easy!
In particular, in two dimensions the Riemann curvature tensor is proportional to the Ricci curvature scalar R, which is (ta da!) equal to -2 times the Gaussian curvature K, just as you guessed. And as you may or may not have learned before, the Gaussian curvature at a point on a surface is inversely proportional to the product of the two "principal radii of curvature": K = 1/(r1 r2).
[Added later: The "radius of curvature" at a point on a curve describes how much the curve is bending at that point by comparing it a circle. A circle with large radius bends slowly, while a circle with small radius bends quickly: curvature is inversely proportional to radius, so a perfectly straight line has infinite radius of curvature. Meanwhile, at a point on a 2D surface, the two "principal radii of curvature" are the largest and smallest radius of curvature possible for any curve on the surface that passes as straight as possible through the point. To describe that in detail would make this an even longer digression!]
So in practice, a surface is not "intrinsically flat" unless at every point, there is at least one direction that is "extrinsically flat". [Added later: For any nitpickers out there, no, "saddle points" aren't allowed: the flat directions at such points are not principal directions.] That's the source of the punchline I gave at the beginning. (And thanks! I don't usually get asked questions this closely related to my work in everyday life!)
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Copyright © 2006 by Steuard Jensen.