Since the plane and the cylinder have zero Gaussian curvature, I'm wondering, is there an "intrinsic" way of telling one from the other?
By "intrinsic" here I loosely mean a property that can be calculated and/or deduced by inhabitants of the manifold itself, without "seeing" it from a higher dimensional space.
Best Answer
Locally there is no difference, but globally there is a difference.
Pick a point on the cylinder; look at a small neighborhood of that point. One can unroll the neighborhood and lay it flat in a plane without stretching the surface, so all distances in the plane are the same as the corresponding intrinsic distances on the cylinder. So you can't tell the difference.
However, if you go far enough in a particular direction on the cylinder, you'll return to where you started. That doesn't happen in the plane.