Gauss presented the theorem this way (translated from Latin):
In more modern language the theorem may be stated this way:
The theorem is remarkable because the definition of Gaussian curvature makes direct use of the imbedding of the surface in space. So it is quite surprising that the end result does not depend on the imbedding.
You can't bend a piece of paper onto a sphere (more formally, the plane and the 2sphere are not locally isometric). The follows immediately from the fact that the plane has Gaussian curvature 0 (at all points) while no point on a sphere always has Gaussian curvature 0. (It is, however, possible to prove this special case more directly.)
Corresponding points on the catenoid[?] and the hellicoid[?] (two very differentlooking surfaces) have the same Gaussian curvature. (The two surfaces are locally isometric.)
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