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Theorema Egregium

The Theorema Egregium ('Remarkable Theorem') is an important theorem of Gauss concerning the curvature of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, that is, it does not depend on how the surface might be imbedded in (3-dimensional) space.

Gauss presented the theorem this way (translated from Latin):

Thus the formula of the preceeding article leads itself to the remarkable
Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.

In more modern language the theorem may be stated this way:

The Gaussian curvature of a surface is invariant under local isometries.

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.

Some simple applications

You can't bend a piece of paper onto a sphere (more formally, the plane and the 2-sphere 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 helicoid (two very different-looking surfaces) have the same Gaussian curvature. (The two surfaces are locally isometric.)