Theorema Egregium

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Theorem

The Gaussian curvature of a surface does not change if one bends the surface without stretching it.

That is, Gaussian curvature can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient $3$-dimensional Euclidean space.

Thus the Gaussian curvature is an intrinsic invariant of a surface.


Proof


Historical Note

The Theorema Egregium was proved by Carl Friedrich Gauss in his $1827$ work Disquisitiones Generales circa Superficies Curvas.


Linguistic Note

Theorema Egregium is Latin for remarkable theorem.

Note that the word egregium, literally meaning outside the flock, or possibly outside the realm, is the root of the English word egregious.

At one time, egregious meant outstandingly good, but the meaning has drifted over time, and now it means appallingly bad.