gauss_celebrated_theorem.pdf

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Math 230a (Fall 2025) Yum-Tong Siu 1 Celebrated Theorem of Gauss §1. Introduction. The Celebrated Theorem of Gauss (Theorema Egregium) says that the Gaussian curvature of a surface in the 3-dimensional Euclidean space, which is defined as the product of the two extremal curvature of curves cut out from a plane containing the normal, depends only on the metric of the surface and not on how the surface is embedded inside the 3-dimensional Euclidean space. Here we give the precise definition of the Gaussian curvature…

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