Theorema Egregium, which is Latin for "Remarkable Theorem," is a fundamental result in differential geometry, particularly in the study of surfaces. It was formulated by the mathematician Carl Friedrich Gauss in 1827. The theorem states that the Gaussian curvature of a surface is an intrinsic property, meaning it can be determined entirely by measurements made within the surface itself, without reference to the surrounding space.
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Gaussian curvature depends only on the first fundamental form and is therefore preserved by local isometries. For a parametrized surface with metric coefficients , decompose its second derivatives by the Gauss formulaTaking tangent inner products determines the lowered Christoffel symbols solely from the metric:Comparing and gives the Gauss equationConsequentlyand the final expression uses only and its first two derivatives.