Theorema Egregium

Differential Geometry · Axiom Academy

Gauss's "Remarkable Theorem" on the intrinsic nature of curvature The Gaussian curvature K of a surface can be expressed entirely in terms of the coefficients E , F , G of the first fundamental form and their derivatives. The first fundamental form measures distances on a surface: where E , F , G are functions that encode how distances are measured in the surface's local coordinates. 2. Intrinsic vs. Extrinsic Properties Before Gauss, curvature was defined using the second fundamental form , which depends on how a surface bends in 3D space (extrinsic). The Theorema Egregium shows that Gaussian curvature is actually intrinsic . Consequence: Gaussian curvature is preserved under isometries —distance-preserving maps. If you can bend a surface without stretching or tearing it, the Gaussian curvature remains unchanged. 3. Proof Outline: Christoffel Symbols The proof relies on Christoffel symbols , which describe how the basis vectors of the surface's coordinate system change as you move along the surface. The Christoffel symbols Γ k ij are defined in terms of E, F, G: The Gaussian curvature can be expressed using these symbols through the Gauss equation : Through careful calculation involving the Christoffel symbols, Gauss derived an explicit formula for curvature in terms of E, F, G and their derivatives: This formula is remarkable because: It contains only E, F, G and their first and second derivatives It requires no reference to the second fundamental form

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