DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 57Analogously, comparing the indicated components of x uvv = x vvu ,wefind:(x u ): (Γuv) u v +ΓuvΓ u uv u +ΓuvΓ v vv u − mc =(Γvv) u u +Γ u(x v ): (Γuv) v v +ΓuvΓ u uv v − md =(Γvv) v u +ΓvvΓ uuu v − nb(n): m v + mΓuv u + nΓuv v = n u + lΓvv u + mΓvv.vThe two equations coming from the normal component give us theCodazzi equationsvvΓ u uu +Γ vvvΓ u uv − nal v − m u = lΓ u uv + m ( Γ v uv − Γ u uu)− nΓvuum v − n u = lΓ uvv + m ( Γ vvv − Γ u uv)− nΓvuv .ln − m2Using K =EG − F 2 and the formulas above for a, b, c, and d, the four equations involving the x uand x v components yield theGauss equationsEK = ( Γuuv )v − ( Γuvv )u +Γu uuΓuv v +ΓuuΓ v vFK = ( Γuvu )u − ( Γuuu )v +Γv uvΓuv u − ΓuuΓ v vvuFK = ( Γuvv )v − ( Γvvv )u +Γu uvΓuv v − ΓvvΓ uuuvGK = ( Γ uvv)u − ( Γ u uv)v +Γu vvΓ u uu +Γ vvvΓ u uv − ( Γ u uvFor example, to derive the first, we use the equation (♦) above:(Γvuu)v − ( Γ v uvvv − Γ u uvΓ v uu − ( Γ v uv)u +Γu uuΓ v uv +Γ v uuΓ vvv − Γ u uvΓ v uu − ( Γ v uv) 2 = ld − mb=) 2) 2 − Γvuv Γ uvv.1 ( ) E(ln − m 2 )l(−mF + nE)+m(lF − mE) =EG − F 2 EG − F 2 = EK.In an orthogonal parametrization (F = 0), we leave it to the reader to check in Exercise 2 thatK = − 1 ( (2 √ Ev) (√EG+ Gu) )(∗)√ .EG v EG uOne of the crowning results of local differential geometry is the followingTheorem 3.1 (Gauss’s Theorema Egregium). The Gaussian curvature is determined by onlythe first fundamental form I. That is, K can be computed from just E, F , G, and their first andsecond partial derivatives.Proof. From any of the Gauss equations, we see that K can be computed by knowing any oneof E, F , and G, together with the Christoffel symbols and their derivatives. But the equations (‡)show that the Christoffel symbols (and hence any of their derivatives) can be calculated in termsof E, F , and G and their partial derivatives. □Corollary 3.2. If two surfaces are locally isometric, their Gaussian curvatures at correspondingpoints are equal.