DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

56 Chapter 2. Surfaces: Local TheoryWhat is quite remarkable about these formulas is that the Christoffel symbols, which tell us aboutthe tangential component of the second derivatives x •• , can be computed just from knowing E, F ,and G, i.e., the first fundamental form.Example 2. Let’s now recompute the Christoffel symbols of the unit sphere and compare ouranswers with Example 1. Since E =1,F =0,and G = sin 2 u,wehave[ ] [ ][ ] [ ]Γuuu 1 0 0 0Γuuv =0 csc 2 =u 0 0[ ] [ ][ ] [ ]Γuvu 1 0 00Γuvv =0 csc 2 =u sin u cos u cot u[ ] [ ][] []Γvvu 1 0 − sin u cos u − sin u cos uΓvvv =0 csc 2 =.u 00Thus, the only nonzero Christoffel symbols are Γuv v =Γvu v = cot u and Γvv u = − sin u cos u, asbefore.▽By Exercise 2.2.2, the matrix of the shape operator S P with respect to the basis {x u , x v } is[ ] [ ] −1 [ ][]a c E F l m 1 lG − mF mG − nF==b d F G m n EG − F 2 .−lF + mE −mF + nENote that these coefficients tell us the derivatives of n with respect to u and v:(††)n u = D xu n = −S P (x u )=−(ax u + bx v )n v = D xv n = −S P (x v )=−(cx u + dx v ).We now differentiate the equations (†) again and use equality of mixed partial derivatives. Tostart, we havex uuv =(Γuu) u v x u +Γuux u uv +(Γuu) v v x v +Γuux v vv + l v n + ln v=(Γuu) u v x u +Γuu( u Γuuv x u +Γuvx v v + mn ) +(Γuu) v v x v +Γuuv (Γuvv x u +Γvvx v v + nn)+ l v n − l(cx u + dx v )= ( (Γuu) u v +ΓuuΓ u uv u +ΓuuΓ v vv u − lc ) x u + ( (Γuu) v v +ΓuuΓ u uv v +ΓuuΓ v vv v − ld ) x v+ ( Γuum u +Γuun v )+ l v n,and, similarly,x uvu = ( (Γ u uv) u +Γ u uvΓ u uu +Γ v uvΓ u uv − ma ) x u + ( (Γ v uv) u +Γ u uvΓ v uu +Γ v uvΓ v uv − mb ) x v+ ( lΓ u uv + mΓ v uv + m u)n.Since x uuv = x uvu ,wecompare the indicated components and obtain:(x u ): (Γuu) u v +ΓuuΓ v vv u − lc =(Γuv) u u +ΓuvΓ v uv u − ma(♦) (x v ): (Γuu) v v +ΓuuΓ u uv v +ΓuuΓ v vv v − ld =(Γuv) v u +ΓuvΓ u uu v +ΓuvΓ v uv v − mb(n): l v + mΓuu u + nΓuu v = m u + lΓuv u + mΓuv.v