DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

54 Chapter 2. Surfaces: Local Theoryc. Suppose M is represented locally near P as in Exercise 20. Show that for small positivevalues of c, the intersection of M with the plane z = c “looks like” the Dupin indicatrix.How can you make this statement more precise?22. Suppose the surface M is given near P as a level surface of a smooth function f : R 3 → R with∇f(P ) ≠0.Aline L ⊂ R 3 is said to have (at least) k-point contact with M at P if, given anylinear parametrization α of L with α(0) = P , the function f = f ◦α vanishes to order k − 1,i.e., f(0) = f ′ (0) = ···= f (k−1) (0) = 0. (Such a line is to be visualized as the limit of lines thatintersect M at P and at k − 1 other points that approach P .)a. Show that L has 2-point contact with M at P if and only if L is tangent to M at P , i.e.,L ⊂ T P M.b. Show that L has 3-point contact with M at P if and only if L is an asymptotic directionat P . (Hint: It may be helpful to follow the setup of Exercise 20.)c. (Challenge) What does it mean for L to have 4-point contact with M at P ?3. The Codazzi and Gauss Equations and the Fundamental Theorem of SurfaceTheoryWe now wish to proceed towards a deeper understanding of Gaussian curvature. We have tothis point considered only the normal components of the second derivatives x uu , x uv , and x vv .Nowlet’s consider them in toto. Since {x u , x v , n} gives a basis for R 3 , there are functions Γuu, u Γuu,vΓuv u =Γvu, u Γuv v =Γvu, v Γvv, u and Γvv v so that(†)x uu =Γ u uux u +Γ v uux v + lnx uv =Γ u uvx u +Γ v uvx v + mnx vv =Γ uvvx u +Γ vvvx v + nn.(Note that x uv = x vu dictates the symmetries Γ uv • =Γ vu.) • The functions Γ •• • are called Christoffelsymbols.Example 1. Let’s compute the Christoffel symbols for the usual parametrization of the sphere.By straightforward calculation we obtainx u = (cos u cos v, cos u sin v, − sin u)x v =(− sin u sin v, sin u cos v, 0)x uu =(− sin u cos v, − sin u sin v, − cos u) =−x(u, v)x uv =(− cos u sin v, cos u cos v, 0)x vv =(− sin u cos v, − sin u sin v, 0) = − sin u(cos v, sin v, 0).(Note that the u-curves are great circles, parametrized by arclength, so it is no surprise that theacceleration vector x uu is inward-pointing of length 1. The v-curves are latitude circles of radiussin u, so, similarly, the acceleration vector x vv points inwards towards the center of the respectivecircle.)