DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

100 Chapter 3. Surfaces: Further Topicsb. Compute ω 12 and dω 12 and verify that K = −1.9. Use moving frames to redoa. Exercise 3.1.9b. Exercise 3.1.1010. a. Use moving frames to reprove the result of Exercise 2.3.13.b. Suppose there are two families of geodesics in M making a constant angle θ. Prove ordisprove: M is flat.11. Recall that locally any 1-form φ with dφ =0can be written in the form φ = df for somefunction f.a. Prove that if a surface M is flat, then locally we can find a moving frame e 1 , e 2 on M sothat ω 12 =0. (Hint: Start with an arbitrary moving frame.)b. Deduce that if M is flat, locally we can find a parametrization x of M with E = G =1and F =0. (That is, locally M is isometric to a plane.)12. (The Bäcklund transform) Suppose M and M are two surfaces in R 3 and f : M → M is asmooth bijective function with the properties that(i) the line from P to f(P )istangent to M at P and tangent to M at f(P );(ii) the distance between P and f(P )isaconstant r, independent of P ;(iii) the angle between n(P ) and n(f(P )) is a constant θ, independent of P .Prove that both M and M have constant curvature K = −(sin 2 θ)/r 2 . (Hints: Write P = f(P ),and let e 1 , e 2 , e 3 (resp. e 1 , e 2 , e 3 )bemoving frames at P (resp. P ) with e 1 = e 1 in the directionof −→P P . Letting x and x = f ◦x be local parametrizations, we have x = x + re 1 . Differentiatethis equation and deduce that ω 12 = cot θω 13 − 1 r ω 2.)4. Calculus of Variations and Surfaces of Constant Mean CurvatureEvery student of calculus is familiar with the necessary condition for a differentiable functionf : R n → R to have a local extreme point (minimum or maximum) at P :Wemust have ∇f(P )=0.Phrased slightly differently, for every vector V, the directional derivativef(P + εV) − f(P )D V f(P )=limε→0 εshould vanish. Moreover, if we are given a constraint set M = {x ∈ R n : g 1 (x) =0,g 2 (x) =0,...,g k (x) =0}, the method of Lagrange multipliers tells us that at a constrained extreme pointP we must havek∑∇f(P )= λ i ∇g i (P )i=1for some scalars λ 1 ,...,λ k . (There is also a nondegeneracy hypothesis here that ∇g 1 (P ),...,∇g k (P )be linearly independent.)Suppose we are given a regular parametrized surface x: U → R 3 and want to find—without thebenefit of the analysis of Section 4 of Chapter 2—a geodesic from P = x(u 0 ,v 0 )toQ = x(u 1 ,v 1 ).