DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

62 Chapter 2. Surfaces: Local Theory2. Use the first Gauss equation to derive the formula (∗) given on p. 57 for Gaussian curvature.3. Check the Gaussian curvature of the sphere using the formula (∗) onp.57.4. Check that for a parametrized surface with E = G = λ(u, v) and F =0,the Gaussian curvatureis given by K = − 12λ ∇2 (ln λ). (Here ∇ 2 f = ∂2 f∂u 2 + ∂2 fis the Laplacian of f.)∂v2 5. Calculate the Christoffel symbols for the following parametrized surfaces. Then check in eachcase that the Codazzi equations and the first Gauss equation hold.a. the plane, parametrized by polar coordinates: x(u, v) =(u cos v, u sin v, 0)b. a helicoid: x(u, v) =(u cos v, u sin v, v)♯ c. a cone: x(u, v) =(u cos v, u sin v, cu), c ≠0♯ *d. a surface of revolution: x(u, v) = ( f(u) cos v, f(u) sin v, g(u) ) , with f ′ (u) 2 + g ′ (u) 2 =16. Prove there is no compact minimal surface M ⊂ R 3 .7. Decide whether there is a parametrized surface x(u, v) witha. E = G =1,F =0,l =1=−n, m =0b. E = G =1,F =0,l = e u = n, m =0c. E =1,F =0,G = cos 2 u, l = cos 2 u, m =0,n =18. a. Modify the proof of Theorem 3.5 to prove that a smooth, compact surface with K>0 andconstant mean curvature is a sphere.b. Give an example to show that the result of Lemma 3.6 fails if we assume k 1 has a localminimum and k 2 has a local maximum at P .9. Give examples of (locally) non-congruent parametrized surfaces x and x ∗ witha. I=I ∗b. II = II ∗ (Hint: Try reparametrizing some of our simplest surfaces.)10. Let x(u, v) =α(u) +vβ(u) beaparametrization of a ruled surface. Prove that the tangentplane is constant along rulings (i.e., the surface is flat) if and only if α ′ (u), β(u), and β ′ (u) arelinearly dependent for every u. (Hint: When is S(x v )=0? Alternatively, consider x u × x v andapply Exercise A.2.1.)11. Prove that α is a line of curvature in M if and only if the ruled surface formed by the surfacenormals along α is flat. (Hint: See Exercise 10.)12. Show that the Gaussian curvature of the parametrized surfacesx(u, v) =(u cos v, u sin v, v)y(u, v) =(u cos v, u sin v, ln u)is the same for each (u, v), and yet the first fundamental forms I x and I y do not agree. (Thus,the converse of Corollary 3.2 is false.)13. Suppose that the surface M is doubly ruled by orthogonal lines (i.e., through each point of Mthere pass two orthogonal lines).