DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

52 Chapter 2. Surfaces: Local Theoryc. (More challenging) Show that, more generally, for any θ and m ≥ 3, we haveH = 1 (( 2π ) ( 2π(m − 1) ) )κ n (θ)+κ n θ + + ···+ κn θ + .mmm10. a. Apply Meusnier’s Formula to a latitude circle on a sphere of radius a to calculate thenormal curvature.b. Prove that the curvature of any curve lying on the sphere of radius a satisfies κ ≥ 1/a.11. Prove or give a counterexample: If M is a surface with Gaussian curvature K>0, then thecurvature of any curve C ⊂ M is everywhere positive. (Remember that, by definition, κ ≥ 0.)♯ 12.Suppose that for every P ∈ M, the shape operator S P is some scalar multiple of the identity,i.e., S P (V) =k(P )V for all V ∈ T P M. (Here the scalar k(P )maywell depend on the pointP .)a. Differentiate the equationsD xu n = n u = − kx uD xv n = n v = − kx vappropriately to determine k u and k v and deduce that k must be constant.b. We showed in Proposition 2.2 that M is planar when k =0. Show that when k ≠0,M is(a portion of) a sphere.13. a. Prove that if α is a line of curvature in M, then (n◦α) ′ (t) =−k(t)α ′ (t), where k(t) istheprincipal curvature at α(t) inthe direction α ′ (t). (More colloquially, differentiating alongthe curve α, wejust write n ′ = −kα ′ .)b. Suppose two surfaces M and M ∗ intersect along a curve C. Suppose C is a line of curvaturein M. Prove that C is a line of curvature in M ∗ if and only if the angle between M andM ∗ is constant along C. (In the proof of ⇐=, be sure to include the case that M and M ∗intersect tangentially along C.)14. Prove or give a counterexample:a. If a curve is both an asymptotic curve and a line of curvature, then it must be planar.b. If a curve is planar and an asymptotic curve, then it must be a line.15. a. How is the Frenet frame along an asymptotic curve related to the geometry of the surface?b. Suppose K(P ) < 0. If C is an asymptotic curve with κ(P ) ≠0,prove that its torsionsatisfies |τ(P )| = √ −K(P ). (Hint: If we choose an orthonormal basis {U, V} for T P (M)with U tangent to C, what is the matrix for S P ? See the Remark on p. 46.)16. Continuing Exercise 15, show that if K(P ) < 0, then the two asymptotic curves have torsionof opposite signs at P .17. Prove that the only minimal ruled surface with no planar points is the helicoid. (Hint: Considerthe curves orthogonal to the rulings. Use Exercises 8b and 1.2.19.)