DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§4. Covariant Differentiation, Parallel Translation, and Geodesics 7312. a. Prove that if a geodesic is planar and not a line, then it is a line of curvature. (Hint: UseExercise 11.)b. Prove that if every geodesic of a (connected) surface is planar, then the surface is containedin a plane or a sphere.13. Prove or give a counterexample:a. A line lying in a surface is both an asymptotic curve and a geodesic.b. If a curve is both an asymptotic curve and a geodesic, then it must be a line.c. If a curve is both a geodesic and a line of curvature, then it must be planar.14. Show that the geodesic curvature at P of a curve C in M is equal (in absolute value) to thecurvature at P of the projection of C into T P M.15. Check using Clairaut’s relation, Proposition 4.4, that great circles are geodesics on a sphere.(Hint: The result of Exercise A.1.3 may be useful.)16. Let M be asurface and P ∈ M. WesayU, V ∈ T P M are conjugate if II P (U, V) =0.a. Let C ⊂ M be a curve. Define the envelope M ∗ of the tangent planes to M along C tobe the ruled surface whose generator at P ∈ C is the limiting position as Q → P of theintersection line of the tangent planes to M at P and Q. Prove that the generator at P isconjugate to the tangent line to C at P .b. Prove that if C is nowhere tangent to an asymptotic direction, then M ∗ is smooth (at leastnear C). Prove, moreover, that M ∗ is tangent to M along C and is a developable (flatruled) surface.c. Apply part b to give a geometric way of computing parallel translation. In particular, dothis for a latitude circle on the sphere. (Cf. Exercise 7.)17. Suppose that on a surface M the parallel translation of a vector from one point to anotheris independent of the path chosen. Prove that M must be flat. (Hint: Fix an orthonormalbasis e o 1 , eo 2 for T P M and define vector fields e 1 , e 2 by parallel translating. Choose coordinatesso that the u-curves are always tangent to e 1 and the v-curves are always tangent to e 2 . SeeExercise 10.)18. Use the Clairaut relation, Proposition 4.4, to describe the geodesics on the torus as parametrizedin Example 1(c) of Section 1. (Start with a geodesic starting at and making angle φ 0 withthe outer parallel. Your description should distinguish between the cases cos φ 0 ≤ a−ba+b andcos φ 0 > a−ba+b. Which geodesics never cross the outer parallel at all? Also, remember thatthrough each point there is a unique geodesic in each direction.)19. Use the proof of the Clairaut relation, Proposition 4.4, to show that a geodesic on a surface ofrevolution is given in terms of the standard parametrization in Example 7 of Section 2 by∫v = cduf(u) √ f(u) 2 − c 2 + const.