DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

74 Chapter 2. Surfaces: Local TheoryNow deduce that in the case of a non-arclength parametrization we obtain∫ √fv = c′ (u) 2 + g ′ (u) 2f(u) √ du + const.f(u) 2 − c2 20. Use Exercise 19 to show that any geodesic on the paraboloid z = x 2 + y 2 that is not a meridianintersects every meridian. (Hint: Show that it cannot approach a meridian asymptotically.)21. Use the Clairaut relation, Proposition 4.4, and Exercise 19 to describe the geodesics on thepseudosphere (see Example 6 of Section 2). Show, in particular, that the only geodesics thatare unbounded are the meridians.22. Consider the surface z = f(u, v). A curve α whose tangent vector projects to a scalar multipleof ∇f(u, v) ateachpoint P =(u, v, f(u, v)) is a curve of steepest ascent (why?). Suppose sucha curve α is also a geodesic.a. Prove that the projection of α into the uv-plane is a geodesic. (Hint: The vector tangentto the surface and orthogonal to α ′ is horizontal. Why? Compare the Darboux framesalong α and its projection.)b. Deduce that α is also a line of curvature. (Hint: See Exercise 12.)c. Show that if all the curves of steepest ascent are geodesics, then f satisfies the partialdifferential equationf u f v (f vv − f uu )+f uv (f 2 u − f 2 v )=0.(Hint: When are the integral curves of ∇f lines?)d. Show that the level curves of f are parallel (see Exercise 1.2.22). (Hint: Show that ‖∇f‖is constant along level curves.)e. Give a characterization of the surfaces with the property that all curves of steepest ascentare geodesics.