Question

Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider for it the parametrization x(u, v) ((v) cos u, p(v) sin u, ^(v) Assume in addition that (a)2 +()21 (a) Prove that a curve a(t) = x(u(t), v(t)) is a geodesic of S if and only if it satisfies dip 1 ü2 dv p dip p(u)2 0, dv where here and in what follows the dot denotes derivative with respect to t 5 marks 1 (b) Prove that meridians (u u0 Constant) are geodesics. Prove that a parallel v vo is a geodesic if and only if vo is a critical point for p(v). 5 marks

Answer 1

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