1. Let S be a regular surface with unit normal vector field N and let a : [0,1] → S be a smooth unit-speed curve. Let kg be the geodesic curvature of a, and 0(s) = -fkgds. Show that w(s) = a' (s) cos 0(s) + (N^ a'(s)) sin 0(s) is a parallel vector field on a. (Hint: there are two ways to prove this one with lots of computations and the other with 2-3 lines.)

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