HW203 - Differential Geometry 2 Homework 03 1. Let S ⊂ V...

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Unformatted text preview: Differential Geometry 2 Homework 03 1. Let S ⊂ V be the unit sphere in a Euclidean space. Recall that if γ is V S a smooth curve in S then Dγ γ = (Dγ γ )T . Use this fact to determine the ˙˙ ˙˙ (unit-speed) geodesics in S explicitly. Do the same for geodesics in hyperbolic space H . 2. Let g be the standard Riemannian metric on the (punctured) plane R2 −{0} on which the scalar function f is strictly positive. Determine f so that for each a > 0 the circle t → (a cos t, a sin t) is a geodesic for the conformally equivalent metric f 2 g . 1 ...
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