HW203sol - Differential Geometry 2 Homework 03 1 Let S ⊂...

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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 a smooth curve in S then D S ˙ γ ˙ γ = ( D V ˙ γ ˙ 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 R 2-{ } on which the scalar function f is strictly positive. Determine f so that for each a > 0 the circle t 7→ ( a cos t,a sin t ) is a geodesic for the conformally equivalent metric f 2 g . Solutions (1) Let a ∈ S and u ∈ S be orthogonal. Let γ be the geodesic in S with γ (0) = a and ˙ γ (0) = u a ; taking the cue, the geodesic γ solves (with these initial conditions) the differential equation 0 = ¨ γ T since the ambient space V ⊃ S is Euclidean. As γ is a unit normal to S at each time, the component of ¨ γ normal to S is precisely ( γ | ¨ γ ) γ . Accordingly, γ satisfies ¨ γ = ( γ | ¨ γ ) γ....
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HW203sol - Differential Geometry 2 Homework 03 1 Let S ⊂...

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