HW202 - Differential Geometry 2 Homework 02 1. Let (M, g )...

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Unformatted text preview: Differential Geometry 2 Homework 02 1. Let (M, g ) be a Riemannian manifold in which γ is a smooth curve. Show that the operator of covariant differentiation along γ Dγ : Vecγ → Vecγ ˙ may be defined by the requirement that if ζ ∈ Vecγ and α ∈ Ω1 (M ) then αγ (t) [(Dγ ζ )t ] = ˙ d [αγ (t) (ζt )] − ( dt ∗ γ (t) α)(ζt ). ˙ Note: This one condition replaces the three standard conditions that were presented in class; in particular, it makes no reference to local expressions. 2. Let γ : I → M be a geodesic in the Riemannian manifold (M, g ). (i) Show that speed along γ is constant in that the real-valued function t → gγ (t) (γ (t), γ (t)) on I is constant. ˙ ˙ (ii) Let φ : J → I be a diffeomorphism of open intervals; show that γ ◦ φ is a geodesic iff φ is affine. 1 ...
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