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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 ] =
[αγ (t) (ζt )] − (
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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- Spring '09