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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 D ˙ γ : Vec γ → Vec γ may be defined by the requirement that if ζ ∈ Vec γ and α ∈ Ω 1 ( M ) then α γ ( t ) [( D ˙ γ ζ ) t ] = d d t [ α γ ( t ) ( ζ t )]- ( ∇ * ˙ γ ( t ) α )( ζ t ) . 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 7→ g γ ( 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. Solutions (1) Experience suggests renaming D ˙ γ simply as D γ . First assume that D γ is defined as in class. If ζ = e ζ ◦ γ locally then ˙...
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- Spring '09
- dΓ, Riemannian geometry, Differentiable manifold