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HW202 - Dierential Geometry 2 Homework 02 1 Let(M g be a...

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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.
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