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Unformatted text preview: Differential Geometry 2 Homework 02 1. Let ( M, g ) be a Riemannian manifold in which is a smooth curve. Show that the operator of covariant differentiation along D : Vec Vec may be defined by the requirement that if Vec and 1 ( M ) then ( t ) [( D ) t ] = d d t [ ( t ) ( t )]- ( * ( t ) )( 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 : I M be a geodesic in the Riemannian manifold ( M, g ). (i) Show that speed along is constant in that the real-valued function t 7 g ( t ) ( ( t ) , ( t )) on I is constant. (ii) Let : J I be a diffeomorphism of open intervals; show that is a geodesic iff is affine. Solutions (1) Experience suggests renaming D simply as D . First assume that D is defined as in class. If = e locally then...
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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
- Spring '09