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Unformatted text preview: 6.1. AFFINE CONNECTION 149 6.1.3 Parallel Transport • Let x and x 1 be two points on a manifold M and C be a smooth curve connecting these points described locally by x i = x i ( t ), where t ∈ [0 , 1] and x (0) = x and x (1) = x 1 . The tangent vector to C is defined by X = ˙ x ( t ) , where the dot denotes the derivative with respect to t . • Let Y be a vector field on M . We say that Y is parallel transported along C if ∇ X Y = . • The vector field Y is parallel transported along C if its components satisfy the linear ordinary di ff erential equation d dt Y i ( x ( t )) + ω i jk ( x ( t ))˙ x k ( t ) Y j ( x ( t )) = . • Problem. Solve the equation of parallel transport in terms of a Taylor series up to terms qubic in the connection coe ffi cients. • A curve C such that the tangent vector to C is trasported parallel along C , that is, ∇ ˙ x ˙ x = , is called the geodesics ....
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
- Spring '10