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differential geometry w notes from teacher_Part_75

differential geometry w notes from teacher_Part_75 - 149...

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6.1. AFFINE CONNECTION 149 6.1.3 Parallel Transport Let x 0 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 0 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 = 0 . 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 )) = 0 . 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 = 0 , is called the geodesics . The coordinates of the geodesics x = x ( t ) satisfy the non-linear second- order ordinary di ff erential equation ¨ x i + ω i jk ( x ( t ))˙ x k ˙ x j = 0 .
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