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As the title suggests, I am trying to find a method of proving that the direction of a parallel transported vector changes direction for Riemann or Pseudo Riemann manifolds.

I can easily show that the length of vector doesnt change when parallel transported by just showing:

$$\frac{dv^2}{du}=0 $$ With $$v^2=v^av_a\ $$ and u= Affine parameter

I am not sure how to show though that the direction does change, and get stumped when directions are involved. This question is for studying and not homework.

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If you have already developed the machinery of the covariant derivative (https://en.wikipedia.org/wiki/Covariant_derivative), then the answer is trivial.

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  • $\begingroup$ I have studied the covariant derivative, but I cannot see how this can be used to prove direction changes on parallel transport of a vector being parallel transported. Would you mind elaborating a small bit? Thanks $\endgroup$ – gline Jul 22 '17 at 19:38

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