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Let $x$ and $y$ denote two points on a Riemannian manifold $M$ and let $\log_xy$ denote the logarithmic map (corresponding to a given metric) applied to $y$ at $x$. Also, let $P^{x\rightarrow y}$ denote a parallel transport operator from $T_xM$ to $T_yM$ over the geodesic connecting $x$ to $y$.

I'm wondering whether it is true that $P^{x\rightarrow y}(\log_xy)=-\log_yx$.

Thanks for any hints!

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Yes. Let $\gamma$ be the geodesic from $x$ to $y$. The defining equation $\nabla_{\gamma^\prime}\gamma^\prime=0$ implies that parallel transport along $\gamma$ maps the tangent vector $\gamma^\prime_x$ to the tangent vector $\gamma^\prime_y$. But the first is a multiple of $log_yx$ and the second is a multiple of $-log_xy$. Since parallel transport preserves length of vectors, and since $\parallel log_xy\parallel=d(x,y)=d(y,x)=\parallel log_yx\parallel$, your claim follows.

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  • $\begingroup$ Thanks a lot for your arguments! If you know of any source with a rigorous proof of it (e.g. a textbook or a paper), please let me know. $\endgroup$ – Simon Jun 18 '16 at 18:33
  • $\begingroup$ I've added some more detail. $\endgroup$ – ThiKu Jun 18 '16 at 18:59

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