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!
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 $\|\log_xy\|=d(x,y)=d(y,x)=\|\log_yx\|$, your claim follows.
