Let $M$ be a completed Riemmannian manifold with bounded curvature. Let $d(\cdot,\cdot):M\times M\rightarrow R$ is the distance function. Suppose that the injective Radius of $M$ is $ injM$. Then the logarithm map $\log_x:M\rightarrow T_x M$ can be defined as follows. For $y\in M$ satisfying $d(x,y)<injM$, then $\log_{x}y=\dot{\gamma}(0)$. Here $\gamma:[0,1]\rightarrow M$ is the shortest geodesic on $M$ connecting $x$ and $y$.
I have the following question.
Let $x(t):[0,\infty)\rightarrow M$ and $y(t):[0,\infty)\rightarrow M$ be two smooth curves on $M$ satisfying $d(x(t),y(t))<injM$, $\|\dot{x}\|$ and $\|\dot{y}\|$ are finte for all $t\in [0,\infty)$. Then $\log_{x(t)}y(t)\in T_{x(t)}M$ is always well defined.
I want to know if there is some expression for $\nabla_{\dot{x}}\log_{x}y$, i.e., the covariant derivative of $\log_{x(t)}y(t)$ along the curve $x(t)$. Or if the norm $\|\nabla_{\dot{x}}\log_{x}y\|$ is always bounded for $t\in [0,\infty)$.
For the Euclidean case, $\nabla_{\dot{x}}\log_{x}y=\dot{y}-\dot{x}$. Since $\dot{y}\in T_{y}M$ and $\dot{x}\in T_{x}M$ in arbitrary manifold, they cannot be compared directly as Eucildean space. But we can comepare them after parallel transport. Let $P_{xy}:T_{x}M\rightarrow T_yM$ be the parallel transport along the shortest geodesic connecting $x$ and $y$. So I wonder to know if there is some relationship between $\nabla_{\dot{x}}\log_{x}y$ and $\dot{y}- p_{xy}\dot{x}$.
