Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau\_{x, w}$ and $\tau\_{y, w}$ be the parallel transports of the tangent planes at $x$ and $y$ to the tangent plane at $w$, respectively. Does there exist a positive constant $C$ such that 


$$\|\tau\_{x,w}\circ Log\_{x}(z) - \tau\_{y,w}\circ Log\_{y}(z)\|\leq C d(x,y),$$
where $C$ is independent of $x, y, z$? 


Thanks in advance!



Parallel transport on a Hadamard manifold
The following counterexample shows that even for fixed $w,x,y$, the norm in your expression need not be bounded in $z$:


Let $X$ be the hyperbolic plane; I'll use the upper half-space model to identify $X$ with a subset of $\mathbb{C}$. Let $w = x = i - 1$, let $y = i + 1$, and let $z\_n = ni$.


As $n \to \infty$, the angle $\theta\_n$ between $\log\_x(z\_n)$ and $\tau\_{y,x} \log\_y(z\_n)$ approaches $\pi /2$. (Parallel translation is easy to compute here since you know the geodesics and parallel translation preserves angle with the tangent to the geodesic.) Moreover, for every $n$ the norms $||\log\_x(z\_n)||$ and $||\tau\_{y,x} \log\_y(z\_n)||$ are equal, say to $L\_n$, and $L\_n \to \infty$. Then
$$
||\log\_x(z\_n) - \tau\_{y,x} \log\_y(z\_n)||^2 = 2D\_n^2(1 - \cos \theta\_n) \to \infty.
$$


