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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!

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1 Answer 1

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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 $$ ||\log_x(z_n) - \tau_{y,x} \log_y(z_n)||^2 = 2D_n^2(1 - \cos \theta_n) \to \infty. $$

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