A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic rays that are asymptotic i.e. two unit-speed geodesic rays $\gamma_1,\gamma_2:[0,\infty)\rightarrow 0$ are members of the same element of $\partial M$ iff $d(\gamma_1(t),\gamma_2(t))$ is bounded on the domain. For each $p\in M$ and $u\in \partial M$, there exists a unique element of $u$ associated with $p$ i.e. a unique unit geodesic ray $\gamma_p$ satisfying $\gamma_p(0)=p$ and $\gamma_p\in u$. Define a vector field $M\rightarrow TM$ by $p\rightarrow (p,\gamma_p'(0))$; that is, $p$ is mapped to the initial velocity of the element of $u$ associated with $p$. Is this vector field smooth?
I feel like this should be a known result but cannot find it in the literature. Any references would be appreciated.