I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I would like to see proven (and clarified).
Let $M$ be a compact, connected Riemannian manifold of nonpositive curvature and let $\tilde{M}$ be its universal cover. Let $\sigma,\tau:[0,T]\to \tilde{M}$ be two unit speed geodesics. Then:
(a) $d(\sigma(t),\tau(t))\leq d(\sigma(0),\tau(0))+d(\sigma(T),\tau(T))$, for all $0\leq t\leq T$
(b) $d(\dot \sigma(t),\dot \tau(t))\leq d(\dot \sigma(0),\dot \tau(0))+d(\dot\sigma(T),\dot\tau(T))$, for all $0\leq t\leq T$
Note: In item (a), $d$ refers to the shortest-path distance induced by the Riemannian metric. In (b) there is a distance $d$ on $T\tilde{M}$, there's no clarification for this on the notes I read, it might be the Sasaki metric or maybe $d(v,w)=\sup\{d(\pi(\phi_tv),\pi(\phi_tw)):\ t\in[0,1]\}$ (where $\phi_t$ is the geodesic flow on $T\tilde{M}$ and $\pi:T\tilde{M}\to\tilde{M}$ is the projection).