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Let $(M,g)$ be a Riemannian manifold of unbounded diameter and let $\gamma_1, \gamma_2$ be two geodesics such that $\gamma_1(0)=\gamma_2(0)=x$. Suppose that $\gamma_1, \gamma_2$ are parametrized so that $||\dot{\gamma_1}||_g=||\dot{\gamma_2}||_g \equiv 1$. What can be said about the asymptotics of the function $d(\gamma_1(t), \gamma_2(t))$, where $d$ is the geodesic distance, in case $M$ has nonpositive sectional curvature? strictly negative sectional curvature?

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  • $\begingroup$ Having unbounded diameter does not help much: if $M$ is the product of hyperbolic plane with a compact negatively curved surface, the geodesics may be bounded, staying within the same compact surface. $\endgroup$ – user147263 Oct 5 '15 at 11:48
  • $\begingroup$ You can find a complete exposition of the relation between curvature and distance between geodesic in Do Carmo's book. As a matter of fact, this is closely related to Jacobi fields along this geodesics. $\endgroup$ – Laz Oct 5 '15 at 23:10
  • $\begingroup$ Try Chapter 5, Riemannian Geometry, Do Carmo. $\endgroup$ – Laz Oct 5 '15 at 23:27

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