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Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic connecting them.

Now let us take a compactification of $X$ such that $\partial X$ is homeomorphic to $S^{n - 1}$. I'm interested in the uniqueness property on $\overline{X} = X \cup \partial X$: for any distinct points $p,q\in \partial X$, is it true that there is a unique geodesic $\gamma \subset X$ connecting the two points at infinity?

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    $\begingroup$ What examples did you check? $\endgroup$
    – Moishe Kohan
    Commented Mar 26 at 15:11
  • $\begingroup$ Hyperbolic space is true. Also, it should be true for manifolds with $\sec \leq -1$ $\endgroup$
    – ZZZ
    Commented Mar 26 at 15:18
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    $\begingroup$ Did you think about the Euclidean plane? Do you know the flat strip theorem? $\endgroup$
    – Moishe Kohan
    Commented Mar 26 at 15:27
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    $\begingroup$ There are two issues here: existence of a geodesic joining any two points at infinity and its uniqueness. Neither one is true for general Cartan-Hadamard manifolds. Those for which any two points at infinity are joined by a geodesic are called visibility manifolds. If the manifold contains a flat strip bounded by two bi-infinite geodesics, uniqueness fails. $\endgroup$
    – Igor Belegradek
    Commented Mar 26 at 15:34
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    $\begingroup$ A standard text for a graduate student would be "Metric Spaces of Non-Positive Curvature" by Bridson and Haefliger, which doesn't limit the discussion to Riemannian manifolds, but in fact the manifold case isn't easier. The books that focus on manifolds are "Manifolds of Nonpositive Curvature" by Ballmann, Gromov and Schroeder and Eberlein's "Geometry of Nonpositively Curved Manifolds". $\endgroup$
    – Igor Belegradek
    Commented Mar 26 at 20:01

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