I'm thinking about some properties of geodesics in visibility spaces. Here I give some definitions:
A Riemannian manifold $(M,g)$ with riemannian distance $d$ is said to be a visibility manifold if given $p\in M$ and $\varepsilon>0$ there exists $r=r(p,\varepsilon)>0$ with this property: if $\sigma:[a,b]\longrightarrow M$ is a geodesic segment such that $d(p,\sigma)\geqslant r$, then $\sphericalangle_p(\sigma(a),\sigma(b))\leqslant\varepsilon$, where $\sphericalangle_p(\sigma(a),\sigma(b))$ is the angle between the tangent vectors of the minimizing geodesics joining $p$ to $\sigma(a)$ and to $\sigma(b)$.
Nice models for visibility manifolds includes negatively-curved spaces, such the hyperbolic plane.
We say that two geodesics $\gamma,\beta:\mathbb{R}\longrightarrow M$ are bi-asymptotic if there exists $C>0$ such that $d(\gamma(t),\beta(t))<C$ for all $t\in\mathbb{R}$.
Consider now non-conpact manifolds. My question is:
If $M$ is a visibility manifold, is it true that there is no bi-asymptotic geodesics besides coincident pairs of geodesics?
I see this happening in negatively curved spaces, but I have no clue if this is a more general property or not.
Any help will be really appreciated. Thanks a lot!
I wasn't sure whether you saw this in chat so I'm posting as an answer here:
Even under the stronger hypothesis that $M$ be simply connected of nonpositive curvature and a visibility manifold, there can exist pairs of bi-asymptotic geodesics.
To construct a counterexample, let $N$ be a negatively curved surface with a single cusp. Chop off the cusp at some finite point and smooth it down so that it ends in a flat cylinder; this gives a nonpositively curved manifold $N'$ with a boundary $\mathbb{S}^1$, such that a neighborhood of this boundary is isometric to a flat cylinder. Let $M$ be the universal cover of the manifold obtained from gluing $N'$ to a copy of its reflection along the flat cylinders.
$M$ is a visibility manifold by the theorem on the first page of [1] (thanks for the reference!) since clearly it is not flat (and hence contains no isometric $\mathbb{R}^2$). But $M$ has lots of pairs of bi-asymptotic geodesics; $M$ even has many flat strips (isometric copies of $\mathbb{R} \times [0,\epsilon]$).
[1] Eberlein : Geodesic flows in certain manifolds without conjugate points. Transactions of the AMS, May 1972.