# Visibility spaces and Gromov hyperbolicity

I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic manifolds? Or even an implication. By visibility I mean:

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 $\sigma(b)$.

The manifold is said to be uniform visibility manifold if $r$ doesn't depend on $p$.

I already know, based on the work of Morse, that the $\delta$-hyperbolic condition implies some sort of shadowind property of quasi-geodesics. Also, based on the work of Bonk, we see that the shadowing property implies $\delta$-hyperbolicity (Bonk, M.: Quasi-geodesic segments and Gromov Hyperbolic Spaces). I'm trying to understand the relation (if there exists some) between visibility, Gromov hyperbolicity and the shadowing property.

• What is your definition of uniform visibility? – Misha Jun 24 '14 at 16:41
• I'll put this on the question for the sake of completeness. Thanks – matgaio Jun 24 '14 at 16:57
• Bridson-Haefliger's textbook has an extensive discussion of these matters. In particular, for proper CAT(0) spaces hyperbolicity and uniform visibility are equivalent (Proposition II.H.1.4), and in a proper, geodesic, hyperbolic space any two points at infinity are joined by a geodesic (Lemma III.H.3.2). – Igor Belegradek Jun 24 '14 at 22:19