Hadamard manifolds and the visibility axiom (Q1105842)

Let H denote a complete, simply connected manifold with sectional curvature \(K\leq 0\). For any two distinct points p, q of H there exists a unique geodesic \(\gamma_{pq}\) joining p to q, and one defines the angle subtended at p by points q, r distinct from p to be the angle between \(\gamma_{pq}'(0)\) and \(\gamma_{pr}'(0)\). The manifold H is said to satisfy the Visibility axiom if for every point p of H and every positive number \(\epsilon\) there exists a positive number \(R=R(p,\epsilon)\) such that if \(\gamma\) : [a,b]\(\to H\) is any geodesic whose distance to p is \(\geq R\), then the angle subtended at p by \(\gamma\) (s) and \(\gamma\) (t) is \(\leq \epsilon\) for all s,t in [a,b]. If \(K\leq c<0\) for some negative constant c, then H satisfies the Visibility axiom, and in fact the constant R in the definition above depends only on \(\epsilon\) and not on p. The author obtains a characterization of the Visibility axiom in terms of the growth rate of Jacobi vector fields. Fix a point p in H and let c: [a,b]\(\to S\) be a Lipschitz curve in the space S of unit tangent vectors at p. For t in [a,b] let \(Y_ t\) be the Jacobi vector field on the geodesic \(s\to \exp_ p(sc(t))\) such that \(Y_ t(0)=0\) and \(Y'_ t(0)=c'(t)\), where \(Y_ t'\) denotes the covariant derivative of \(Y_ t\) and \(c'(t)\in T_{c(t)}S\) is identified in the natural way with a vector at p. Define \(\Phi_ c(r)=\int^{b}_{a}\| Y_ t(r)\| dt\) for each \(r>0\). For example, \(\Phi_ c(r)\geq \sinh (r) length(c)\) if \(K\leq - 1\). The author proves the follwing theorem. A manifold H as above satisfies the Visibility axiom if and only if \(\Phi_ c(r)\to \infty\) as \(r\to \infty\) for all Lipschitz curves c: [a,b]\(\to S\).