On asymptotes in a metric space with non-positive curvature
Yasuo Nasu · Tohoku Mathematical Journal · 1957
We wish to thank Prof. H Busemann fcr his kind advice in the investigation.1) Numbers in brackets refer to the references cited at the end of the pεper.4. The relation between ray and coray is symmetric and transitive [ §3J. 5.The set K(l) consists of finite number of unbounded and continuous curves no two of which has common points.If the set K{\) has branch points, then the number of these points is finite.At each branch point the number of branch curves is equal to that of asymptotes issuing from this point [ §4, §5].1.In this paragraph we explain some preliminary concepts.In a metric space the distance between two points x and y will be denoted by p(x,y).The axioms for a metric space 3t to be an Zs-space are follows:A. 9t is metric with distance p{x,y) not necessarily symmetric.for any two distinct points a, b in S(x, cέ(x)) and any positive number £ there exist positive numbers B^a, b) (k = 1,2) not greater than £ for which a point a λ with p(a 1} a) + p(a, b) = p(aι,b) and another point b x with p(a, b) + ρ{b, bι) = p{a, b τ ) exist and are unique.If the metric is symmetric, then 9ΐ is said a G-space.If further R has dimension 2 in the sense of Mehger-Uryson, 3t is said a G-surface and is topologically a connected manifold.The axioms A, B and C guarantee the existence of a segment T{p, q) from p to q (or T(q, p) from q to p) whose length equals the distance p(p, q) (or p(q,p)).The prolongation of a segment is locally possible and unique under the axiom D. The whole prologation of a segment is said a geodesic.A geodesic @ has a parametric representation x{τ), -00 be two directions.Let a,ι and a 2 be the initial points of Dι and D t and bι and b 2 the end points of Dι and A Following H. Busemann (2 §7], the distance of A and A is defined as Ί, A) = -τr(p(tfi> 0*) + P(*i> b-2 )\The set of all directions on 3t is finitely compact under the above metric.The distance of two half geodesies (or two geodesic subarcs) is defined as that of their initial directions.Let x(τ),[0^τ< +00, be a parametric representation of a ray I, and let