Can a Busemman space be CAT(1)?



CAT(K) and Busemann
Answering a sensible question appeared in comments: If a geodesic space is Busemann and CAT(1), then it must be CAT(0).


Indeed, CAT(1) implies that the space has well-defined metric angles between geodesic segments. And if a space is Busemann and has well-defined angles, then it is CAT(0).


Indeed, let $X$ be the space in question and $\alpha,\beta:[0,1]\to X$ constant-speed minimizing geodesics with $\alpha(0)=\beta(0)=p$. Then, by the Busemann definition, the function $t\mapsto d(\alpha(t),\beta(t))/t$ is non-decreasing. Hence the comparison angle at $p$ of the triangle $p\alpha(t)\beta(t)$ is non-decreasing. Hence its limit at $t=0$ (which is by definition the angle between $\alpha$ and $\beta$) is no greater than the comparison angle at $p$ of the triangle $p\alpha(1)\beta(1)$. This means that Toponogov's hinge comparison holds. It remains to recall that existence of angles plus hinge comparison is one of the standard definitions of CAT(0).


