Isometries in Aleksandrov spaces of curvature bounded above

Abstract

We study the isometry groups of Aleksandrov spaces with curvature bounded above. We prove that the metric of any finitely compact geodesically complete $\operatorname{CAT}(K)$-space $(K \lt 0)$, all of whose spheres are arcwise connected, can be recovered from the family of all closed balls of a given positive radius. As a corollary, we obtain that every bijection of such a space onto itself which preserves this family is an isometry. In particular, these results hold for any simply connected Riemannian space with sectional curvature at most $K$, where $K \lt 0$, and of dimension greater than $1$.