Isometries in Aleksandrov spaces of curvature bounded above
By studying isometries of Aleksandrov spaces of curvature bounded above by a negative number, the author proves the following theorem: if \(M\) is a locally compact complete and geodesically complete \(\Re_{K}\)-domain (or \(\text{CAT}( K) \)-space), \(K<0\), in which all spheres are arcwise connected, then every bijection \(f\) of the space \(M\) onto itself such that \(f\) and \(f^{-1}\) map any closed ball of some fixed radius \(r>0\) onto some closed ball of radius \(r\), is an isometry. For hyperbolic spaces, a stronger result by \textit{A. V. Kuz'minykh} [Sib. Mat. Zh. 20, 597-602 (1979; Zbl 0427.51008)] is known.
scientific article; zbMATH DE number 1839967