  An analogue of the Alexandrov-Zeeman theorem, based on hyperboloid
preservation, as opposed to light cone preservation, is provided. This
characterizes exactly the Poincar\'e group, as opposed to the Poincar\'e group
extended by dilations. The hyperbolic analogue, as opposed to the cone-based
Alexandrov-Zeeman theorem, is also valid in the case of a single space
dimension. An orthochronous version also holds, based on the preservation of
forward hyperboloid shells.
