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An analogue of the Alexandrov-Zeeman theorem, based on hyperboloidpreservation, as opposed to light cone preservation, is provided. Thischaracterizes exactly the Poincar\'e group, as opposed to the Poincar\'e groupextended by dilations. The hyperbolic analogue, as opposed to the cone-basedAlexandrov-Zeeman theorem, is also valid in the case of a single spacedimension. An orthochronous version also holds, based on the preservation offorward hyperboloid shells.

Hyperboloid preservation implies the Lorentz and Poincar\'e groups without dilations