Distance Preservers for Lobachevsky Space

We obtain a complete description of the class of entrywise preservers of Lorentz–Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings. These preservers admit a Lévy-Khintchine-type representation and their asymptotic characteristics are related to Krein’s classification of screw lines in Lobachevsky space. Connections with complete Nevanlinna-Pick kernels and Bochner subordination are also obtained.