ON THE GEOMETRY INDUCED BY LORENTZ TRANSFORMATIONS IN PSEUDO-EUCLIDEAN SPACES

The Lorentz transformations of order (m, n) in pseudo-Euclidean spaces with indefinite inner product of signature (m; n) are extended in this work from m = 1 and n >= 1 to all m, n >= 1. A parametric realization of the Lorentz transformation group of any order (m, n) is presented, giving rise to generalized gyrogroups and gyrovector spaces called bi-gyrogroups and bi-gyrovector spaces. The latter, in turn, form the setting for generalized analytic hyperbolic geometry that underlies generalized balls called eigenballs.