Gyrovector spaces in the service of hyperbolic geometry (Q2711305)

Gyrogroup theory is modeled on the relativity groupoid of all relativistically admissible velocities with their Einstein addition and Thomas precession. As an introduction to gyrovector space theory, the author presents three isomorphic gyrovector spaces named Möbius, Einstein and Ungar, and shows that these respectively form the setting for the Poincaré ball, the Klein-Beltrami ball, and the Ungar space model of hyperbolic geometry.NEWLINENEWLINENEWLINEWhile demonstrating the power and elegance of the resulting super-theories in their capability to capture analogies, the author shows that the innocuous gyrotion that the gyrocommutative and the gyroassociative laws of Möbius addition suggest stores mathematical symmetries and shares analogies with the relativistic effect known as the Thomas precession.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00011].