Regular vs. classical Möbius transformations of the quaternionic unit ball

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular Möbius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular Möbius transformations and the Poincaré metric of B, which is preserved by the classical Möbius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.