Möbius Transformations and the Poincaré Distance in the Quaternionic Setting

In the space H of quaternions, we investigate a natural, invariant geometry of the open, unit disc ∆H and of the open half-space H . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Möbius transformations of ∆H and H + and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Möbius transformations, and use it to define the analogous of the Poincaré distances on ∆H and H . We then show that this natural, quaternionic Poincaré distance of ∆H does not coincide with the Kobayashi distance inherited by ∆H as a domain of C 2 .