M\"obius transformations and the Poincar\'e distance in the quaternionic setting

In the space $\hh$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $\Delta_{\hh}$ and of the open half-space $\hh^{+}$. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of $\Delta_{\hh}$ and $\hh^{+}$ 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 M\"obius transformations, and use it to define the analogous of the Poincar\'e distances on $\Delta_{\hh}$ and $\hh^{+}$. We easily deduce that there exists no isometry between the quaternionic Poincar\'e distance of $\Delta_{\hh}$ and the Kobayashi distance inherited by $\Delta_{\hh}$ as a domain of $\mathbb{C}^{2}$, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.