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.
Categories: math.CV