Moebius Transformations and the Poincarè distance in the quaternionic setting.

In the space $\mathbb{H}$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $\Delta_{\mathbb{H}}$ and of the open half-space $\mathbb{H}^+.$ These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of $\Delta_{\mathbb{H}}$ and $\mathbb{H}^+$ 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 M\"obius transformations, and use it to define the analog of the Poincare' distances and differential metrics on $\Delta_{\mathbb{H}}$ and $\mathbb{H}^+.$