Möbius transformations and the Poincaré distance in the quaternionic setting
Abstract
Description
In the space $\hh$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $Δ_{\hh}$ 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 Möbius transformations of $Δ_{\hh}$ 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 Möbius transformations, and use it to define the analogous of the Poincaré distances on $Δ_{\hh}$ and $\hh^{+}$. We easily deduce that there exists no isometry between the quaternionic Poincaré distance of $Δ_{\hh}$ and the Kobayashi distance inherited by $Δ_{\hh}$ as a domain of $\mathbb{C}^{2}$, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.
Details added in proof of Theorem 6.4. Exposition improved in some ambiguous points. References added. 28 pages
Details added in proof of Theorem 6.4. Exposition improved in some ambiguous points. References added. 28 pages