Möbius transformations and the Poincaré distance in the quaternionic setting
| dc.creator | Bisi, Cinzia | |
| dc.creator | Gentili, Graziano | |
| dc.date | 2008-05-03 | |
| dc.date | 2008-06-01 | |
| dc.date.accessioned | 2026-07-07T09:41:48Z | |
| dc.date.available | 2026-07-07T09:41:48Z | |
| dc.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. | |
| dc.description | Details added in proof of Theorem 6.4. Exposition improved in some ambiguous points. References added. 28 pages | |
| dc.identifier | https://arxiv.org/abs/0805.0357 | |
| dc.identifier | http://arxiv.org/abs/0805.0357 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161957 | |
| dc.subject | Complex Variables | |
| dc.subject | 30G35, 30C20, 30F45 | |
| dc.title | Möbius transformations and the Poincaré distance in the quaternionic setting | |
| dc.type | text |