Moebius transformations and the Poincare distance in the quaternionic setting
Cinzia BisiGraziano Gentili
30G3530C2030F45functions of hypercomplex variablesquaternionicMoebius transformationsquaternionic Poincare distance and metric
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 Poincar\'e distances and differential metrics on $\Delta_{\mathbb{H}}$ and $\mathbb{H} ^{+}$.
Indiana University Mathematics Journal
2009
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10.1512/iumj.2009.58.3706
10.1512/iumj.2009.58.3706
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Indiana Univ. Math. J. 58 (2009) 2729 - 2764
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