In this paper, we generalize the Cayley transform in three-dimensional homogeneous geometries which are fiber bundles over the hyperbolic plane. Obtained generalizations are isometries between existing models in corresponding homogeneous geometries. | Turk J Math (2018) 42: 2942 – 2952 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Generalization of the Cayley transform in 3D homogeneous geometries Zlatko ERJAVEC∗, University of Zagreb, Faculty of Organization and Informatics, Pavlinska 2, HR-42000 Varaždin, Croatia Received: • Accepted/Published Online: • Final Version: Abstract: The Cayley transform maps the unit disk onto the upper half-plane, conformally and isometrically. In this paper, we generalize the Cayley transform in three-dimensional homogeneous geometries which are fiber bundles over the hyperbolic plane. Obtained generalizations are isometries between existing models in corresponding homogeneous geometries. Particularly, constructed isometry between two models of ^R) geometry is nontrivial and enables SL(2, comparison and transfer of known and even future results between these two models. Key words: Cayley transform, homogeneous geometry, isometry 1. Introduction Homogeneous geometries came into focus in 1982 when Thurston formulated the geometrization conjecture for three-manifolds. The Thurston conjecture states that every compact orientable three-manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure from among the 8 maximal simply connected homogeneous Riemannian three-dimensional geometries. ^R) are specific because they are Among these eight 3D homogeneous geometries, H2 × R and SL(2, ^R) is the least researched and structured as line bundles over the hyperbolic plane. Particularly, the SL(2, generally, because of its unique features, presents a rich area for future investigation. The Cayley transform is an isometry (differentiable bijection with a differentiable inverse which preserves distance) between the disk model and the upper half-plane model of the hyperbolic plane. In this paper, we ^R) geometries. In the literature, .
