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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 SL(2, R) geometry is nontrivial and enables comparison and transfer of known and even future results between these two models.

Generalization of the Cayley transform in 3D homogeneous geometries