Vector parameters in classical hyperbolic geometry. (English) Zbl 1369.51004

Summary: Here we use an extension of Rodrigues’ vector parameter construction for pseudo-rotations in order to obtain explicit formulae for the generalized Euler decomposition with arbitrary axes for the structure groups in the classical models of hyperbolic geometry. Although the construction is projected from the universal cover \(\mathrm{SU}(1,1) \simeq \mathrm{SL}(2,\mathbb R)\), most attention is paid to the \(2+1\) Minkowski space model, following the close analogy with the Euclidean case, and various decompositions of the restricted Lorentz group \(\mathrm{SO}^+(2,1)\) are investigated in detail. At the end we propose some possible applications in special relativity and scattering theory.


51M10 Hyperbolic and elliptic geometries (general) and generalizations
53A45 Differential geometric aspects in vector and tensor analysis
22E70 Applications of Lie groups to the sciences; explicit representations