Cycles in de Sitter's world
The \(n\)-dimensional de Sitter's world is the geometry of Lie cycles \({\mathcal C}^n\) of the \(n\)-sphere \({\mathcal S}^n\) (the sphere of radius 1 in Euclidean \((n+1)\)-space with respect to the Lorentz distance), that is, points and oriented hyperspheres of \({\mathcal S}^n\) with respect to oriented contact between Lie cycles defined in the usual way. A Lie transformation is a permutation \(f\) of \({\mathcal C}^n\) such that two Lie cycles are in oriented contact if and only if their images under \(f\) are in oriented contact. It is well known that Lie cycles can be algebraically described as points of the Lie quadric \({\mathbb Q}^n\) in \((n+2)\)-dimensional projective space and that the Lie transformations are precisely the projective linear maps that leave the Lie quadric invariant. The author shows that a map \(f\) from \({\mathcal C}^n\) into \({\mathcal C}^n\) such that two Lie cycles are in oriented contact if and only if their images under \(f\) are in oriented contact is already a Lie transformation. Furthermore, a representation of Lie transformations is given in terms of Lorentz matrices acting on centres of hyperspheres and of real numbers modifying signed radii of hyperspheres. As a consequence, the geometry of all Lie transformations acting on the Lie cycles is isomorphic, in the sense of Klein's Erlangen programme, to the conformal geometry of the Lorentz manifold with respect to the Lorentz metric of \((n+1)\)-dimensional Minkowski spacetime.
oriented contact preserving transformation
scientific article; zbMATH DE number 1346914
scientific article