Cycles in de Sitter's world
type: Item
instance of: scholarly article
title: Cycles in de Sitter's world
zbMATH Open document ID: 0943.51013
DOI: 10.1007/BF03322102
author: Roland Höfer
published in: Results in Mathematics
publication date: 2000-08-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51B25
Mathematics Subject Classification ID: 51B15
Mathematics Subject Classification ID: 83C20
zbMATH DE Number: 1346914
zbMATH Keywords: oriented hypersphere
zbMATH Keywords: oriented contact preserving transformation
zbMATH Keywords: Lie geometry
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1346914