Cayley-Klein's model of dimension-free hyperbolic geometry via projective mappings

• Autores: Walter Benz
• Localización: Aequationes mathematicae, ISSN 0001-9054, Vol. 75, Nº. 3, 2008, págs. 226-238
• Idioma: inglés
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• Resumen

  • With respect to notation and notions we will follow our book Classical Geometries in Modern Contexts, Birkhäuser, 2005. If (X, d), dim X = 2, is a real inner product space, exactly the subsets H(a, a) = {x ? X | d(a, x) = a} of X with 0 ? a ? X and are called Euclidean hyperplanes of (X, d). Concerning the notion of a quasi-hyperplane of (X, d) see in our book, p. 50. In this note we characterize all d-affine mappings of (X, d), i.e. all bijections of X such that images and inverse images of Euclidean hyperplanes are Euclidean hyperplanes, by d-linear mappings. As in our book we do not assume that X is finite-dimensional. Furthermore, we introduce d-projective mappings and characterize Cayley�Klein�s model dimension-free by those mappings.