Non-euclidean geometries: the Cayley-Klein approach
type: Item
instance of: scholarly article
title: Non-euclidean geometries: the Cayley-Klein approach
zbMATH Open document ID: 1213.51018
author: Horst Struve
author: Rolf Struve
published in: Journal of Geometry
publication date: 2011-02-03T00:00:00Z
review text: The following abstract given by the authors explains the content of this article very nicely: ``A. Cayley and F. Klein discovered in the nineteenth century that Euclidean and non-Euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (`begründeten') in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces over arbitrary fields and of arbitrary dimensions requires two steps, the introduction of a metric in a Pappian projective space and the definition of substructures as Cayley-Klein geometries. While the first step is taken in [\textit{H. Struve} and \textit{R. Struve}, J. Geom. 81, No.~1--2, 155--167 (2004; Zbl 1078.51001)], the second step is made in this article. We show that the concept of a Cayley-Klein geometry leads to a unified description and classification of a wide range of non-Euclidean geometries including the main geometries studied in the foundations of geometry be D. Hilbert, J. Hjelmslev, F. Bachmann, R. Lingenberg, H. Karzel et al.''
reviewed by: Rüstem Kaya
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 06B25
Mathematics Subject Classification ID: 51F10
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 5844613
zbMATH Keywords: Cayley-Klein geometry
zbMATH Keywords: non-Euclidean geometry
zbMATH Keywords: projective-metric spaces
zbMATH Keywords: Klein's model
Wikidata QID: Q56873869
MaRDI profile type: Publication
full work available at URL: s00022-010-0053-z
OpenAlex ID: W2069994587
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DOI: 10.1007/S00022-010-0053-Z
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description: scientific article; zbMATH DE number 5844613