Non-euclidean geometries: the Cayley-Klein approach
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.''
scientific article; zbMATH DE number 5844613
scientific article