Non-euclidean geometries: the Cayley-Klein approach

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 H. Struve and R. Struve (J Geom 81:155–167, 2004), 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 by D. Hilbert, J. Hjelmslev, F. Bachmann, R. Lingenberg, H. Karzel et al. Mathematics Subject Classification (2010). 06B25, 51F10, 20G15.