A note on Hilbert and Beltrami systems

Abstract

Hilbert and Beltrami (line-) systems were introduced by H. Mohrmann, Math. Ann. 85 (1922) p.177-183. These systems give examples of non-desarguesian affine planes, in fact, the earliest known examples are of this type. We describe a construction for "generalized Beltrami systems", and show that every such system defines a topological affine plane with point set R 2. Since our construction uses only the topological structure of R 2-planes, it is possible to iterate this process. As an application, we obtain an embeddability theorem for a class of two-dimensional stable planes, including Strambach's exceptional SL2R-plane.