A note on Hilbert and Beltrami systems
The author defines a compact disk to be a linear space \(\mathbb{D}=(D,{\mathcal D})\) with point set \(D\) homeomorphic to the closed unit disk in \(\mathbb{R}^ 2\) and line set \({\mathcal D}\) such that each line \(L\in{\mathcal D}\) is homeomorphic to the closed unit interval and such that the boundary \(\partial L\) is contained in \(\partial D\) and \(L\backslash\partial D\) is connected. One can always find a model of \(\mathbb{D}\) in the real affine plane \(\mathbb{A}(\mathbb{R})\) in the sense that the induced geometry on the boundary of some compact convex subset of \(\mathbb{A}(\mathbb{R})\) is isomorphic to the induced geometry on \(\partial D\). Starting off with an affine \(\mathbb{R}^ 2\)-plane \(\mathbb{A}=(\mathbb{R}^ 2,{\mathcal A})\) and a compact disk \(\mathbb{D}=(D,{\mathcal D})\) the author considers a homeomorphism \(\gamma:D\to E\) from \(D\) onto some \({\mathcal A}\)-convex subset \(E\subset\mathbb{R}^ 2\) that induces an isomorphism of the incidence structures that are induced on \(\partial D\) (from \(\mathbb{D})\) and on \(\partial E\) (from \(\mathbb{A})\) respectively. He shows that one can construct a new affine \(\mathbb{R}^ 2\)-plane \(\mathbb{A}^ \mathbb{D}_ \gamma\) by replacing the parts of lines that pass through \(E\) by suitable \(\gamma\)-images of lines in \(\mathbb{D}\) which are uniquely determined by the intersection with \(\partial E\). Hilbert and Beltrami systems as defined by \textit{H. Mohrmann} [Hilbertsche und Beltramische Liniensysteme, Math. Ann. 85, 177-183 (1922)] are certain subclasses of the planes \(\mathbb{A}(\mathbb{R})^ \mathbb{D}_ \gamma\) where one uses the real affine plane. The planes \(\mathbb{A}^ \mathbb{D}_ \gamma\) also comprise D. Hilbert's example of a non- Desarguesian plane in Grundlagen der Geometrie, Teubner, Leipzig (1899) and the semiclassical projective planes constructed by the reviewer [Arch. Math. 44, 472-480 (1985; Zbl 0564.51006)] and many other constructions where two pieces of projective planes are glued together along a nice common boundary. Using this pasting process one sees that each compact disk can be embedded in a two-dimensional projective plane. In particular, this applies to the extension of Strambach's exceptional \(SL_ 2\mathbb{R}\)-plane [\textit{K. Strambach}, Math. Ann. 179, 15-30 (1968; Zbl 0164.513)] onto the closed unit disk. In this way one obtains an embedding of this plane which cannot be \(SL_ 2\mathbb{R}\)-equivariant.
scientific article; zbMATH DE number 484148
scientific article