Hyperbolic spaces
For any commutative ordered field \((K,\leq)\), the hyperbolic plane over \(K\) is defined as the incidence space induced on the interior points of a conic given by a standard symmetric bilinear form of Witt index \(1\). Such an analytic model allows the introduction of a betweenness function \(\zeta\) on collinear triplets and a congruence relation for pairs of points, as usual. In the paper under review, a hyperbolic space is defined as an ordered incidence space \((M,{\mathfrak G},\zeta)\) with an additional relation \(\equiv\) on \(M\times M\) such that each plane \(E\) in \((M,{\mathfrak G})\) is isomorphic to a hyperbolic plane over some ordered commutative field \(K_E\), with the appropriate restrictions of \(\zeta\) and \(\equiv\). It is known [\textit{K. Sörensen}: `Projektive Einbettung hyperbolischer Räume', TUM, Math. Inst., Beitr. Geom. Algebra M 8612 (15), 8-35 (1986)] that \((M,{\mathfrak G},\zeta)\) allows a locally complete embedding into an ordered projective space \((P,{\mathfrak L},\tau)\) over some ordered field \(K\). Up to isomorphism, this embedding is unique, and one obtains \(K_E\cong K\) for each plane \(E\). The authors introduce ends for each of the (hyperbolic) planes in \((M,{\mathfrak G},\zeta,\equiv)\) and show that the set \({\mathfrak K}\) of all these ends is a regular quadric of index \(1\) in \((P,{\mathfrak L})\). The hyperbolic space \((M,{\mathfrak G},\zeta)\) is thus isomorphic to the incidence space induced on the interior points of \({\mathfrak K}\).
scientific article; zbMATH DE number 1146255
scientific article