A characterization of miquelian Minkowski planes by means of transitive properties
The author investigates Minkowski planes that admit many homotheties. More precisely, a \([p,q]\)-homothety is an automorphism that fixes precisely the two non-parallel points \(p\) and \(q\) and induces a homothety in the derived affine plane at \(p\). A Minkowski plane \({\mathcal M}\) is called \([p,q]\)-transitive if the group of all \([p,q]\)-homotheties acts transitively on \(C \backslash \{p,q\}\) for each circle \(C\) through \(p\) and \(q\). Moreover, \({\mathcal M}\) is called strongly \(p\)-transitive if \({\mathcal M}\) is \([p,q]\)-transitive for each point \(q\) not parallel to \(p\). With this notation the main result of the paper is as follows. A Minkowski plane \({\mathcal M}\) is Miquelian, that is, is isomorphic to the geometry of plane sections of a ruled quadric in 3-dimensional projective space over a field, if and only if there are two points \(p \neq q\) for which \({\mathcal M}\) is strongly \(p\)-transitive and strongly \(q\)-transitive. This result improves a characterization of Miquelian Minkowski planes by \textit{E. Hartmann} [Result. Math. 5, 136-148 (1982; Zbl 0496.51003)] where the strong transitivity was postulated for each point. Consequently, class 22 in the classification of Minkowski planes with respect to transitive subgroups of central automorphisms by \textit{M. Klein} [J. Geom. 43, No. 1/2, 116-128 (1992; Zbl 0746.51009)] is empty (when the full automorphism group of the plane is considered).
scientific article; zbMATH DE number 806299