On 2-dimensional Laguerre planes of shift type
A planar function \(f: \mathbb{R}\to\mathbb{R}\) is one whose graph \(G_f\) generates an affine (`shift') plane, the lines being all vertices and all translates of \(G_f\). A sufficient condition is that \(f\) is continuously differentiable with bijective derivative. Let \(h\) be a primitive of such a function and apply the group \(\Gamma\) of all translations and all shears with vertical axes to \(G_h\), then you get the Laguerre planes under consideration. This amounts to apply \textit{B. Polster's} integration process [ibid. 64, No. 1, 75-85 (1995; Zbl 0813.51007)] to the shift plane generated by \(f\). The group \(\Gamma\) acts sharply transitively on the set of circles. The author shows that the planes generated by two functions \(h\) and \(\widetilde h\), respectively, are isomorphic iff there is a relation of the form \(\widetilde h(x)= rh(sx+ t)+ ax^2+ bx+ c\). In particular, \(h\) generates the classical plane iff \(h\) is a cubic polynomial. In all other cases, the plane is non-ovoidal and has an automorphism group of dimension 3 or 4. The automorphism group is at least 4-dimensional iff \(h\) is a semi-multiplicative homeomorphism, i.e., \(h(xy)= h(x) h(y)\) holds for positive \(x\). Such maps have the form \(h(x)= q(x)|x|^m\), where \(q(x)= 1\) for \(x\geq 0\) and \(q(x)= q\) for \(x< 0\); here, \(m\) and \(q\) are positive constants. The derivative of such a function is planar iff \(m> 2\).
2-dimensional Laguerre planes of shift type
scientific article; zbMATH DE number 1798903