Half-ovoidal flat Laguerre planes
The classical 2-dimensional Laguerre plane can be obtained as the geometry of non-trivial plane sections of a cylinder in the euclidean three-space with a flat ellipse as a base. Generalizing this concept by allowing the base to be a topological oval in \(\mathbb{R}^2\), this gives examples of 2-dimensional topological Laguerre planes. These are called \textit{ovoidal}. Topological Laguerre planes have been introduced by \textit{H. Groh} [Abh. Math. Semin. Univ. Hamb. 32, 216-231 (1968; Zbl 0165.23603), Abh. Math. Semin. Univ. Hamb. 34, 11-21 (1969; Zbl 0187.42302), and Arch. Math. 20, 219-224 (1969; Zbl 0176.17904)]. The authors classify all 2-dimensional topological Laguerre planes which can be decomposed into two halves of ovoidal Laguerre planes glued together along two parallel classes. Such planes are called \textit{half-ovoidal} by the authors. For any differentiable function \(f:\mathbb{R}\to\mathbb{R}\) such that \(f(0)=f'(0)=0\), \(f'\) is a strictly increasing homeomorphism, and the maps \(h_1\), \(h_2\), \(h_3\) are orientation-preserving homeomorphisms of \(\mathbb{R}\), they construct an incidence structure \(L(f,h_2,h_1,h_0)\) on the cylinder \(P:=(\mathbb{R}\cup\{\infty\})\times\mathbb{R}\). The incidence structure \(L(f,h_2,h_1,h_0)\) is a Laguerre plane, although it is not a topological one, since its circles are not homeomorphic to \(1\)-spheres. This can be achieved by applying a specific map \(F:P\to P\) to the cylinder. This leads to a 2-dimensional \textit{topological} Laguerre plane \(\widehat L(f,h_2,h_1,h_0)\). Moreover, it turns out that \(\widehat L(f,h_2,h_1,h_0)\) is, in fact, a half-ovoidal Laguerre plane. Conversely, they prove the striking fact that every half-ovoidal Laguerre plane is isomorphic to some plane \(\widehat L(f,h_2,h_1,h_0)\). A characterization in terms of the functions \(h_i\) of those planes \(\widehat L(f,h_2,h_1,h_0)\) which are ovoidal rounds up the results. The beautiful paper ends with an investigation of isomorphisms and automorphisms of half-ovoidal Laguerre planes.
scientific article; zbMATH DE number 1106607
scientific article