Half-ovoidal flat Laguerre planes

We classify the 2-dimensional Laguerre planes which can be obtained by gluing together halves of ovoidal Laguerre planes along two parallel classes.This research was supported by a Feodor Lynen fellowship and a DAAD fellowship.     

Stiffness in four-dimensional stable planes

We present stiffness results for four-dimensional stable planes showing that a non-projective plane is suffer than a projective one. Whereas the best possible result in projective planes is that the isotropy group of a non-degenerate quadrangle has order at most two, we obtain that certain degenerate quadrangles with two non-compact sides have a zero-dimensional isotropy group. In [2] our stiffness results help to determine all four-dimensional stable planes with an at least nine-dimensional automorphism group. Our results improve those of Löwen [12], 5.2. We introduce the Freudenthal compactification as a new tool.     

Sisterhoods of flat Laguerre planes

Every flat Laguerre plane of shear type over a pair of skew parabolae is related to a flat Laguerre plane of translation type over a pair of skew parabolae and vice versa. The relationship is defined using the connection between flat Laguerre planes and three-dimensional generalized quadrangles.Dedicated to Prof. H. R. Salzmann on his 65th birthday     

Some automorphisms of Laguerre planes

Results in Mathematics -     

The Apollonius contact problem and Lie contact geometry

A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries—Laguerre geometry and oriented M?bius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model.     

Three-dimensional quadrangles and flat Laguerre planes

Every three-dimensional generalized quadrangle can be constructed from flat Laguerre planes.Dedicated to Prof. H. Salzmann on his 60th birthday     

A symmetry theorem for Laguerre planes

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Berührsatz). The same special Pascal theorem now leads to an incidence theorem, , for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.     

Laguerre and Minkowski planes produced by dilatations

We show that each parabolic curve f in R² produces a Laguerre plane if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f₁,f₂ produce a Minkowski planeM(f₁,f₂). We determine for which curves is miquelian resp. ovoidal, and for which pairs f₁,f₂,M(f₁,f₂) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.     

Generalized quadrangles constructed from topological Laguerre planes

The Lie geometry of a finite-dimensional locally compact connected Laguerre plane is a topological generalized quadrangle.