Günter F. Steinke

Chapter 24 – Topological Circle Geometries

Publisher Summary This chapter focuses on topological circle geometries. The geometry of all circles on the real 2-sphere in 3-dimensional Euclidean space (i.e. the intersection of the sphere with Euclidean planes) has since long been investigated under various aspects. Another model of the same geometry is obtained by stereographic projection from one point of the sphere onto a plane not passing through the point of projection. This leads to the geometry of Euclidean lines extended by an infinite point and Euclidean circles in the real plane. Similarly, one considers extensions of the geometry of Euclidean lines and Euclidean parabolae with given direction of axis or Euclidean hyperbolae with given direction of asymptotes. An algebraic description of these geometries leads to chain geometries. It discusses the planar situation and takes a more geometric point of view. An axiomatization is given and thus a generalization of these geometries which leads to so-called circle planes. Since affine and projective planes occur as derived incidence structures, the structure theory and classification of topological locally compact connected finite-dimensional affine and projective planes plays a crucial role.

Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classes

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along two parallel classes suitably. Isomorphism classes and automorphisms of these planes are determined.

Some Minkowski planes with 3-dimensional automorphism group

A new rather large family of 2-dimensional locally compact topological Minkowski planes with an at least 3-dimensional automorphism group is introduced here. Isomorphism classes and automorphisms of these planes are determined.

Semiclassical Topological Flat Laguerre Planes Obtained by Pasting Along A Circle

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along a circle suitably. Isomorphism classes and automorphism groups of these planes are determined. Together with [9] this gives a complete classification of all semicalssical topological flat Lguerre planes.

On the structure of finite elation Laguerre planes

The miquelian Laguerre plane of order q (q being a prime power) is obtained as the geometry of non-trivial plane sections of a quadratic cone in the 3-dimensional projective space over GF(q). Similarly, an ovoidal Laguerre plane of order q is obtained as the geometry of non-trivial plane sections of a cone over an oval (not necessarily a conic) in the 3-dimensional projective space over GF(q).

On the Kleinewillinghöfer types of flat Laguerre planes

Kleinewillinghöfer classified in [7] Laguerre planes with respect to central automorphisms and obtained a multitude of types. For finite Laguerre planes many of these types are known to be empty. In this paper we investigate the Kleinewillinghöfer types of flat Laguerre planes with respect to the full automorphism groups of these planes and completely determine all possible types of flat Laguerre planes with respect to Laguerre translations.

4-Dimensional point-transitive groups of automorphisms of 2-dimensional laguerre planes

It is shown that a 2-dimensional Laguerre plane that admits a closed connected 4-dimensional point-transitive group of automorphisms must be classical. Further, up to conjugacy in the automorphism group of the classical real Laguerre plane, all closed connected 4-dimensional point-transitive subgroups are determined.

The inner and outer space of 2-dimensional Laguerre planes

The classical 2-dimensional Laguerre plane is obtained as the geometry of non-trivial plane sections of a cylinder in R 3 with a circle in R 2 as base. Points and lines in R 3 define subsets of the circle set of this geometry via the affine non-vertical planes that contain them. Furthemore, vertical lines and planes define partitions of the circle set via the points and affine non-vertical lines, respectively, contained in them. We investigate abstract counterparts of such sets of circles and partitions in arbitrary 2-dimensional Laguerre planes. We also prove a number of related results for generalized quadrangles associated with 2-dimensional Laguerre planes.

A family of 2-dimensional Laguerre planes of generalised shear type

We construct a family of 2-dimensional Laguerre planes that generalises ovoidal Laguerre planes and the Laguerre planes of shear type, as described by Lowen and Pfuller, by gluing together circle sets from up to eight different ovoidal Laguerre planes. Each plane in this family admits all maps ( x, y ) ↦ ( x , ry ) for r > 0 as central automorphisms at the circle y = 0.

A Family Of 2-Dimensional Minkowski Planes with Small Automorphism Groups

This paper concerns 2-dimensional (topological locally compact connected) Minkowski planes. It uses a construction of J. Jakóbowski [4] of Minkowski planes over half-ordered fields and applies it to the field of reals. This generalizes a construction by A. Schenkel [7] of 2-dimensional Minkowski planes admitting a 3-dimensional kernel. It is shown that most planes in this family of Minkowski planes have 0-dimensional and even trivial automorphism groups.