More on Kleinewillinghöfer Types of Flat Laguerre Planes

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillingh?fer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Laguerre homotheties that can occur in flat Laguerre planes. Building on results by M?urer and Hartmann to characterize ovoidal or miquelian Laguerre planes we further characterize certain flat Laguerre planes in terms of their Kleinewillingh?fer types. Received: January 16, 2007. Revised: July 26, 2007.     

Schliessungssätze in Laguerre-Ebenen

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration , consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

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Gewidmet Herrn Professor Benz zum 60. Geburtstag     

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.     

Eine Kennzeichnung der miquelschen Minkowski-Ebenen durch Transitivitätseigenschaften

A characterization of miquelian Minkowski planes by means of transitivity properties is given. As a corollary we obtain that in the classification of M. Klein, class 22 is empty.

Herrn Helmut Salzmann zum 65. Geburtstag gewidmet     

On M?bius Spaces Admitting Automorphism Groups of Type II.2

In 1987 the first author extended C. Hering??s classification for M?bius planes (cf. Hering in Math Z 87:252?C262, 1965) to higher dimensional M?bius spaces (cf. Kroll in Result Math 12:357?C365, 1987). Actually the classification was done for groups of automorphisms of a M?bius space. In this paper we are concerned with M?bius spaces admitting a group of type II.2. Our main result is a generalization of a result of Yaqub (Math Z 142:281?C292, 1975, Theorem 1).     

The Symmetry Axioms in Laguerre Planes

We introduce two axioms in Laguerre geometry and prove that they provide a characterization of miquelian planes over fields of the characteristic different from 2. They allow to describe an involutory automorphism that sheds some new light on a Laguerre inversion as well as on a symmetry with respect to a pair of generators.     

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     

A non-miquelian Laguerre plane satisfying a theorem of miquelian type

This note shows that a theorem of miquelian type known as (M2) holds in a certain non miquelian Laguerre plane of shear type as defined by Löwen and Pfüller[1].Dedicated to Professor H. Karzel on the occasion of his 70th birthday     

Flat Circle Planes as Integrals of Flat Affine Planes

In a previous article (Arch. Math. {64} (1995), 75–85) we showed that flat Laguerre planes can be constructed by'integrating' flat affine planes. It turns out that'most' of the known flat Laguerre planes arise in this manner. In this paper we show that similar constructions are also possible in the case of the other two kinds of flat circle planes, that is, the flat Möbius planes and the flat Minkowski planes. In particular, we show that many of the known flat Möbius planes can be constructed by integrating a closed strip taken from a flat affine plane. We also show how flat Minkowski planes arise as integrals of two flat affine planes. All known flat Minkowski planes can be constructed in this manner.     

Flat weakly miquelian laguerre planes are ovoidal

Every flat Laguerre plane that satisfies a certain variation of the Miquel Condition is ovoidal. Equivalently, in flat Laguerre planes a certain special version of the Bundle Theorem already implies the Bundle Theorem.     

Laguerre planes of even order and translation ovals

Translation Laguerre planes of even order are represented in high dimensional projective space over GF(2) by a collection of subspaces that satisfies a very simple condition.This research was supported for the respective authors by a grant from the University of Canterbury and by a Feodor Lynen Fellowship.     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

On the Minkowski planes constructed by Artzy and Groh

The Minkowski planes constructed by R. Artzy and H. Groh [1] are characterized among the locally, connected and finite dimensional Minkowski planes as strongly semi-(p, w)-transitive Minkowski planes (see Theorem 2). The types of the Artzy-Groh planes in the typification of the Minkowski planes by M. Klein are determined (see Proposition 4). The second author was supported by a DAAD scholarship for a research visit at TU München. He sincerely thanks the Zentrum Mathematik der TU München for their hospitality.     

Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Let M ⁿ be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere , then M ⁿ is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M ⁿ under the M?bius transformation group of . A classical theorem of M?bius geometry states that M ⁿ (n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, we determine all M?bius isoparametric hypersurfaces in by proving the following classification theorem: If is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in ; or (ii) the pre-image of the stereographic projection of the cone in over the Cartan isoparametric hypersurface in with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in . The classification of hypersurfaces in with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen. Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM. Partially supported by the Zhongdian grant No. 10531090 of NSFC. Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.     

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.     

A family of flat Laguerre planes of Kleinewillingh?fer type IV.A

Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillingh?fer types reflect transitivity properties of certain groups of central automorphisms in Laguerre planes. In the case of flat Laguerre planes, Polster and Steinke have shown that some of the conceivable types cannot exist, and they gave models for most of the other types. Only few types are still in doubt. Two of them are types IV.A.1 and IV.A.2, whose existence we prove here. In order to construct our models, we make systematic use of the restrictions imposed by the group generated by all central automorphisms guaranteed in type IV. With these models all simple Kleinewillingh?fer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillingh?fer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.     

The gradient flow of the M?bius energy near local minimizers

In this article we show that for initial data close to local minimizers of the M?bius energy the gradient flow exists for all time and converges smoothly to a local minimizer after suitable reparametrizations. To prove this, we show that the heat flow of the M?bius energy possesses a quasilinear structure which allows us to derive new short-time existence results for this evolution equation and a ?ojasiewicz-Simon gradient inequality for the M?bius energy.     

Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q ² is precisely (q–1)/2 when q is an odd prime and precisely (q–1)/2e when q=p ^e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.This work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

Dense subgroups of <Emphasis Type="Italic">n</Emphasis>-dimensional Möbius groups

We will derive a new discreteness condition for n-dimensional M?bius subgroups as well as obtain some results concerning classification of such groups. We will also discuss dense subgroups of n-dimensional M?bius groups. The main result is that any dense group of an n-dimensional M?bius group contains a dense subgroup which is generated by at most n elements if . Received: 5 June 2001 / Published online: 24 February 2003 RID="*" ID="*" The research was partly supported by FNS of China, grant number 19801011