Helmut Mäurer

Die Bedeutung des Spiegelungsbegriffs in der Möbius- und der Laguerre-Geometrie

Die Frage, inwieweit geometrische Strukturen durch Symmetrieeigenschaften charakterisiert werden konnen, spielte in der Geometrie stets eine zentrale Rolle. In den letzten Jahrzehnten wurde z.B. die Forschungsrichtung im Bereich der metrischen Geometrie fast vollig durch diese Fragestellung bestimmt (vgl. [1]).

Involutorische Automorphismen von Laguerre-Ebenen mit genau zwei Fixpunkten

A miquelian Laguerre-plan of characteristic ≠2 can be characterized by the property that for each pair {A, B} of non parallel points there exists an involutoric automorphism whose set of fixed points is exactly {A, B}.

A Characterization of Pappian Affine Hjelmslev Planes

In [7], Pickert showed that an affine plane is isomorphic to an affine plane over a field if the dual Pappus theorem holds for two fixed pencils of parallel lines (see also [2]). Here we ask whether a similar result is true for an affine Hjelmslev plane Ω. The geometric structure Ω will be defined in such a generality that the algebraic models of affine Hjelmslev planes over commutative local rings (not only those over Hjelmslev rings) are included. We consider two distinguished not-neighbouring pencils V, W of parallel lines and assume the validity of the configuration conditions (DP), (SD). The main result is the following: If the epimorphic image a(Ω) of Ω is not a Fano plane then the incidence structure of Ω is isomorphic to that of an affine Hjelmslev plane Ω(R) over a local ring R. In R the element 2 is invertible. All pencils of parallel lines in Ω correspond to pencils of parallel lines in Ω(R) with the possible exception of those neighbouring V or W.

Parabolische Kollineationen, die eine ebene konvexe Menge invariant lassen

AbstractCollineations τ1, τ2 of PG(2, ℝ) leaving invariant a compact convex setK

Die Automorphismengruppe der Möbiusgeometrie einer Körpererweiterung

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Eine Charakterisierung der zwischen PS-L (2, K ) und PGL (2, K ) liegenden Permutationsgruppen

ℝ2 are called parabolic if |∂K ∩ Fix τi|=1. Conditions are stated under which the existence of τ1, τ2 imply that ∂K is an ellipse.

Permutationsgruppen ohne Pseudoinvolutionen, in deren standgruppen die involutionen regulär operieren

Let (ℝ2,ℝ[x]) denote the incidence structure consisting of the point set ℝ2 and the set {{(x, f(x)) ¦ x ∈ ℝ} ¦ f ∈ ℝ[x]} of blocks. It will be shown, that every automorphism of this geometry has the form (x, y) ↦ (ax + b, cy + d(x)), where a, b, c are real numbers with a · c ≠ 0 and where d is a real polynomial.