B�schels�tze zur Charakterisierung projektiv darstellbarer Zykelebenen

Herzer, Armin

doi:10.1007/bf01182268

Cited by 13 publications

(8 citation statements)

References 11 publications

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“…The theorem above is just a re-formulation of what is usually called the Büschelsatz, see Benz (1973aBenz ( , p. 133, 1973b, and Herzer (1979). It is a (8 3 , 6 4 )-configuration of eight not necessarily real points and six i-Möbius circles, but can also be seen as a (4 3 , 6 2 )-configuration of six i-Möbius circles and four pencils of i-Möbius circles.…”

Section: Theorem 5 Let Be Given Three I-möbius Circles C I and A Poinmentioning

confidence: 94%

On elementary circle-geometry in Cayley–Klein planes

Weiss

Hamann

2011

Beitr Algebra Geom

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This contribution is sort of an addendum to a recently published paper on circle-geometries in Cayley-Klein planes, see Martini and Spirova (Publicationes Mathematicae Debrecen 72:371-383, 2008b), as it deals with further generalisations and extensions of the author's results to circle-geometries in all Cayley-Klein planes. The main methods in this paper are the interpretation of planar figures in space and the dualizing according to the duality principle of projective spaces. There are, in principle, only three types of R 2 -ring structures and, thus, only three types of corresponding circle-geometries, see Benz (Geometrie der Algebren, 1973a). Therefore, each generalisation to non-Euclidean planes must turn out to be just another representation of the classical Euclidean cases. This point of view gives more insight into why some elementary geometric theorems remain valid when changing the place of action from the Euclidean plane to non-Euclidean circle planes and makes explicit proofs of such elementary geometric theorems in non-Euclidean circle planes superfluous. We believe that even the Euclidean cases of circle-geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle-geometries might also be of interest for their own. For example, among the planar Cayley-Klein geometries the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated, similar to the isotropic Möbius geometry, by suitable projections of the Blaschke cylinder.

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Section: Theorem 5 Let Be Given Three I-möbius Circles C I and A Poinmentioning

confidence: 94%

On elementary circle-geometry in Cayley–Klein planes

Weiss

Hamann

2011

Beitr Algebra Geom

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“…PROLOGUE The question of embedding locally generalized projective geometries in generalized projective geometries has been considered by a number of people, and in various settings ( [1], [2], [6]- [14]). It is known that in the rank-4 case the same results cannot be obtained; counterexamples appear in [8] and [10]. Traditionally ({-1], [2], [10]- [14]), the assumption of rank strictly greater than 4 has been made in embedding theorems of this type, and then a natural geometric constructive argument is possible.…”

Section: Locally Generalized Projective Lattices Satisfying a Bundle mentioning

confidence: 99%

“…The most popular cadre seems to have been the lattice-theoretic one. Under additional assumptions, the rank-4 case has been handled by Herzer [6]- [8] and Kahn [9] more recently and in quite different ways. It is known that in the rank-4 case the same results cannot be obtained; counterexamples appear in [8] and [10].…”

Section: Locally Generalized Projective Lattices Satisfying a Bundle mentioning

confidence: 99%

Locally generalized projective lattices satisfying a bundle theorem

Batten

1987

Geom Dedicata

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The question of embedding locally generalized projective geometries in generalized projective geometries has been considered by a number of people, and in various settings ([1], [2], [6]-[14]). The most popular cadre seems to have been the lattice-theoretic one. Traditionally ({-1], [2], [10]-[14]), the assumption of rank strictly greater than 4 has been made in embedding theorems of this type, and then a natural geometric constructive argument is possible. It is known that in the rank-4 case the same results cannot be obtained; counterexamples appear in [8] and [10]. Under additional assumptions, the rank-4 case has been handled by Herzer [6]-[8] and Kahn [9] more recently and in quite different ways. In [7] Herzer supposes a lattice situation allowing lines (rank-2 elements) with a single point (rank-1 element). He supposes, however, that all 1-point lines on any given point are contained in a single plane on the point. Under this condition a rank-4 semimodular locally projective (of order >-4) lattice is shown to be embeddable in a modular one precisely when a certain configuration holds throughout the lattice. It is of interest to note that central collineations are introduced in the proof.Kahn [9] supposes a weaker condition on the 1-point lines, but this leads, it seems, to a more complex proof in which at one point the finite and infinite cases must be handled differently. However, the configuration he shows to be necessary and sufficient for embeddability is much simpler (in a geometrical sense) than that considered by Herzer. He also assumes that the lattice is locally projective.In this article, we present an embedding theorem which unifies the rank-4 and rank >4 cases. We prove that a particular situation is necessary and sufficient in very general circumstances. We suppose: rank >-4, locally generalized projective, outside each plane (rank-3 element) there be at least one point, and outside each rank-4 element there be at least one point unless the lattice itself has rank-4. Thus elements of rank >`2 may contain a single point. We note that the conditions used by Herzer and Kahn imply that there is a point outside each plane.The proof of the embedding is geometrically natural and very similar to that given by Kantor [10] and also that used in [1] and [2]. We divide our

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“…Herzer betrachtet in [12] S. 221f/ihnlich definierte Gruppen, welche dort auf einem Geradenbiischel in einem Verband dreifach transitiv operieren; weil man diesen Verband erst durch Hinzunehmen vori Idealebenen und Tangenten aus einer M6bius-bzw. Herzer betrachtet in [12] S. 221f/ihnlich definierte Gruppen, welche dort auf einem Geradenbiischel in einem Verband dreifach transitiv operieren; weil man diesen Verband erst durch Hinzunehmen vori Idealebenen und Tangenten aus einer M6bius-bzw.…”

Section: (B) Affine Projektivitlitenunclassified

“…1.5. Auch Herzer[12] betrachtet Projektivit//tengruppen, welche auf Geradenbfischeln operieren. M sei eine ovoidale M6bius-oder Laguerreebene zu einer quadratischen Menge in PG(3, K).…”

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Projektivit�tengruppen von ovoidalen M�bius-und Laguerreebenen

Grundhöfer

1982

Geom Dedicata

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THEO GRUNDHOFER PROJEKTIVITATENGRUPPEN VON OVOIDALEN MOBIUS-UND LAGUERREEBENEN ZUSAMMENFASSUNG. Die Untersuchungen fiber Projektivitiitengruppen in MSbius-und Laguerreebenen yon Freudenthal-Strambach [7], Kroll [16] und Funk [9] zeigten, dal3 die Sch/irfe der dreifachen Transitivit/it dieser Gruppen gerade die miquelschen Ebenen charakterisiert. Die vorliegende Arbeit zielt vor allem darauf, fiir endliche nichtmiquelsche M6bius-und Laguerreebenen die Projektivitfitengruppe anzugeben. Dazu identifizieren wir zunfichst einige Untergruppen H(P) der Projektivitg.tengruppe, und untersuchen dann im zweiten Abschnitt die Beziehungen dieser Gruppen zueinander, wobei sich Zusammenh/inge mit der Steinerschen Erzeugung yon Kegelschnitten ergeben. Nach diesen Vorbereitungen ergibt sich dann Satz 3.1, der besagt, dab die Projektivit~tengruppe jeder endlichen ovoidalen MSbius-oder Laguerreebene, welche nicht miquelsch ist, die alternierende Gruppe enth~ilt. Abschnitt 4 liefert schliel31ich Methoden zur Beantwortung der Frage, ob eine solche Pro-jektivit~itengruppe ungerade Permutationen enth~lt. Damit kann man z.B. die genauen Projektivitfitengruppen der MSbiusebenen vom Suzuki-Tits-Typ angeben (Korollar 4.9). 0. BEGRIFFE UND BEZEICHNUNGEN Sei M eine M6bius-oder Laguerreebene. Zur Definition vgl. etwa Freudenthal-Strambach [7] oder Halder-Heise [11] S. 228f. Der Einheitlichkeit halber heigen die Zykel einer Laguerreebene hier auch Kreise. Als affine Ableitung M e von M im Punkt P bezeichnen wir die affine Ebene, welche man folgendermal3en erh/ilt: Ist M eine M6biusebene, so bestehe Mp aus den yon P verschiedenen Punkten yon M und aus den Mengen k\{P}, wobei k ein Kreis von M durch P ist. Ist M eine Laguerreebene, so bestehe M e aus denjenigen Punkten von M, welche nicht zu P parallel sind, d.h. nicht auf der Erzeugenden durch P liegen, und aus den Mengen k\{P}, k ein Kreis von M durch P, sowie allen Erzeugenden, welche nicht durch P gehen (vgl. etwa Halder-Heise [11] S. 228). Die Ordnung von Mist die Ordnung der affinen Ebenen Mp. Wir betrachten in dieser Arbeit nur kommutative KSrper K (obwohl 1.4 auch noch ffir Schiefk6rper K gilt) und die zugeh6rigen projektiven Geometrien PG(V) oder PG(n, K), welche durch den Unterraumverband eines Vektorraums V vom Rang n + 1 fiber K gegeben sind. Ein Ovoid ~ in einer projektiven Geometrie ist eine Punktmenge ~ mit (i) Keine drei Punkte von ~ sind kollinear. (ii) Ffir jeden Punkt Pe~ ist die Vereinigung aller Geraden g mit g c~ ~? = {P} eine Hyperebene. Ein Ovoid in einer projektiven Ebene PG(2, K) werden wir als Oval bezeichnen (wie Dembowski [5] S. 48 und S. 147, abweichend yon der Terminologie bei Hirschfeld El3] S. 163). Die ebenen Schnitte eines Ovoids Geometriae Dedicata 13 (1982) 125-147. 0046-5755/82/0132-0125503.45.

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B�schels�tze zur Charakterisierung projektiv darstellbarer Zykelebenen

Cited by 13 publications

References 11 publications

On elementary circle-geometry in Cayley–Klein planes

On elementary circle-geometry in Cayley–Klein planes

Locally generalized projective lattices satisfying a bundle theorem

Projektivit�tengruppen von ovoidalen M�bius-und Laguerreebenen

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