Homogene kompakte projektive Ebenen

Salzmann, Helmut

doi:10.2140/pjm.1975.60.217

Cited by 44 publications

(17 citation statements)

References 19 publications

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“…By the main result of Salzmann [20] a compact connected projective plane admitting a flag-transitive group is a Moufang plane; see also LOwen [16], Grundh6fer et al [9]. The collineation group of every Moufang plane acts transitively on the set of ordered quadrangles, cp.…”

Section: Resultsmentioning

confidence: 99%

“…Salzmann [20] has proved that all compact connected flag-homogeneous planes are classical, see also L(iwen [16]. The compact connected topological generalized polygons with a flag-transitive collineation group have been determined by GrundhOfer et al [9] -with the exception of quadrangles with different topological parameters.…”

Section: I) N = 3 and ~3 Is Isomorphic To One Of The Classical Planesmentioning

confidence: 99%

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Generalized polygons with highly transitive collineation groups

Joswig

1995

Geom Dedicata

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Abstract. It will be proved that the compact connected topological generalized quadrangles which admit a collineation group that acts transitively on ordered pentagons are precisely the real or complex orthogonal quadrangles, up to duality. Classitieations (1991): 51E 12, 51H 15, 51H20. Mathematics SubjectIt is natural to characterize classes of geometries which have a lot of symmetry. Usually this symmetry is expressed in terms of collineation groups. So we want to examine geometries -here we are interested in generalized polygons -with large collineation groups. There are several ways to make precise what 'large' means. In this paper we investigate the situation where the collineation group of generalized n-gon acts transitively on the set of ordered (n + 1)-gons. This condition is of special interest because it is related to questions of coordinatization (cp. Hughes and Piper [12, §V.3]

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Section: Resultsmentioning

confidence: 99%

Section: I) N = 3 and ~3 Is Isomorphic To One Of The Classical Planesmentioning

confidence: 99%

Generalized polygons with highly transitive collineation groups

Joswig

1995

Geom Dedicata

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“…Condition (R2) implies flag homogeneity. Point homogeneous compact connected projective planes are always flag homogeneous [15], [9]. c) In the case of affine translation planes, our result is known already (though stated differently): cp.…”

Section: --(P L F) Be a Stable Plane And Assume That A Closed Subgmentioning

confidence: 86%

Reconstruction of incidence geometries from groups of automorphisms

Stroppel

1992

Arch. Math

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“…Hence ~ is an injective immersion, and in fact an embedding, because 92 is compact. Proof The group of all continuous automorphisms is locally compact by Bums and Spatzier [7, 2.1], see Grundh6fer [12] and Salzmann [32] for n = 3. Being a locally compact group of diffeomorphisms, it is a smooth Lie transformation group (cp.…”

Section: Smooth Polygons and Their Homomorphismsmentioning

confidence: 99%

On homomorphisms between generalized polygons

Bödi

Krämer

1995

Geom Dedicata

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Abstract. We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphismis either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Mathematics Subject Classifications (1991): 51A25, 51 E24, 51H20, 51 H25.The aim of this paper is to investigate homomorphisms of abstract, topological, and smooth generalized polygons. The first two sections deal with generalized polygons and their homomorphisms. We have tried to make this paper self-contained, so we give a short exposition of the coordinatization and the algebraic operations (+, -, o,/) of a generalized polygon. The most important theorems at this stage are the characterization (2.7) of injective homomorphisms, and the proof (2.9) of Pasini's theorem [27] that the fibers of a non-injective homomorphism are infinite.Topological polygons are introduced in Section 3. Here, the main result is that a homomorphism is either injective or locally constant (3.4); in particular, a connected polygon admits only injective homomorphisms. For topological projective planes, this has been proved by Breitsprecher [4]. This result may be compared to Pasini's theorem [27] that finite polygons admit only injective homomorphisms.In the last section, we introduce smooth polygons. The main result of this paper states that a continuous homomorphism between smooth polygons is always a smooth embedding (4.7). In particular, every continuous automorphism of a smooth polygon is smooth, and thus the topological automorphism group of the polygon, endowed with the compact-open topology, is a smooth Lie transformation group (4.9). As an application, we prove a strong inhomogeneity theorem for a class of isoparametric hypersurfaces.

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Homogene kompakte projektive Ebenen

Cited by 44 publications

References 19 publications

Generalized polygons with highly transitive collineation groups

Generalized polygons with highly transitive collineation groups

Reconstruction of incidence geometries from groups of automorphisms

On homomorphisms between generalized polygons

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