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Title: Collineations of smooth stable planes
Authors : Bödi, Richard
Published in : Forum mathematicum
Volume(Issue) : 10
Issue : 6
Pages : 751
Pages to: 773
Publisher / Ed. Institution : Berlin
Publisher / Ed. Institution: de Gruyter
Issue Date: 1-Nov-1998
License (according to publishing contract) : Licence according to publishing contract
Type of review: Peer review (Publication)
Language : English
Subject (DDC) : 500: Natural sciences and mathematics
Abstract: Smooth stable planes have been introduced in [4]. We show that every continuous collineation between two smooth stable planes is in fact a smooth collineation. This implies that the group Γ of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set ℒ of lines. In particular, this shows that the point and line sets of a (topological) stable plane ℐ admit at most one smooth structure such that ℐ becomes a smooth stable plane. The investigation of central and axial collineations in the case of (topological) stable planes due to R. Löwen ([25], [26], [27]) is continued for smooth stable planes. Many results of [26] which are only proved for low dimensional planes (dim ℐ ≤ 4) are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers Γ[c,c] 1 and Γ[A,A] 1 (see (3.2) Notation) are closed, simply connected, solvable subgroups of Aut(ℐ) (Corollary (4.17)). Moreover, we show that Γ[c,c] is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections.
Further description : Erworben im Rahmen der Schweizer Nationallizenzen (
Departement: School of Engineering
Publication type: Article in scientific Journal
DOI : 10.1515/form.10.6.751
ISSN: 0933-7741
Appears in Collections:Publikationen School of Engineering

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