Collineations of smooth stable planes

Bödi, Richard

doi:10.21256/zhaw-1743

Please use this identifier to cite or link to this item: https://doi.org/10.21256/zhaw-1743

Publication type:	Article in scientific journal
Type of review:	Peer review (publication)
Title:	Collineations of smooth stable planes
Authors:	Bödi, Richard
DOI:	10.21256/zhaw-1743 10.1515/form.10.6.751
Published in:	Forum Mathematicum
Volume(Issue):	10
Issue:	6
Page(s):	751
Pages to:	773
Issue Date:	1-Nov-1998
Publisher / Ed. Institution:	De Gruyter
Publisher / Ed. Institution:	de Gruyter
ISSN:	0933-7741
Language:	English
Subject (DDC):	510: 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 (http://www.nationallizenzen.ch)
URI:	https://digitalcollection.zhaw.ch/handle/11475/3260
Fulltext version:	Published version
License (according to publishing contract):	Licence according to publishing contract
Departement:	School of Engineering
Appears in collections:	Publikationen School of Engineering

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Bödi, R. (1998). Collineations of smooth stable planes. Forum Mathematicum, 10(6), 751–773. https://doi.org/10.21256/zhaw-1743

Bödi, R. (1998) ‘Collineations of smooth stable planes’, Forum Mathematicum, 10(6), pp. 751–773. Available at: https://doi.org/10.21256/zhaw-1743.

R. Bödi, “Collineations of smooth stable planes,” Forum Mathematicum, vol. 10, no. 6, pp. 751–773, Nov. 1998, doi: 10.21256/zhaw-1743.

BÖDI, Richard, 1998. Collineations of smooth stable planes. Forum Mathematicum. 1 November 1998. Bd. 10, Nr. 6, S. 751–773. DOI 10.21256/zhaw-1743

Bödi, Richard. 1998. “Collineations of Smooth Stable Planes.” Forum Mathematicum 10 (6): 751–73. https://doi.org/10.21256/zhaw-1743.

Bödi, Richard. “Collineations of Smooth Stable Planes.” Forum Mathematicum, vol. 10, no. 6, Nov. 1998, pp. 751–73, https://doi.org/10.21256/zhaw-1743.