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Title: Smooth stable planes
Authors : Bödi, Richard
Published in : Results in Mathematics
Volume(Issue) : 31
Issue : 3-4
Pages : 300
Pages to: 321
Publisher / Ed. Institution : Birkhäuser
Publisher / Ed. Institution: Basel
Issue Date: May-1997
License (according to publishing contract) : Licence according to publishing contract
Type of review: Peer review (Publication)
Language : English
Subjects : Stable plane; Smooth; Differentiable incidence geometry; Tangent translation plane
Subject (DDC) : 500: Natural sciences and mathematics
Abstract: This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes [7]. It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6)). Moreover, the flag space is a closed submanifold of the product manifold P×L (Theorem (1.14)), and the smooth structure on the set P of points and on the set L of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point p \te P the tangent space TpP carries the structure of a locally compact affine translation plane A p , see Theorem (2.5). Dually, we prove in Section 3 that for any line L∈L the tangent space T L L together with the set S L ={T L L p ∣p∈L} gives rise to some shear plane. It turned out that the translation planes A p are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes.
Further description : Erworben im Rahmen der Schweizer Nationallizenzen (
Departement: School of Engineering
Publication type: Article in scientific Journal
DOI : 10.1007/BF03322167
ISSN: 1422-6383
Appears in Collections:Publikationen School of Engineering

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