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Publication type: Article in scientific journal
Type of review: Peer review (publication)
Title: Smooth stable planes
Authors: Bödi, Richard
DOI: 10.21256/zhaw-1746
Published in: Results in Mathematics
Volume(Issue): 31
Issue: 3-4
Page(s): 300
Pages to: 321
Issue Date: May-1997
Publisher / Ed. Institution: Springer
Publisher / Ed. Institution: Basel
ISSN: 1420-9012
Language: English
Subjects: Stable plane; Smooth; Differentiable incidence geometry; Tangent translation plane
Subject (DDC): 510: Mathematics
Abstract: This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes. 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 (
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. (1997). Smooth stable planes. Results in Mathematics, 31(3-4), 300–321.
Bödi, R. (1997) ‘Smooth stable planes’, Results in Mathematics, 31(3-4), pp. 300–321. Available at:
R. Bödi, “Smooth stable planes,” Results in Mathematics, vol. 31, no. 3-4, pp. 300–321, May 1997, doi: 10.21256/zhaw-1746.
BÖDI, Richard, 1997. Smooth stable planes. Results in Mathematics. Mai 1997. Bd. 31, Nr. 3-4, S. 300–321. DOI 10.21256/zhaw-1746
Bödi, Richard. 1997. “Smooth Stable Planes.” Results in Mathematics 31 (3-4): 300–321.
Bödi, Richard. “Smooth Stable Planes.” Results in Mathematics, vol. 31, no. 3-4, May 1997, pp. 300–21,

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