On the existence of topological ovals in flat projective planes
type: Item
instance of: scholarly article
title: On the existence of topological ovals in flat projective planes
zbMATH Open document ID: 0876.51002
published in: Archiv der Mathematik
publication date: 1997-07-06T00:00:00Z
full work available at URL: http://hdl.handle.net/10092/11282
review text: Any closed oval (= smooth convex set) in a compact, connected projective plane is a topological oval; it is homeomorphic to a sphere. Such ovals exist only in \(2\)- and \(4\)-dimensional planes [\textit{H. Salzmann, D. Betten, Th. Grundhöfer, H. Hähl, R. Löwen} and \textit{M. Stroppel}, `Compact projective planes', Berlin : de Gruyter (1995; Zbl 0851.51003), Theorem 55.14]. The tangents of a closed oval form an oval in the dual plane. All known examples in \(2\)-dimensional (= flat) non-Desarguesian planes consist of the absolute points of a polarity (and hence are analogous to conics). By a clever construction, the authors show that in any flat plane \(\mathcal P\) each triangle is contained in a topological oval. A set \(\mathcal Q\) of flags in \(\mathcal P\) is called a quasi-oval, if it is contained in a suitable affine subplane of \(\mathcal P\) such that for each flag \((p,L)\) in \(\mathcal Q\) the other points of \(\mathcal Q\) are contained in one half-plane with respect to \(L\). Any finite quasi-oval can be extended successively in such a way that the closure of the point set is a full oval.
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 1028213
zbMATH Keywords: topological ovals
zbMATH Keywords: flat projective plane
author: Burkard Polster
author: Nils Rosehr
author: Günter F. Steinke
reviewed by: Helmut Salzmann
MaRDI profile type: Publication
OpenAlex ID: W1967087099
cites work: Topologische Ovale
cites work: Topological planes
cites work: Compact projective planes. With an introduction to octonion geometry
cites work: Q4764160
DOI: 10.1007/S000130050074
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description: scientific article; zbMATH DE number 1028213