A note on isoparametric generalized quadrangles
type: Item
instance of: scholarly article
title: A note on isoparametric generalized quadrangles
zbMATH Open document ID: 1106.51002
author: Stefan Immervoll
published in: Archiv der Mathematik
publication date: 2006-12-06T00:00:00Z
review text: An arbitrary isoparametric hypersurface with four distinct principal curvatures in a sphere defines a generalized quadrangle, the point and lines sets of which are the respective two associated focal manifolds \(\mathcal{P}\) and \(\mathcal{L}\). Incidence is defined by the point-line pairs that are projected from the same point of the compact connected hypersurface. In the paper under review, the author proves that two points \(p,q\) are collinear if and only if the difference \(p-q\) divided by the norm \(\| p-q\| \) belongs to \(\mathcal{L}\), and similarly for lines. Since collinearity completely defines the geometric structure of the generalized quadrangle (in particular, the incidence relation), this gives a simple algebraic description of the generalized quadrangle. For the Clifford type isoparametric generalized quadrangles, this result was previously proved by \textit{L. Kramer} [Geom. Dedicata 79, 321--339 (2000; Zbl 0958.51006)].
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 51H25
Mathematics Subject Classification ID: 51H30
zbMATH DE Number: 5077610
zbMATH Keywords: Isoparametric hypersurfaces
zbMATH Keywords: generalized quadrangles
zbMATH Keywords: compact connected quadrangles
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: s00013-006-1758-y
OpenAlex ID: W2030675677
DOI: 10.1007/S00013-006-1758-Y
description: scientific article; zbMATH DE number 5077610