Discussing Knarr's 2-surface of \(\mathbb{R}^4\) which generates the first single shift plane
type: Item
instance of: scholarly article
title: Discussing Knarr's 2-surface of \(\mathbb{R}^4\) which generates the first single shift plane
zbMATH Open document ID: 0969.51004
DOI: 10.1023/A:1005283924339
author: Dieter Betten
author: Rolf Riesinger
published in: Geometriae Dedicata
publication date: 2001-09-27T00:00:00Z
review text: \textit{N. Knarr} [Topologische Differenzenflächenebenen, Diplomarbeit, Kiel (1983)] showed that a 4-dimensional shift plane admitting a 7-dimensional collineation group is isomorphic to the shift plane generated by the planar function \(f:{\mathbb R}^2\to{\mathbb R}^2\) given by \(f(s,t)=(st-{1\over 3}s^3,{1\over 2}t^2-{1\over 12}s^4)\). The derivatives of \(f\) give rise to a topological spread of \({\mathbb R}^4\) and thus to a 4-dimensional translation plane. A long-standing open problem whether a differentiable map \({\mathbb R}^2\to{\mathbb R}^2\) whose derivatives lead to a topological spread are planar (and so generate a shift plane) or whether additional properties are needed. As a step in that direction the authors investigate Knarr's plane algebraically. To begin with the affine part of Knarr's surface \(\Lambda^a_{Kn}\), that is, the graph of \(f\), is investigated. All automorphic affinities of \(\Lambda^a_{Kn}\) are computed and it is shown that \(\Lambda^a_{Kn}\) is equiaffinely homogeneous. Furthermore, \(\Lambda^a_{Kn}\) is convex with respect to 1-dimensional affine subspaces of \({\mathbb R}^4\). The authors then determine the projective closure \(\Lambda_{Kn}\) of \(\Lambda^a_{Kn}\) as an algebraic variety in 4-dimensional projective space \(\Pi_4\). It is shown that this closure is unique and that one has to add the points on the line at infinity \(\ell_{34}=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid x_0=x_1=x_2=0\}\). In fact, \(\ell_{34}\) is the only line of \(\Pi_4\) entirely contained in \(\Lambda_{Kn}\). Moreover, Knarr's surface \(\Lambda_{Kn}\) can algebraically be described as the intersection of the cones \(C_3=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid -12x_0^3x_4+6x_0^2x_2^2-x_1^4=0\}\) and \(C_4=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid -3x_0^2x_3+3x_0x_1x_2-x_1^3=0\}\) and the cubic hypervariety \(V_3=\{(x_0,\ldots,x_4){\mathbb R}\in\Pi_4\mid 4x_0x_2x_4-3x_0x_3^3-4x_1^2x_4+5x_1x_2x_3-2x_2^3=0\}\). Projection from \((0,0,0,0,1){\mathb…
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51H20
zbMATH DE Number: 1546677
zbMATH Keywords: planar function
zbMATH Keywords: shift plane
zbMATH Keywords: Cayley's surface
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
full work available at URL: a:1005283924339
OpenAlex ID: W2912270379
description: scientific article; zbMATH DE number 1546677