On affine normal forms and a classification of homogeneous surfaces in affine three-space
type: Item
instance of: scholarly article
title: On affine normal forms and a classification of homogeneous surfaces in affine three-space
zbMATH Open document ID: 0999.53008
DOI: 10.1023/A:1005083518793
author: Vladimir Ejov
author: M. Eastwood
published in: Geometriae Dedicata
publication date: 2002-07-16T00:00:00Z
review text: The authors contribute to the homogeneity problem of affine hypersurfaces by establishing a new method based on the Taylor expansion for the defining graph function with respect to the Blaschke/Berwald normal as last axis. Specifically, they classify all the affinely homogeneous surfaces in \(3\)-space, thus confirming the result of \textit{B. Doubrov, B. Komrakov} and \textit{M. Rabinovich} [Geometry and Topology of Submanifolds VIII, World Scientific, Singapore, 168-178 (1996; Zbl 0934.53007)]. The main difference is that instaed of normalizing the symmetry algebras now the graph functions are normalized. Actually, this is done by a clever induction procedure on the Taylor polynomials of these functions. The authors believe that their approach is more advantegeous. As they realized after the completion of their work, yet another method for the classification of these surfaces has been proposed by \textit{A. V. Loboda} [Math. Notes 65, No. 5, 668-672 (1999); translation from Mat. Zametki 65, 793-797 (1999; Zbl 0976.53012) and Proc. `Pontrjagin's Readings-6', Voronezh-98, 124 (1998)]. Several computations related to the problem of homogeneity are nonlinear and rather complex, including enormous branching phenomena. So machine computing is sometimes inevitable. On the other hand, there doesn't exist an automatism to attack the problem as a whole. For the next higher dimension (hypersurfaces in \(4\)-space) there are recent works by the same authors in the complex domain [Asian J. Math. 5, 721-740 (2001; Zbl 1022.53043)] and by \textit{M. Wermann} with respect to real classification [Dissertation Univ. Dortmund 2001, Shaker Verlag (2001; Zbl 1028.53010)]. Codimension two homogeneity problems in \(4\)-space have been solved by normalizing Lie subalgebras in the reviewer's papers [Geom. Dedicata 71, 129-178 (1998; Zbl 0973.53511), Contrib. Algebra Geom. 41, No. 1, 159-180 (2000; Zbl 0976.53013)], and forthcoming [Banach Center Publications, Proc. Conf. `PDEs, submanifol…
reviewed by: Rolf Walter
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1348241
zbMATH Keywords: affine hypersurfaces
zbMATH Keywords: homogeneous surfaces
zbMATH Keywords: homogeneity problem
zbMATH Keywords: classification
zbMATH Keywords: normal forms
MaRDI profile type: Publication
full work available at URL: a:1005083518793
OpenAlex ID: W1516143337
description: scientific article; zbMATH DE number 1348241