Singularities and duality in the flat geometry of submanifolds of Euclidean spaces
type: Item
instance of: scholarly article
title: Singularities and duality in the flat geometry of submanifolds of Euclidean spaces
zbMATH Open document ID: 1066.58021
author: D. K. H. Mochida
author: Carmen Romero-Fuster
author: Maria Aparecida Soares Ruas
published in: Beiträge zur Algebra und Geometrie
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/223927
full work available at URL: 8.html
review text: In [\textit{J. W. Bruce} and \textit{M. C. Romero-Fuster}, Q. J. Math., Oxf. Ser. (2) 42, 433--441 (1991; Zbl 0780.53006)] it has been shown that the subset of normals over the parabolic set of a generic surface \(M\) in \(\mathbb R^ 3\) (i.e. the caustic of the family of height functions) and that of principal asymptotic directions are duals as subset of the 2-sphere and a similar duality result was obtained for surfaces in \(\mathbb R^ 4\) by \textit{J. W. Bruce} and \textit{A. C. Nogueira} [Q. J. Math., Oxf. (2) 49, 433--443 (1998; Zbl 0982.53006)]. In the paper under review the authors extend these duality results to submanifolds of any dimension in Euclidean space. They prove that the dual of the set of binormals of a submanifold \(M\) is the set of its asymptotic directions and obtain a duality relation concerning the lowest codimensional multilocal strata. To accomplish this they characterize the singularities of the height functions and orthogonal projections on generic submanifolds from the geometric viewpoint and utilize the canal hypersurfaces.
reviewed by: Ivko Dimitrić
Mathematics Subject Classification ID: 58K30
Mathematics Subject Classification ID: 57R45
zbMATH DE Number: 1549906
zbMATH Keywords: Duality
zbMATH Keywords: singularities
zbMATH Keywords: generic contact
zbMATH Keywords: submanifolds
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1549906