Singularities of the affine chord envelope for two surfaces in four-space
type: Item
instance of: scholarly article
title: Singularities of the affine chord envelope for two surfaces in four-space
zbMATH Open document ID: 1301.58018
DOI: 10.1134/S0081543812040165
author: Graham M. Reeve
author: V. M. Zakalyukin
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-09-18T00:00:00Z
review text: This article continues a series of previous ones devoted to singular varieties associated in a natural way to a pair of submanifolds in an affine space [\textit{P. J. Giblin} et al., Proc. Steklov Inst. Math. 267, 59--75 (2009; Zbl 1205.37060); \textit{P. J. Giblin} and \textit{V. M. Zakalyukin}, Proc. Lond. Math. Soc. (3) 90, No. 1, 132--166 (2005; Zbl 1068.57028); Geom. Dedicata 130, 43--58 (2007; Zbl 1148.58016)]. For the given two 2-surfaces \(M\) and \(N\) in 4-dimensional affine space, pairs of points \(a \in M\) and \(b \in N\) are considered such that the tangent planes \(T_aM\) and \(T_bN\) are not in general position. Two 2-planes in \(\mathbb{R}^4\) are in general position when their parallels, being shifted to the origin, span \(\mathbb{R}^4\) as a linear space: \(T_aM \oplus T_bN = \mathbb{R}^4\). When \(T_aM \oplus T_bN \neq \mathbb{R}^4\) the pair of points \(a, b\) is called special. A straight line through a special pair is called a chord. The envelope of these chords for two generic 2-surfaces is called the Minkowski set. In this article the chords for special pairs of two surface germs in \(\mathbb{R}^4\) are considered and the generic local singularities of its Minkowski set are classified. The obtained results can be used in studies of wave propagation in \(\mathbb{R}^4\) and in a range of variational problems.
Mathematics Subject Classification ID: 58K05
zbMATH DE Number: 6345588
zbMATH Keywords: surface
zbMATH Keywords: germs in four-space
zbMATH Keywords: chord envelope
zbMATH Keywords: singularities
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
cites work: Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnol'd
cites work: Semi-classical mechanics in phase space: A study of Wigner’s function
cites work: Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
cites work: Bitangencies on surfaces in four dimensions
cites work: Q4952525
cites work: Geometry of curves and surfaces through the contact map
cites work: Bifurcations of affine equidistants
cites work: Singularities of centre symmetry sets
cites work: Recognition of centre symmetry set singularities
cites work: Q4115402
cites work: Bifurcations of the center of symmetry
cites work: Differentiable invariants
cites work: Singularities of the Minkowski set and affine equidistants for a curve and a surface
full work available at URL: s0081543812040165
OpenAlex ID: W2114659711
description: scientific article; zbMATH DE number 6345588