Singularities of the affine chord envelope for two surfaces in four-space
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.
scientific article; zbMATH DE number 6345588