Singularities and duality in the flat geometry of submanifolds of Euclidean spaces
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.
scientific article; zbMATH DE number 1549906