Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space

Abstract

Abstract. For a generic embedding of a smooth closed surface M into R 4 , the subset of R 4 which is the affine λ−equidistant of M appears as the discriminant set of a stable mapping M × M → R 4 , hence their stable singularities are A k , k = 2, 3, 4, and C ± 2,2 . In this paper, we characterize these stable singularities of λ−equidistants in terms of the bi-local extrinsic geometry of the surface, leading to a geometrical study of the set of weakly parallel points on M .