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

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 .W. Domitrz and S.

Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

“…Proof. Without loss of generality we may assume that locally f (s) = (s, F (s)), (4.8) where F (s) = as 3 + G(s), a = 0 and s 0 = 0, where G(s) ∈ m 4 1 , where m n is the maximal ideal of smooth function-germs R n → R vanishing at 0. Let us notice that (s, F (s)), (t, F (t)) is a parallel pair of M nearby f (0) if and only if s = t and…”

“…It leads to the construction of bi-dimensional improper affine spheres ( [3]). The Wigner caustic is an example of an affine λ-equidistant, which is the locus of points dividing chords connecting points on M with parallel tangent lines in a fixed ratio λ ( [5,8,10,23]).…”

“…Local properties of singularities of the Wigner caustic and affine equidistants were studied in many papers [2,4,5,6,7,13,17,19]. In this paper we study global properties of the Wigner caustic of a generic planar closed curve.…”

The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

“…The singularities and the geometry of affine λ-equidistants were very widely studied in many papers [1,5,6,8,13,17,33,40]. The envelope of affine diameters (the Centre Symmetry Set) was studied in [7,12,14,15,16].…”

The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

“…Local properties of singularities of the Wigner caustic and affine equidistants were studied in many papers [5,[9][10][11][12]19,23,25]. In this paper we study global properties of the Wigner caustic of a generic planar closed curve.…”