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

TL;DR

This paper studies the stable singularities of affine equidistants of surfaces in four-dimensional space, linking them to the extrinsic geometry and weakly parallel points of the surface.

Contribution

It characterizes stable singularities of affine equidistants in terms of the surface's extrinsic geometry, providing a geometric perspective on their structure.

Findings

Stable singularities are classified as A_k and C_{2,2}^{ ext{±}}.

Singularities are characterized via bi-local extrinsic geometry.

The study links singularities to weakly parallel points on the surface.

Abstract

For a generic embedding of a smooth closed surface $M$ into $\mathbb R^4$, the subset of $\mathbb R^4$ which is the affine $\lambda$-equidistant of $M$ appears as the discriminant set of a stable mapping $M \times M \to \mathbb R^4$, hence their stable singularities are $A_k, \, k=2, 3, 4,$ and $C_{2,2}^{\pm}$. In this paper, we characterize these stable singularities of $\lambda$-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 $M$.