Affine focal sets of codimension $2$ submanifolds contained in hyper surfaces

TL;DR

This paper investigates the affine focal set of submanifolds within hypersurfaces, providing conditions for regularity, describing singularities in 3D, and characterizing umbilic and flat immersions.

Contribution

It offers new conditions for the regularity of affine focal sets, describes their singularities, and characterizes umbilic and normally flat immersions in affine differential geometry.

Findings

Affine focal set regularity conditions established.

Stable singularities for curves in 3-space described.

Characterization of umbilic and normally flat immersions provided.

Abstract

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds $N^n$ contained in hypersurfaces $M^{n+1}$ of the $(n+2)$-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in $3$-space, we describe its stable singularities. For a given Darboux vector field $\xi$ of the immersion $N\subset M$, one can define the affine metric $g$ and the affine normal plane bundle $\mathcal{A}$. We prove that the $g$-Laplacian of the position vector belongs to $\mathcal{A}$ if and only if $\xi$ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For $N$ contained in a hyperplane $L$, we show that $N\subset M$ is umbilic if and only if $N\subset L$ is an affine sphere and the envelope of tangent spaces is a cone. For $M$ hyperquadric, we prove that $N\subset M$ is umbilic if and only if $N$ is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: Given a hypersurface $f$ and a point $O$ in the $(n+1)$-space, the immersion $(\nu,\nu\cdot(f-O))$, where $\nu$ is the co-normal of $f$, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.