The flat geometry of the $I_{1}$ singularity: $(x,y)\mapsto(x,xy,y^{2},y^{3})$

TL;DR

This paper investigates the geometric properties of the $I_1$ singularity in $\,\mathbb{R}^4$, providing a normal form, classifying hyperplane contacts, and analyzing height function singularities.

Contribution

It introduces a generic normal form for the $I_1$ singularity invariant under source diffeomorphisms and target isometries, and classifies hyperplane contacts via height function singularities.

Findings

Established a normal form for the $I_1$ singularity.

Classified hyperplane contact types through submersion analysis.

Analyzed singularities of the height function related to the $I_1$ singularity.

Abstract

We study the flat geometry of the least degenerate singularity of a singular surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by $(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when projecting a regular surface in $\mathbb R^5$ orthogonally to $\mathbb R^4$ along a tangent direction. We obtain a generic normal form for $I_1$ invariant under diffeomorphisms in the source and isometries in the target. We then consider the contact with hyperplanes by classifying submersions which preserve the image of $I_1$. The main tool is the study of the singularities of the height function.