The flat geometry of the $I_1$ singularity: $(x,y)\mapsto (x,xy,y^2,y^3)$

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.

16 pages. arXiv admin note: text overlap with arXiv:1801.06380