Functions on a swallowtail

TL;DR

This paper classifies certain smooth functions on a swallowtail surface and explores its flat geometry through contact and singularity analysis, providing a detailed understanding of its geometric properties.

Contribution

It introduces a classification of submersions from 3D space to 1D space that preserve the swallowtail, linking singularity theory with geometric analysis.

Findings

Classification of submersions preserving the swallowtail

Analysis of the flat geometry via contact with planes

Identification of singularities of height functions

Abstract

We classify submersions from $(\mathbb{R}^3,0)$ to $(\mathbb{R},0)$ up to diffeomorphisms which preserve the swallowtail and use this classification to study its flat geometry. The flat geometry is derived from the contact of the swallowtail with planes, which is measured by the singularities of the height function.