A generalization of Zakalyukin's lemma, and symmetries of surface singularities

TL;DR

This paper extends Zakalyukin's lemma to frontals and explores its applications to surface singularities, enhancing understanding of wave front coincidences and their implications in differential geometry.

Contribution

The paper generalizes Zakalyukin's lemma for frontals and applies it to analyze symmetries and singularities of surfaces.

Findings

Extended Zakalyukin's lemma to frontals

Identified new symmetries in surface singularities

Provided applications to surface wave front analysis

Abstract

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.