Geometric equivalence among smooth map germs

TL;DR

This paper explores generalized geometric equivalence relations among smooth map germs related to G-structures, extending classical A-equivalence, with notable applications despite some limitations in subgroup properties.

Contribution

It introduces a new class of geometric equivalence relations among smooth map germs based on G-structures, broadening the scope of classical A-equivalence.

Findings

These equivalence relations generalize A-equivalence.

They are not always geometric subgroups in Damon's sense.

Applications demonstrate their usefulness despite limitations.

Abstract

We consider equivalence relations among smooth map germs with respect to geometry of G-structures on the target space germ. These equivalence relations are natural generalization of right-left equivalence (i.e., A-equivalence) in the sense of Thom-Mather depending on geometric structures on the target space germ. Unfortunately, these equivalence relations are not necessarily geometric subgroups in the sense of Damon (1984). However, we have interesting applications of these equivalence relations.