Normal forms for the sub-Riemannian exponential map of $\mathbb{G}_\alpha$, $\mathrm{SU}(2)$, and $\mathrm{SL}(2)$

TL;DR

This paper applies singularity theory to classify the local behavior of the sub-Riemannian exponential map in specific geometric structures, revealing normal forms near critical points for the $ ext{SU}(2)$, $ ext{SL}(2)$, and $ ext{α}$-Grushin plane.

Contribution

It introduces a method to find normal forms of the exponential map's singularities in sub-Riemannian geometry, with detailed analysis of key examples.

Findings

Normal forms for fold and saddle singularities identified

Techniques applicable to various sub-Riemannian structures

Discussion of implications and open problems included

Abstract

The goal of this paper is to use singularity theory to find normal forms near the critical points of the sub-Riemannian exponential map. Three cases are studied: the $\alpha$-Grushin plane with fold singularities, and the special unitary group $\mathrm{SU}(2)$ and special linear group $\mathrm{SL}(2)$ with fold and saddle-like singularities. They serve as examples of different sub-Riemannian structures and the techniques presented can be applied to other contexts. The paper also includes a discussion of the implications of this approach, as well as open problems.