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

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