On the classification of sub-Riemannian structures on a 5D two-step nilpotent Lie group

Abstract

We classify the left-invariant sub-Riemannian structures on the unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D group is the first two-step nilpotent Lie group beyond the three- and five-dimensional Heisenberg groups. Alongside, we also present a classification, up to automorphism, of the subspaces of the associated Lie algebra (together with a complete set of invariants).