On $$\mathrm {G}_2$$ and sub-Riemannian model spaces of step and rank three

Abstract

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. Model spaces in this context refer to spaces where any infinitesimal isometry between horizontal tangent spaces can be integrated to a full isometry. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces. In particular, there are no nontrivial sub-Riemannian model spaces of step and rank three with free nilpotentization. We also realize both the compact real form $${\mathfrak {g}}_2^c$$ and the split real form $${\mathfrak {g}}_2^s$$ of the exceptional Lie algebra $${\mathfrak {g}}_2$$ as isometry algebras of different model spaces.