Mathematische Zeitschrift | 2019
On \n \n \n \n $$\\mathrm {G}_2$$\n \n \n G\n 2\n \n \n 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 $${\\mathfrak {g}}_2^c$$ g 2 c and the split real form $${\\mathfrak {g}}_2^s$$ g 2 s of the exceptional Lie algebra $${\\mathfrak {g}}_2$$ g 2 as isometry algebras of different model spaces.