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.
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E. Grong is supported by the Research Council of Norway (project number 249980/F20). The authors were partially supported by the joint NFR-DAAD project 267630/F10. Results are partially based on the first author’s Master Thesis at the University of Bergen, Norway.
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Berge, E., Grong, E. On \(\mathrm {G}_2\) and sub-Riemannian model spaces of step and rank three. Math. Z. 298, 1853–1885 (2021). https://doi.org/10.1007/s00209-020-02653-y
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DOI: https://doi.org/10.1007/s00209-020-02653-y