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.

中文翻译：

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在 $$\mathrm {G}_2$$ 和阶跃和秩 3 的亚黎曼模型空间上

我们给出了所有子黎曼模型空间的完整分类，包括阶梯和秩三。此上下文中的模型空间是指水平切线空间之间的任何无穷小的等距可以集成为完整等距的空间。它们将根据幂零化分为三个家族。每个系列将取决于不同数量的参数，这使得结果与已知的步骤二模型空间情况截然不同。特别是，不存在具有自由幂零化的阶跃和秩 3 的非平凡子黎曼模型空间。我们还实现了异常的紧凑实型 $${\mathfrak {g}}_2^c$$ g 2 c 和分裂实型 $${\mathfrak {g}}_2^s$$ g 2 s李代数 $${\mathfrak {g}}_2$$ g 2 作为不同模型空间的等距代数。