论文标题
cartan理论分类乘以$（2,3,5）$ - 分布
A Cartan-theoretic classification of multiply-transitive $(2,3,5)$-distributions
论文作者
论文摘要
ÉlieCartan在他的1910年论文中，为5个manifolds上的通用等级2分布（即$（2,3,5）$分布提供了（本地）等价问题的旅行解决方案。从现代的角度来看，这些结构将等效的描述视为以$ g_2 $的商为模型的（常规，正常）抛物线的几何形状，但这在他的文章中并不透明：的确，1910年的纸箱“连接”不是现代意义上的“ cartan连接”。 We revisit the classification of multiply-transitive $(2,3,5)$-distributions from a modern Cartan-geometric perspective, incorporating $G_2$ structure theory throughout, obtaining: (i) the complete (local) classifications in the complex and real settings, phrased "Cartan-theoretically", and (ii) the full curvature and infinitesimal holonomy of all these models.此外，从理论上讲，我们证明了两个2个2秒互相滚动的$ 3：1 $比率的异常性，而不会扭曲或滑倒，从而产生了$（2,3,5）$的$（2,3,5）$ - 与Symemertry的分配$ g_2 $的分裂真实形式的lie代数。
In his 1910 paper, Élie Cartan gave a tour-de-force solution to the (local) equivalence problem for generic rank 2 distributions on 5-manifolds, i.e. $(2,3,5)$-distributions. From a modern perspective, these structures admit equivalent descriptions as (regular, normal) parabolic geometries modelled on a quotient of $G_2$, but this is not transparent from his article: indeed, the Cartan "connection" of 1910 is not a "Cartan connection" in the modern sense. We revisit the classification of multiply-transitive $(2,3,5)$-distributions from a modern Cartan-geometric perspective, incorporating $G_2$ structure theory throughout, obtaining: (i) the complete (local) classifications in the complex and real settings, phrased "Cartan-theoretically", and (ii) the full curvature and infinitesimal holonomy of all these models. Moreover, we Cartan-theoretically prove exceptionality of the $3:1$ ratio for two 2-spheres rolling on each other without twisting or slipping, yielding a $(2,3,5)$-distribution with symmetry the Lie algebra of the split real form of $G_2$.
