Homogeneous $2$-nondegenerate CR manifolds of hypersurface type in low dimensions
Homogeneous $2$-nondegenerate CR manifolds of hypersurface type in low dimensions
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低维超曲面型齐次$2$-非简并CR流形
DOI：
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
D. Sykes
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文献类型：
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作者：
D. Sykes
在最近的一篇论文中，作者和我。Zelenko引入了修改的CR符号的概念，用于组织$2$-非退化CR结构的局部不变量。本文考虑$mathbb{C}^4$中的齐次超曲面，它是CR超曲面埃尔兰根程序中的自然边界，并对$mathbb{C}^4$中的局部齐次2 $-非退化超曲面进行了局部等价分类，其对称群维数在所有这样的结构中是最大的，这些结构具有相同的局部不变量编码在它们各自的修改符号中.在所考虑的维度，我们表明，在均匀的结构与给定的修改CR符号，最对称的结构（称为模型结构）是唯一的。然后通过对$mathbb{C}^4 $中齐次超曲面的修改符号进行分类来间接实现分类，得到（直到局部等价）9个模型结构。用于获得这种分类的方法，然后应用到寻找齐次超曲面在高维空间。在$mathbb{C}^5 $中描述了总共20 $局部非等价的极大对称齐次$2$-非退化超曲面，在$mathbb{C}^6 $中描述了40 $这样的超曲面，其中一些在其他著作中描述过，而许多是新的。最后，给出了$mathbb{C}^{n+1}$中两个新的齐次2 $-非退化超曲面序列（标号为$n$）.值得注意的是，所有来自后一序列的例子都可以实现为幂零李群上的左不变结构。
In a recent paper, the author and I. Zelenko introduce the concept of modified CR symbols for organizing local invariants of $2$-nondegenerate CR structures. In this paper, we consider homogeneous hypersurfaces in $mathbb{C}^4$, a natural frontier in the CR hypersurface Erlangen programs, and classify up to local equivalence the locally homogeneous $2$-nondegenerate hypersufaces in $mathbb{C}^4$ whose symmetry group dimension is maximal among all such structures with the same local invariants encoded in their respective modified symbols. In the considered dimension, we show that among homogeneous structures with given modified CR symbols, the most symmetric structures (termed model structures) are unique. The classification is then achieved indirectly through classifying the modified symbols of homogeneous hypersurfaces in $mathbb{C}^4$, obtaining (up to local equivalence) nine model structures. The methods used to obtain this classification are then applied to find homogeneous hypersurfaces in higher dimensional spaces. In total $20$ locally non-equivalent maximally symmetric homogeneous $2$-nondegenerate hypersurfaces are described in $mathbb{C}^5$, and $40$ such hypersurfaces are described in $mathbb{C}^6$, of which some have been described in other works while many are new. Lastly, two new sequences, indexed by $n$, of homogeneous $2$-nondegenerate hypersurfaces in $mathbb{C}^{n+1}$ are described. Notably, all examples from one of these latter sequences can be realized as left-invariant structures on nilpotent Lie groups.
超曲面型2-非简并CR结构和Levi叶空间上旗形结构的几何
影响因子：
1.7
作者：
Sykes, David;Zelenko, Igor
通讯作者：
Zelenko, Igor
具有一维Levi核的超曲面型齐次2-非简并CR结构对称群的最大维数
影响因子：
0.7
作者：
Sykes, David;Zelenko, Igor
通讯作者：
Zelenko, Igor