Sub-Riemannian geometry of parallelizable spheres
Sub-Riemannian geometry of parallelizable spheres
复制标题
可平行球体的亚黎曼几何
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
I. Markina
中科院分区：
文献类型：
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作者：
M. G. Molina;I. Markina
本文的第一个目的是比较三维球S上各种次黎曼结构起源于不同的构造。也就是说，我们描述了S的次黎曼几何，它的右作用是C上的李群，它继承于C中开单位球的自然复结构，以及通过Hopf映射被认为是主S Hopf丛时出现的几何。这种比较的主要结果是，事实上，这三种结构是一致的。我们给出了阶数为6和4的七维球面S的两个括号生成分布。第二个分布给出了S的亚黎曼结构，这种结构到目前为止还没有广泛存在。其中一种分布可以通过考虑S的CR几何得到，这种几何继承自C中开放单位球的自然复数结构。另一种分布源于四元数类Hopf映射。
The first aim of the present paper is to compare various subRiemannian structures over the three dimensional sphere S originating from different constructions. Namely, we describe the sub-Riemannian geometry of S arising through its right action as a Lie group over itself, the one inherited from the natural complex structure of the open unit ball in C and the geometry that appears when it is considered as a principal S−bundle via the Hopf map. The main result of this comparison is that in fact those three structures coincide. We present two bracket generating distributions for the seven dimensional sphere S of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for S that is not widely present in the literature until now. One of the distributions can be obtained by considering the CR geometry of S inherited from the natural complex structure of the open unit ball in C. The other one originates from the quaternionic analogous of the Hopf map.