论文标题
关于韦尔的类型定理和射射线次刚性几何形状中的较刚性的通用性
On Weyl's type theorems and genericity of projective rigidity in sub-Riemannian Geometry
论文作者
论文摘要
H. Weyl在1921年表明，对于一个大于$ 1 $的连接的层次，如果两个riemannian指标是共形的，并且具有相同的测量学指标，则可以进行重新训练，那么一个指标是另一个度量的恒定标准。在本文中，我们调查了伊曼尼亚次级指标的类似特性。特别是，我们证明了称为Weyl射击刚度的类似陈述在所有分析次分析类别中都属于其分布的分布，其特定属性具有其复杂的异常极端秩序（称为最小秩序），或在所有分布中平滑类别，例如，所有分布都具有所有复杂的极端异常的非正态异常，其nillopotent近似值是Migimal isal Mistal Mistail Mistail Mistail Mistail Mistail Mistail Mistail Mist的。这也表明，在实际的分析类别中，所有次摩nanian次级指标的分布的通用性是螺旋出的刚性，而在给定的支架生成分布上，Weyl的weyl均具有较刚性的固定性，并且在投射刚性的刚性次 - 利马尼亚指标上。最后，这使我们能够获得亚riemannian指标的投影刚度的类似概要结果，即，当唯一具有相同的次摩nanian次测量的亚riemannian指标，最多具有给定的量子化，是给定的，是一个给定的恒定尺度。这是我们在最近的论文ARXIV中证明的较弱刚性特性的一般结果的改善：1801.04257 [Math.dg]。
H. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than $1$, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper arXiv:1801.04257[math.DG].
