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].

中文翻译：

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亚黎曼几何中外尔类型定理和射影刚性的一般性

H. Weyl 在 1921 年证明，对于维度大于 $1$ 的连通流形，如果两个黎曼度量是共形的并且在重新参数化之前具有相同的测地线，那么一个度量是另一个度量的恒定缩放。在本文中，我们研究了子黎曼度量的类似属性。特别是，我们证明了类似的陈述，称为 Weyl 射影刚性，对于具有其复杂异常极值的特定属性的分布的所有亚黎曼度量在实分析范畴中成立，称为最小阶次，或者在所有的平滑范畴中成立。分布使得其幂零近似的所有复杂异常极值都是最小阶的。这也表明，在实解析范畴中，在给定的括号生成分布上，所有子黎曼度量都是 Weyl 投影刚性的分布的通用性和 Weyl 投影刚性亚黎曼度量的通用性。最后，这允许我们获得亚黎曼度量的投影刚性的类似通用结果，即当唯一具有相同亚黎曼测地线的亚黎曼度量，直到重新参数化，给定一个，是这个的常数缩放给一个。这是我们在最近的论文 arXiv:1801.04257[math.DG] 中证明的较弱刚性属性的通用性的结果的改进。e.当唯一的子黎曼度量具有相同的子黎曼测地线时，直到重新参数化，给定的，是这个给定的一个常数缩放。这是我们在最近的论文 arXiv:1801.04257[math.DG] 中证明的较弱刚性属性的通用性的结果的改进。e.当唯一的子黎曼度量具有相同的子黎曼测地线时，直到重新参数化，给定的，是这个给定的一个常数缩放。这是我们在最近的论文 arXiv:1801.04257[math.DG] 中证明的较弱刚性属性的通用性的结果的改进。