[论文解读] On projective and affine equivalence of sub-Riemannian metrics

Frédéric Jean, Sofya Maslovskaya+1 more|arXiv (Cornell University)|Jan 12, 2018

Geometric Analysis and Curvature Flows参考文献 21被引用 3

一句话总结

本文证明了非共形刚性的子黎曼度量在余切丛纤维上至少存在一个非平凡的二次积分，并在其幂零逼近中诱导出乘积结构。本文证明了在射影等价下，一般子黎曼度量是共形刚性的；在仿射等价下，其刚性更强，通过几何控制与Jacobi曲线分析，将经典黎曼几何结果推广至子黎曼几何领域。

ABSTRACT

Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and affinely equivalent metrics is classical (Levi-Civita, Eisenhart). In particular, a Riemannian metric which is not rigid satisfies the following two special properties: its geodesic flow possesses nontrivial integrals and the metric induces certain canonical product structure on the ambient manifold. These classification results were extended to contact and quasi-contact distributions by Zelenko. Our general goal is to extend these results to arbitrary sub-Riemannian manifolds, and we establish two types of results toward this goal: if a sub-Riemannian metric is not projectively conformally rigid, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results: first, we show that a generic sub-Riemannian metric on a fixed pair $(M,D)$ is projectively conformally rigid. Second, we prove that, except for special pairs $(m,n)$, every sub-Riemannian metric on a rank $m$ generic distribution in an $n$-dimensional manifold is projectively conformally rigid. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity.

研究动机与目标

• 将经典黎曼几何中关于射影等价与仿射等价的结果推广至子黎曼几何领域。
• 通过识别几何与代数障碍，刻画非刚性子黎曼度量的特征。
• 建立子黎曼设定下共形刚性与标准刚性的泛性结果。
• 分析Jacobi曲线的结构及其在确定子黎曼度量等价类中的作用。
• 证明非共形刚性度量必须具有非平凡的二次首次积分，并且其幂零逼近可分解。

提出的方法

• 利用庞特里亚金最大值原理与Jacobi曲线分析测地线与正规极值。
• 应用轨道微分同胚技术研究子黎曼度量的射影等价与仿射等价。
• 利用基本代数系统推导等价性与可积性的条件。
• 采用幂零逼近研究分布的局部结构及其等价类。
• 运用子主丛论证方法，证明充分余向量与2-决定集的开性与稠密性。
• 利用指数映射与摄动理论，构造满足几何与代数条件（H.1）–（H.3）的有限点集。

实验结果

研究问题

• RQ1 在何种条件下，两个子黎曼度量是射影等价的？
• RQ2 非共形刚性子黎曼度量必须具备何种几何与代数结构？
• RQ3 二次首次积分的存在性如何影响幂零逼近的结构？
• RQ4 子黎曼度量在射影等价与仿射等价下的泛性行为如何？
• RQ5 子黎曼度量的刚性能否通过其幂零逼近的可分解性来刻画？

主要发现

• 若一个子黎曼度量在射影等价下非共形刚性，则其余切丛纤维上至少存在一个非平凡的二次积分。
• 任意点处，非共形刚性子黎曼度量的幂零逼近均具有规范的乘积结构。
• 对于流形上固定的分布，一般子黎曼度量在射影等价下为共形刚性。
• 对于n维流形上秩为$m$的泛分布（排除特定$(m,n)$对），所有子黎曼度量在射影等价下均为共形刚性。
• 在仿射等价下，共形刚性在一般情形下可加强为标准刚性。
• 存在$ N = n(n+1)/2 $ 个2-决定点，可确保关于二次首次积分与等价性的关键命题成立。