非正曲率秩一流形上测地流的遍历性
The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
- College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）
- Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非正曲率秩一流形上测地流的遍历性，通过引入由曲率消失定义的子集刻画奇异集，证明其零体积时测地流关于刘维尔测度遍历，并推广至实解析度量。
AI中文摘要：
本文研究了闭的非正截面曲率秩一流形上测地流的遍历性。我们引入了流形的一个子集，该子集由二维情形下高斯曲率的无穷阶消失或任意维数情形下约化雅可比行列式的纤维第二矩的无穷阶消失所定义，并基于该子集推导出奇异集的刘维尔测度和豪斯多夫维数的界。特别地，若该子集具有零体积，则奇异集具有零刘维尔测度，且测地流关于刘维尔测度是遍历的。通过将该方法推广到实解析度量，我们将奇异集刻画为由曲率算子构造的非平凡实解析行列式的零点集，从而在此情形下无需任何额外假设即可建立遍历性。
英文摘要：
In this paper, we study the ergodicity of geodesic flows on closed rank-one manifolds of nonpositive sectional curvature. We introduce the infinite-order vanishing set of the Gaussian curvature in dimension two and of the fiberwise second moment of the reduced Jacobi determinant in higher dimensions. We bound the Liouville measure of the singular set in terms of the volume of this subset and, for surfaces, obtain a corresponding Hausdorff dimension bound. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic. We also give a geometric criterion for finitely many exceptional regions whose fundamental groups have virtually Abelian images of rank smaller than the dimension of the manifold. Under strict negativity of sectional curvature outside these regions, we obtain a Hausdorff dimension bound for the singular set and prove that the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real-analytic metrics, we characterize the singular set as the zero set of a nontrivial real-analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without further assumptions.
