本原测地线长度与（几乎）等差数列
Primitive geodesic lengths and (almost) arithmetic progressions
摘要：
本文研究了黎曼流形上的本原测地线长度集何时具有任意长的等差数列。我们证明了在负曲度量空间中，具有这样的等差数列的度量是相当罕见的。我们引入几乎等差数列，它是等差数列的一个粗化，并证明了每一个负曲闭黎曼流形在其本原长度谱中都有任意长的几乎等差数列.关于真正的等差数列，我们证明了每一个非紧的，局部对称的，算术流形有任意长的等差数列在其本原长度谱。最后，我们用本原长度谱中的等差数列来刻画算术性。我们还提出了一种方法来解决一个著名的谱刚性问题的基础上的流形的稀缺性与等差数列。
英文摘要：
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
