标题： 类时共轭点在洛伦兹长度空间中 显示英文标题

标题： Timelike conjugate points in Lorentzian length spaces

摘要： 我们研究在洛伦兹(预-)长度空间的合成设定中沿类时测地线的共轭点概念，这受到Shankar–Sormani早期关于度量空间工作的启发。 在对类时和因果测地线收敛性的初步考虑之后，我们引入并比较了沿类时测地线的一侧、对称、不可达和终极共轭点。 我们证明了所有这些概念在光滑（强因果）时空设定中与通常的概念相容。 作为应用，我们证明了一个类时Rauch比较定理，以及一个与Erös–Gieger最近建立的洛伦兹Cartan–Hadamard定理密切相关的结果。 在附录中，我们详细处理了在重新参数化下非停止曲线空间上的Fréchet距离，这是一个在整个论文中使用的技术工具。 显示英文摘要

摘要： We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Er\"{o}s--Gieger. In the appendix, we give a detailed treatment of the Fr\'{e}chet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper.