洛伦兹叶状结构中的横向稳定因果性
Transverse stable causality in Lorentzian foliations
AI总结：
该研究拓展了横向因果性框架，定义了稳定因果性的横向类似物，在简单叶状结构中证明了其等价性，一般叶状结构的完全等价性仍待研究。
AI中文摘要：
我们引入并研究一种类似于稳定因果性的洛伦兹叶状结构概念，极大拓展了近期工作中提出的横向因果性框架。在时空几何中，稳定因果性由若干等价条件刻画，例如度规扰动下无因果环的稳定性、光滑时间函数的存在性，以及通过Seifert和$K^+$关系的关系论表述。我们为叶状结构定义了这些不同表述的自然横向类似物，并在叶状结构场景中建立了它们之间的部分逻辑蕴含关系。对于一般叶状结构，能否以及在多大程度上能建立完全等价性仍是一个开放问题，部分原因在于任意叶空间最终的非豪斯多夫性质和其他拓扑复杂性。此外，我们证明，对于由特定浸没定义的重要类“简单”叶状结构，几乎所有稳定因果性的横向类似物的等价性确实成立。
英文摘要：
We introduce and investigate a notion analogous to stable causality for Lorentzian foliations, considerably extending the framework of transverse causality initiated in a recent work. It is well-known that in spacetime geometry stable causality is characterized by several equivalent conditions, such as the stability of non-existence of causal loops under metric perturbations, the existence of smooth temporal functions, and relation-theoretic formulations via the so-called Seifert and $K^+$ relations. We define natural transversal analogues of these distinct formulations and establish partial logical implications between them in the foliation setting. Whether and to what extent the full equivalences can be established remains an open question for general foliations, partly due to the eventual non-Hausdorff nature and other topological complexities of arbitrary leaf spaces. Furthermore, we demonstrate that for the important class of \textit{simple} foliations, which are defined by certain submersions, the equivalences of almost all the transverse analogues of stable causality are indeed recovered.
