Kunzinger Michael, Sämann Clemens
Faculty of Mathematics, University of Vienna, Vienna, Austria.
Ann Glob Anal Geom (Dordr). 2018;54(3):399-447. doi: 10.1007/s10455-018-9633-1. Epub 2018 Oct 5.
We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way, we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT()-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.
我们将长度空间理论的一个类似物引入到洛伦兹几何和因果理论的背景中。度量的作用由时间分离函数取代，所有基本概念都据此来表述。通过这种方式，我们以更高的普遍性恢复了许多基本结果，同时阐明了该理论基本构建块的最低要求及其相互依赖性。这项工作的一个主要重点是基于三角形比较引入类似于亚历山德罗夫理论和CAT()空间的合成曲率界。其应用包括具有低正则性度量的洛伦兹流形、闭锥结构以及某些量子引力方法。
