We present a short review of geometric and algebraic approach to causal cones and describe cone preserving transformations and their relationship with causal structure related to special and general theory of relativity. We describe Lie groups, especially matrix Lie groups, homogeneous and symmetric spaces and causal cones and certain implications of these concepts in special and general theory of relativity related to causal structure and topology of space-time. We compare and contrast the results on causal relations with those in the literature for general space-times and compare these relations with K-causal maps. We also describe causal orientations and their implications for space-time topology and discuss some more topologies on space-time which arise as an application of domain theory.
Causality theory for closed cone structures with applications
我们发展了流形上锥的上半连续分布的因果关系理论，在两个方向上推广了数学相对论的结果：非圆锥和非正则可微性假设。利用光滑时间函数证明了正则Lorentzian因果关系理论的大多数结果的有效性，包括因果阶梯、Fermat原理、因果公式中的显著奇异性定理、Avez-Seifert定理、稳定因果关系的刻画和全局双曲性.例如，我们给这些结构的等价性之间的稳定的因果关系，$K$ -因果关系和存在的时间函数的第一个证明。这一结果意味着允许连续增函数的闭锥结构也允许光滑的闭锥结构。我们还研究了真锥的结构，即真锥的纤维束模拟。对于他们，我们得到的大多数结果的依赖域。此外，我们证明了horismos和柯西视界产生的类光测地线，后者被定义通过的非时性属性。因果测地线和陡峭的时间函数得到了一个强大的产品技巧。本文还研究了Lorentz-Minkowski空间在非常弱的正则性条件下的性质。最后，我们引入了稳定距离和稳定时空的概念，解决了两个著名的问题：（a）可嵌入Minkowski时空的Lorentz流形的特征，它们是稳定时空;（b）证明拓扑、序和距离（用一个公式a la Connes）可以用光滑陡峭的时间函数表示。本文自成体系，实际上没有使用任何数学相对论的先进结果。
Rigidity and non-existence of spacelike submanifolds with causal mean curvature vector field in spacetimes and the Cauchy problem in General Relativity
