A类时空
Class A Spacetimes
摘要：
我们引入了A类时空，即紧的恶性时空$(M,g)$，使得阿贝尔覆盖$(\bar{M},\bar{g})$是全局双曲的。我们使用类似于D. Sullivan“Cycles for the dynamic study of foliated manifolds and复流形“（Invent.数学、36，225-255（1976））和D. Yu. Burago“周期度量”（表示论和动力系统（Adv. Soviet Math.），9，205-210（1992））。因此，我们能够刻画流形承认A类度量完全映射环面。进一步我们证明了A类时空的概念等价于在加洛韦的“空间闭合时空的分裂定理”（Comm Math Phys 96：423-429，1984）中引入的SCTP（空间紧致时间周期）时空的概念。证明了A类时空在洛伦兹度规集上的$C^0$ -拓扑中是开的.作为应用，我们证明了阿贝尔覆盖的时间分离的一个粗糙Lipschitz性质。这种粗糙的Lipschitz性质是洛伦兹几何中Aubry-Mather理论研究的重要部分。
英文摘要：
We introduce class A spacetimes, i.e. compact vicious spacetimes $(M,g)$ such that the Abelian cover $(\bar{M},\bar{g})$ is globally hyperbolic. We study the main properties of class A spacetimes using methods similar to the one introduced in D. Sullivan "Cycles for the dynamical study of foliated manifolds and complex manifolds" (Invent. Math.,36, 225-255 (1976)) and D. Yu. Burago "Periodic metrics" (Representation theory and dynamical systems (Adv. Soviet Math.), 9, 205-210 (1992)). As a consequence we are able to characterize manifolds admitting class A metrics completely as mapping tori. Further we show that the notion of class A spacetime is equivalent to that of SCTP (spacially compact time-periodic) spacetimes as introduced in Galloway "Splitting theorems for spatially closed spacetimes" (Comm Math Phys 96:423-429, 1984). The set of class A spacetimes is shown to be open in the $C^0$-topology on the set of Lorentzian metrics. As an application we prove a coarse Lipschitz property for the time separation of the Abelian cover. This coarse Lipschitz property is an essential part in the study of Aubry-Mather theory in Lorentzian geometry.
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