论文标题
通过Lorentzian几何形状来区分Riemannian指标的障碍物
Obstructions to distinguished Riemannian metrics via Lorentzian geometry
论文作者
论文摘要
我们通过考虑洛伦兹（Lorentzian）的指标，从适当的意义上考虑它们是双重的，从而解决了弯曲曲率的问题。然后对后者的障碍物给前者产生障碍物。该框架在本地和全球范围内都适用，包括紧凑的歧管，对曲率的各个方面敏感。在这里，我们以两种不同的方式应用它。首先，通过将Riemannian歧管嵌入Lorentzian One并利用Penrose的“平面波限制”，我们发现，仅在一个功能的Hessian方面，就发现了必要的局部条件，用于其中的大量Riemannian指标来包含其中的那些人，而这些指标内部的那些人在其中包含的那些局部条件，而这些指标与ricci tensor或ricci-flaT tensor或ricci-flaT the complatection consement of ricci tensor或Local-flatlat，或者是相称的。其次，通过考虑通过一种灯芯旋转的类型的Riemannian指标到恒定的曲率Lorentzian指标，我们能够排除在精确意义上偏离恒定曲率的紧凑型利曼尼亚歧管（在所有维度上）的存在。
We approach the problem of finding obstructions to curvature distinguished Riemannian metrics by considering Lorentzian metrics to which they are dual in a suitable sense. Obstructions to the latter then yield obstructions to the former. This framework applies both locally and globally, including to compact manifolds, and is sensitive to various aspects of curvature. Here we apply it in two different ways. First, by embedding a Riemannian manifold into a Lorentzian one and utilizing Penrose's "plane wave limit," we find necessary local conditions, in terms of the Hessian of just one function, for large classes of Riemannian metrics to contain within them those that have parallel Ricci tensor, or are Ricci-flat, or are locally symmetric. Second, by considering Riemannian metrics dual to constant curvature Lorentzian metrics via a type of Wick rotation, we are able to rule out the existence of a family of compact Riemannian manifolds (in all dimensions) that deviate from constant curvature in a precise sense.
