论文标题
杀死Riemannian和Lorentzian 3-manifolds上的矢量场
Killing vector fields on Riemannian and Lorentzian 3-manifolds
论文作者
论文摘要
我们提供了所有Riemannian 3-manifolds $（m，g）的完整本地分类，并承认不存在杀人杀伤矢量场$ t $。然后，我们将此分类扩展到Lorentzian 3 manifolds上的及时杀死矢量场，这些载体实际上是非危险的。我们分类中所需的两个关键成分是$ g $的标量曲率$ s $和功能$ \ text {ric}（t，t，t）$，其中$ \ text {ric} $是ricci张量；实际上，它们的总和似乎是从$ t $的动作获得的商标的高斯曲率。我们的分类概括了Sasakian结构，这是$ \ text {ric}（t，t，t）= 2 $的特殊情况。我们还提供了必要的，单独的，足够的条件，均以$ \ text {ric}（t，t）$表示，$ g $在本地固定。然后，我们从本地设置转移到全局设置，并证明了两个结果：如果$ t $具有单位长度，并且在我们的分类中得出的坐标是在$ \ mathbb {r}^3 $上全球定义的，我们给出了$ s $完全确定指标何时完成的条件。如果3-manifold $ m $紧凑，我们会提供一个条件，说明当它承认恒定的正截面曲率指标时。
We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field $T$. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature $S$ of $g$ and the function $\text{Ric}(T,T)$, where $\text{Ric}$ is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of $T$. Our classification generalizes that of Sasakian structures, which is the special case when $\text{Ric}(T,T) = 2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of $\text{Ric}(T,T)$, for $g$ to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that $T$ has unit length and the coordinates derived in our classification are globally defined on $\mathbb{R}^3$, we give conditions under which $S$ completely determines when the metric will be geodesically complete. In the event that the 3-manifold $M$ is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.
