论文标题
洛伦兹太空形式的里曼尼亚人
Riemannian counterparts to Lorentzian space forms
论文作者
论文摘要
在带有$ n \ geq 3 $的平滑$ n $ -manifold $ m $中，我们研究对$（g，t）$，由Riemannian Metric $ g $和单位长度闭合矢量字段$ t $组成。由RICCI孤子如何通过杰出的向量字段概括爱因斯坦指标的动机，我们建议通过考虑那些对相应的lorentzian指标$ g _ {\ scriptScriptScriptScriptstyle l} = G -2T^{\ 2T^{\ flat flate flat cuntantim tlanant tlanant flats tlanant tlant cuntance来概括$（g，t）$来概括空间形式。我们通过示例表明，当$ m $不划定时，存在这种对，并且其中存在完整的指标。但是，当$ m $紧凑时，情况更加严格。在紧凑的设置中，我们证明唯一的对$（g，t）$其相应的Lorentzian Metric $ g _ {\ scriptscriptScriptScriptScriptScriptScriptScriptScriptScriptScriptStyleL} $是一个空间形式，是$（m，g）$是平坦的，其通用覆盖范围是iSometrys insometrys insomortical insomelys insomelys grops as sproude $ \ nathbb {r} r} r} \ times n $ n $ n $ n $。紧凑的洛伦兹球形空间形式的不存在在我们的证明中起着关键作用。
On a smooth $n$-manifold $M$ with $n \geq 3$, we study pairs $(g,T)$ consisting of a Riemannian metric $g$ and a unit length closed vector field $T$. Motivated by how Ricci solitons generalize Einstein metrics via a distinguished vector field, we propose to generalize space forms by considering those pairs $(g,T)$ whose corresponding Lorentzian metric $g_{\scriptscriptstyle L} = g - 2T^{\flat} \otimes T^{\flat}$ has constant curvature. We show by examples that such pairs exist when $M$ is noncompact, and that complete metrics exist among them. When $M$ is compact, however, the situation is more rigid. In the compact setting, we prove that the only pairs $(g,T)$ whose corresponding Lorentzian metric $g_{\scriptscriptstyle L}$ is a space form are those where $(M,g)$ is flat and its universal covering splits isometrically as a product $\mathbb{R} \times N$. The nonexistence of compact Lorentzian spherical space forms plays a key role in our proof.
