论文标题
在全球双曲线洛伦兹四个manifolds上的平行纺纱器
Parallel spinors on globally hyperbolic Lorentzian four-manifolds
论文作者
论文摘要
我们研究了全球双曲线四个manifolds $（m，g）的差异几何形状和拓扑结构，并承认平行的真实旋转器$ \ varepsilon $。使用最近在Arxiv：1911.08658中介绍的抛物线对理论，我们首先为$ m $上的$ \ varepsilon $在$ m $上的平行条件制定为部分微分方程的系统，平行旋转旋转方程，用于任何给定的Cauchy表面$σ\ unghookrightArrow m $ $ $。 $（m，g）$上的平行旋转器的存在可引起$σ$的约束部分微分方程系统，我们被证明等同于外部微分系统，该系统涉及嵌入$σ\ jkingrightarrow m $的嵌入形状运算符的共同体条件。该差分系统的解决方案正是平行旋转器的演化问题的允许的初始数据，并定义了平行的Cauchy Pair Pair $（\ Mathfrak {e}，θ）$的概念，其中$ \ Mathfrak {e} $是coframe，$θ$是对称的两次t镜头。我们表征了所有平行的Cauchy对在简单地连接的Cauchy表面上，并完善了Baum，Leistner和Lischewski的结果。此外，我们对所有紧凑型三个manifolds进行了分类，这些三个manifolds承认并行cauchy对，证明它们在规范上配备了$ \ mathbb {r}^2 $的本地免费动作，并且对$ s^1 $ yemannian结构的某些torus bundles是同构的，我们的riemannian结构是我们的详细表征。此外，我们将所有剩余的cauchy对在简单连接的谎言组上进行了分类，并指定何时允许ricci平面方程的初始数据以及何时形状运算符为codazzi。最后，我们给出了一类平行旋转旋转流并在几个示例中解决的新型几何解释，并获得了带有平行纺纱物的四维洛伦兹歧管的明确家族。
We investigate the differential geometry and topology of globally hyperbolic four-manifolds $(M,g)$ admitting a parallel real spinor $\varepsilon$. Using the theory of parabolic pairs recently introduced in arXiv:1911.08658 , we first formulate the parallelicity condition of $\varepsilon$ on $M$ as a system of partial differential equations, the parallel spinor flow equations, for a family of polyforms on any given Cauchy surface $Σ\hookrightarrow M$. Existence of a parallel spinor on $(M,g)$ induces a system of constraint partial differential equations on $Σ$, which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding $Σ\hookrightarrow M$. Solutions of this differential system are precisely the allowed initial data for the evolution problem of a parallel spinor and define the notion of parallel Cauchy pair $(\mathfrak{e},Θ)$, where $\mathfrak{e}$ is a coframe and $Θ$ is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of Baum, Leistner and Lischewski. Furthermore, we classify all compact three-manifolds admitting parallel Cauchy pairs, proving that they are canonically equipped with a locally free action of $\mathbb{R}^2$ and are isomorphic to certain torus bundles over $S^1$, whose Riemannian structure we characterize in detail. Moreover, we classify all left-invariant parallel Cauchy pairs on simply connected Lie groups, specifying when they are allowed initial data for the Ricci flat equations and when the shape operator is Codazzi. Finally, we give a novel geometric interpretation of a class of parallel spinor flows and solve it in several examples, obtaining explicit families of four-dimensional Lorentzian manifolds carrying parallel spinors.
