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 [22], 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 an appropriate Cauchy surface \(\Sigma \hookrightarrow M\). The existence of a parallel spinor on (M, g) induces a system of constraint partial differential equations on \(\Sigma \), which we prove to be equivalent to an exterior differential system involving a cohomological condition on the shape operator of the embedding \(\Sigma \hookrightarrow M\). 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 \(({\mathfrak {e}},\Theta )\), where \({\mathfrak {e}}\) is a coframe and \(\Theta \) is a symmetric two-tensor. We characterize all parallel Cauchy pairs on simply connected Cauchy surfaces, refining a result of 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.

我们研究了允许平行实旋量\(\varepsilon \)的全局双曲四流形 ( M ,  g )的微分几何和拓扑结构。使用最近在 [22] 中引入的抛物线对理论，我们首先将M上的\(\varepsilon \)的并行性条件公式化为偏微分方程组，平行旋量流方程，对于适当的多形族柯西表面\(\Sigma \hookrightarrow M\)。( M ,  g )上平行旋量的存在导致了约束偏微分方程组在\(\Sigma \)，我们证明它相当于一个外部微分系统，涉及嵌入\(\Sigma \hookrightarrow M\)的形状算子上的上同调条件。这个微分系统的解正是平行旋量演化问题所允许的初始数据，并定义了平行柯西对 \(({\mathfrak {e}},\Theta )\) 的概念，其中\({\mathfrak {e}}\)是一个共框和\(\Theta \)是一个对称的二张量。我们在简单连接的柯西表面上表征所有平行柯西对，改进 Leistner 和 Lischewski 的结果。此外，我们对所有允许平行柯西对的紧凑三流形进行分类，证明它们规范地配备了\({\mathbb {R}}^2\)的局部自由动作，并且与\(S ^1\)，我们详细描述了其黎曼结构。此外，我们在简单连接的李群上对所有左不变平行柯西对进行分类，指定何时允许使用 Ricci 平面方程的初始数据以及何时形状算子为 Codazzi。最后，我们给出了一类平行旋量流的新几何解释，并在几个例子中解决它，获得携带平行旋量的四维洛伦兹流形的明确族。