Parallel spinors on globally hyperbolic Lorentzian four-manifolds

Abstract

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 $\Sigma \hookrightarrow M$. 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 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.