Parallel spinors on globally hyperbolic Lorentzian four-manifolds

We investigate the differential geometry and topology of globally hyperbolic four-manifolds (M, g) admitting a parallel real spinor $$\varepsilon $$ . Using theory parabolic pairs recently introduced in [22], we first formulate parallelicity condition on M as system partial equations, flow for family polyforms an appropriate Cauchy surface $$\Sigma \hookrightarrow M$$ The existence induces constraint equations , which prove to be equivalent exterior involving cohomological shape operator embedding Solutions this are precisely allowed initial data evolution problem define notion pair $$({\mathfrak {e}},\Theta )$$ where $${\mathfrak {e}}$$ is coframe $$\Theta symmetric two-tensor. characterize all simply connected surfaces, refining result Leistner Lischewski. Furthermore, classify compact three-manifolds pairs, proving that they canonically equipped with locally free action $${\mathbb {R}}^2$$ isomorphic certain torus bundles over $$S^1$$ whose Riemannian structure detail. Moreover, left-invariant Lie groups, specifying when Ricci flat Codazzi. Finally, give novel geometric interpretation class flows solve it several examples, obtaining explicit families four-dimensional Lorentzian manifolds carrying spinors.