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We investigate the differential geometry and topology of four-dimensionalLorentzian manifolds $(M,g)$ equipped with a real Killing spinor $\varepsilon$,where $\varepsilon$ is defined as a section of a bundle of irreducible realClifford modules satisfying the Killing spinor equation with non-zero realconstant. Such triples $(M,g,\varepsilon)$ are precisely the supersymmetricconfigurations of minimal four-dimensional supergravity and necessarily belongto the class Kundt of space-times, hence we refer to them as supersymmetricKundt configurations. We characterize a class of Lorentzian metrics on$\mathbb{R}^2\times X$, where $X$ is a two-dimensional oriented manifold, towhich every supersymmetric Kundt configuration is locally isometric, provingthat $X$ must be an elementary hyperbolic Riemann surface when equipped withthe natural induced metric. This yields a class of space-times that vastlygeneralize the Siklos class of space-times describing gravitational waves inAdS$_4$. Furthermore, we study the Cauchy problem posed by a real Killingspinor and we prove that the corresponding evolution problem is equivalent to asystem of differential flow equations, the real Killing spinorial flowequations, for a family of functions and coframes on any Cauchy hypersurface$\Sigma\subset M$. Using this formulation, we prove that the evolution flowdefined by a real Killing spinor preserves the Hamiltonian and momentumconstraints of the Einstein equation with negative curvature and is thereforecompatible with the latter. Moreover, we explicitly construct allleft-invariant evolution flows defined by a Killing spinor on a simplyconnected three-dimensional Lie group, classifying along the way all solutionsto the corresponding constraint equations, some of which also satisfy theconstraint equations associated to the Einstein condition.

Supersymmetric Kundt four manifolds and their spinorial evolution flows