  We investigate the differential geometry and topology of four-dimensional
Lorentzian manifolds $(M,g)$ equipped with a real Killing spinor $\varepsilon$,
where $\varepsilon$ is defined as a section of a bundle of irreducible real
Clifford modules satisfying the Killing spinor equation with non-zero real
constant. Such triples $(M,g,\varepsilon)$ are precisely the supersymmetric
configurations of minimal four-dimensional supergravity and necessarily belong
to the class Kundt of space-times, hence we refer to them as supersymmetric
Kundt configurations. We characterize a class of Lorentzian metrics on
$\mathbb{R}^2\times X$, where $X$ is a two-dimensional oriented manifold, to
which every supersymmetric Kundt configuration is locally isometric, proving
that $X$ must be an elementary hyperbolic Riemann surface when equipped with
the natural induced metric. This yields a class of space-times that vastly
generalize the Siklos class of space-times describing gravitational waves in
AdS$_4$. Furthermore, we study the Cauchy problem posed by a real Killing
spinor and we prove that the corresponding evolution problem is equivalent to a
system of differential flow equations, the real Killing spinorial flow
equations, for a family of functions and coframes on any Cauchy hypersurface
$\Sigma\subset M$. Using this formulation, we prove that the evolution flow
defined by a real Killing spinor preserves the Hamiltonian and momentum
constraints of the Einstein equation with negative curvature and is therefore
compatible with the latter. Moreover, we explicitly construct all
left-invariant evolution flows defined by a Killing spinor on a simply
connected three-dimensional Lie group, classifying along the way all solutions
to the corresponding constraint equations, some of which also satisfy the
constraint equations associated to the Einstein condition.
