  This paper is devoted to the classification of 4-dimensional Riemannian spin
manifolds carrying skew Killing spinors. A skew Killing spinor $\psi$ is a
spinor that satisfies the equation $\nabla$X$\psi$ = AX $\times$ $\psi$ with a
skew-symmetric endomorphism A. We consider the degenerate case, where the rank
of A is at most two everywhere and the non-degenerate case, where the rank of A
is four everywhere. We prove that in the degenerate case the manifold is
locally isometric to the Riemannian product R x N with N having a skew Killing
spinor and we explain under which conditions on the spinor the special case of
a local isometry to S 2 x R 2 occurs. In the non-degenerate case, the existence
of skew Killing spinors is related to doubly warped products whose defining
data we will describe.
