Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $$\psi $$ ψ is a spinor that satisfies the equation $$\nabla _X\psi =AX\cdot \psi $$ ∇ X ψ = A X · ψ 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 $${\mathbb {R}}\times N$$ R × 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 $${\mathbb {S}}^2\times {\mathbb {R}}^2$$ S 2 × 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.