Skew Killing spinors in four dimensions

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