Nicolas Ginoux, Georges Habib, Ines Kath

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 \(\nabla _X\psi =AX\cdot \psi \) 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 \({\mathbb {R}}\times N\) with *N* having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to \({\mathbb {S}}^2\times {\mathbb {R}}^2\) occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.