Skew Killing spinors in four dimensions (Q2033054)

Let \((M,g)\) be a \(4\)-dimensional Riemannian manifold, assume \(M\) to be spin with fixed spin structure. Moreover assume \(M\) to carry a skew Killing spinor field \(\psi\) corresponding to a skew-symmetric endomorphism \(A\) of the tangent bundle \(TM\), that is \(\nabla_X\psi=AX\cdot\psi\) for all \(X\in TM\), where \(\nabla\) is the Levi-Civita connection of \(g\). The main purpose of this paper is to classify \(4\)-dimensional Riemannian spin manifolds with skew Killing spinors. If the rank of \(A\) is at most two everywhere (degenerate case) it is proved that \((M,g)\) is locally isometric to the Riemannian product of a line by a \(3\)-dimensional Riemannian manifold carrying a skew Killing spinor. Moreover a precise description is given based on special assumption on the spinor field or completeness of \((M,g)\), (Theorem 4.12). If the rank of \(A\) is four everywhere (non-degenerate case) \((M,g)\) is related to doubly warped products (Theorem 5.5 and Corollary 5.8).