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.

本文专门研究带有偏斜Killing旋子的4维黎曼自旋流形的分类。偏斜杀伤旋量\（\ PSI \）是一个旋量满足方程式\（\ nabla _X \ PSI = AX \ CDOT \ PSI \）具有斜对称自同态甲。我们考虑退化的情况，其中A的等级在任何地方最多为2，而非退化的情况下，A的等级在任何地方都为4。我们证明在退化情况下，流形对于黎曼乘积\ {{\ mathbb {R}} \乘以N \}是局部等距的，其中N具有Killing spinor斜率，并且我们解释了在Spinor上的哪种条件下的特殊情况局部等距\（{{mathbb {S}} ^ 2 \ times {\ mathbb {R}} ^ 2 \）发生。在非简并的情况下，歪斜Killing旋转子的存在与我们将描述其定义数据的双翘产品有关。