Semi-Riemannian $$\text {spin}^c$$ manifolds carrying generalized Killing spinors and the classification of Riemannian $$\text {spin}^c$$ manifolds admitting a type I imaginary generalized Killing spinor

Abstract We classify Riemannian $$\text {spin}^c$$ spin c manifolds carrying a type I imaginary generalized Killing spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian $$\text {spin}^c$$ spin c manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian $$\text {spin}^c$$ spin c manifolds. We carry out much of the work in the setting of semi-Riemannian $$\text {spin}^c$$ spin c -manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study semi-Riemannian manifolds carrying closed and conformal vector fields.