On generalized imaginary $\mathrm{Spin}^c$-Killing spinors

A non-trivial spinor field $\psi$ is called a generalized imaginary $\mathrm{Spin}^c$-Killing spinor if $\nabla^{g,A} _X \psi = i\mu X \cdot \psi$ for all vector fields $X$, where $\mu$ is a real function that is not identically zero and $\nabla^{g,A}$ is the $\mathrm{Spin}^c$ Levi-Civita connection with $\mathrm{U}(1)$-connection $A$. Associated with $\psi$ is a vector field $V$, the Dirac current, defined by $g(V,X) = i \langle X\cdot \psi, \psi \rangle$. We prove that if $V$ vanishes somewhere and $\operatorname{dim} M \geq 3$, the manifold is locally isometric to real hyperbolic space. When $V$ never vanishes and $\operatorname{dim} M \geq 3$, we obtain a global geometric description of all $\mathrm{Spin}^c$-Riemannian manifolds carrying such spinors, under the assumption that either the normalized Dirac current $\xi = \frac{V}{|V|}$ is complete or the leaves of $\mathcal{D} = \ker(\xi^\flat)$ are complete. Finally, we reinterpret the case of type~I generalized imaginary $\mathrm{Spin}^c$-Killing spinors in terms of parallel spinors for a suitable connection with vectorial torsion.