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

Abstract

A non-trivial spinor field $\ddot{I}ˆ$ is called a generalized imaginary ${\mathrm{Spin}}^c$-Killing spinor if ${\nabla }_X^{g,A}\ddot{I}ˆ=i\hat{I}¼X\cdot \ddot{I}ˆ$ for all vector fields $X$, where $\hat{I}¼$ 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 $\ddot{I}ˆ$ is a vector field $V$, the Dirac current, defined by $g(V,X)=i⟨X\cdot \ddot{I}ˆ,\ddot{I}ˆ⟩$. We prove that if $V$ vanishes somewhere and $\dim M\ge 3$, the manifold is locally isometric to real hyperbolic space. When $V$ never vanishes and $\dim M\ge 3$, 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 $\hat{I}¾=\frac{V}{|V|}$ is complete or the leaves of $\mathcal{D}=\ker (\hat{I}{¾}^{♭})$ 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.