Parallel and Killing spinors on \(\text{Spin}^ c\) manifolds (Q1367000)

The relation between the equations for parallel and real Killing spinors for \(\text{Spin}\) structures on the one hand and for \(\text{Spin}^c\) structures on the other hand is investigated. The author proves that a simply connected \(\text{Spin}^c\) manifold carrying a parallel spinor is isometric to the Riemannian product of a Kähler manifold and a spin manifold carrying a parallel spinor. The second main result of the paper is that a \(\text{Spin}^c\) manifold with a real Killing spinor is a Sasakian manifold unless it is a spin manifold with a real Killing spinor.