\(\text{Spin}^c\) manifolds and complex contact structures (Q1270154)

The author extends the notion of a projectable spinor on a spin manifold to a manifold with \(\text{Spin}^c\) structure and shows relations of spinors on the base and on the total space of a Riemannian submersion with totally geodesic one-dimensional fibres. As an application, it is shown that a compact Kähler manifold \(M\) of positive scalar curvature and complex dimension \(4m+3\) is a twistor space of a quaternionic Kähler manifold if and only if \(M\) is Kähler-Einstein and admits a complex contact structure.