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\(\text{Spin}^c\) manifolds and complex contact structures. (English) Zbl 0908.53024
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.

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C27 Spin and Spin\({}^c\) geometry
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