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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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