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\(Sp(n)U(1)\)-connections with parallel totally skew-symmetric torsion. (English) Zbl 1107.53012

Author’s abstract: The author considers the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. He proves that if the torsion is parallel and the holonomy is \(\text{Sp}(n)U(1)\subset U(2n)\times U(1)\). Then the manifold is locally isomorphic to the twistor space of a quaternionic Kähler manifold with positive scalar curvature. Furthermore if the manifold is complete, then it is globally isomorphic to such a twistor space.

MSC:

53B05 Linear and affine connections
53B35 Local differential geometry of Hermitian and Kählerian structures
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C28 Twistor methods in differential geometry
53C05 Connections (general theory)
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References:

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