Sp(n)U(1)-connections with parallel totally skew-symmetric torsion

TL;DR

This paper classifies Hermitian manifolds with a special connection having parallel skew-symmetric torsion and holonomy Sp(n)U(1), showing they are locally or globally isomorphic to twistor spaces of positive quaternionic Kähler manifolds.

Contribution

It establishes a classification result linking Hermitian manifolds with specific torsion and holonomy to twistor spaces of quaternionic Kähler manifolds.

Findings

Manifolds with parallel skew-symmetric torsion and Sp(n)U(1) holonomy are locally twistor spaces.

Complete manifolds with these properties are globally twistor spaces.

The work characterizes the geometric structure of such Hermitian manifolds.

Abstract

We consider the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. We prove that if the torsion is parallel and the holonomy is Sp(n)U(1), considered as a subgroup of U(2n) x U(1), then the manifold is locally isomorphic to the twistor space of a quaternionic Kaehler manifold with positive scalar curvature. If the manifold is complete, then it is globally isomorphic to such a twistor space.