Curvature of quaternionic Kahler manifolds with S1-symmetry

We study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kahler side in terms of the initial hyper-Kahler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kahler manifolds arising from flat hyper-Kahler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.