Curvature of quaternionic K\"ahler manifolds with $S^1$-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 K\"ahler side in terms of the initial hyper-K\"ahler 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 K\"ahler manifolds arising from flat hyper-K\"ahler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.