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We study the behavior of connections and curvature under the HK/QKcorrespondence, proving simple formulae expressing the Levi-Civita connectionand Riemann curvature tensor on the quaternionic K\"ahler side in terms of theinitial hyper-K\"ahler data. Our curvature formula refines a well-knowndecomposition theorem due to Alekseevsky. As an application, we compute thenorm of the curvature tensor for a series of complete quaternionic K\"ahlermanifolds arising from flat hyper-K\"ahler manifolds. We use this to deducethat these manifolds are of cohomogeneity one.

Curvature of quaternionic K\"ahler manifolds with $S^1$-symmetry