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 Kähler side in terms of the initial hyper-Kähler 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 Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

中文翻译：

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具有$$ S ^ 1 $$ S 1-对称性的四元离子Kähler流形的曲率

我们研究了HK / QK对应关系下的连接和曲率行为，并证明了用原始的超Kähler数据表达四元离子Kähler侧的Levi-Civita连接和Riemann曲率张量的简单公式。由于Alekseevsky，我们的曲率公式完善了众所周知的分解定理。作为应用，我们计算了由平面超Kähler流形产生的一系列完整四元离子Kähler流形的曲率张量范数。我们用它来推论这些流形是同质的。