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arXiv · 2001.10032

Curvature of quaternionic Kähler manifolds with $S^1$-symmetry

V. Cortés·A. Saha·D. Thung
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Abstract

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.

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V. Cortés, A. Saha, D. Thung. 2021-03-31. Curvature of quaternionic Kähler manifolds with $S^1$-symmetry. https://doi.org/10.1007/s00229-021-01294-7

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