On positive quaternionic Kähler manifolds with certain symmetry rank. (English) Zbl 1172.53030
Summary: Let \(M\) be a positive quaternionic Kähler manifold of real dimension \(4m\). In this paper we show that if the symmetry rank of \(M\) is greater than or equal to \([m/2] + 3\), then \(M\) is isometric to \(\mathbb HP^m\) or \(\text{Gr}_2(\mathbb C^{m+2})\). This is sharp and optimal, and will complete the classification result of positive quaternionic Kähler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kähler manifolds with a group action and the induction arguments on the dimension of the manifold.
MSC:
| 53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |
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