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On positive quaternionic Kähler manifolds with \(b_{4} = 1\). (English) Zbl 1268.53059

A quaternionic Kähler manifold is called positive if it has positive scalar curvature. An interesting problem is to classify positive quaternionic Kähler manifolds in terms of their symmetry rank, i.e., the rank of their isometry groups. It is well known that the symmetry rank of a \(4m\)-dimensional quaternionic Kähler manifold \(M\) is less than or equal to \(m + 1\).
A classification of \(4m\)-dimensional positive quaternionic Kähler manifolds with isometry rank equal to \(m + 1\) was obtained by R. Bielawski in [Math. Ann. 314, No. 3, 505–528 (1999; Zbl 0952.53024)]. In the present paper, the authors show that if a \(4m\)-dimensional positive quaternionic Kähler manifold \(M\) with \(b_4 (M) = 1\) has symmetry rank at least \( [m/2]+2\) for \(m \geq 5\), then M is isometric to the quaternionic projective space \({\mathbb HP}^m\).
Reviewer: Anna Fino (Torino)

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry

Citations:

Zbl 0952.53024
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Full Text: Euclid

References:

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