Rigidity of nonnegatively curved compact quaternionic-Kähler manifolds (Q1118180)

The main purpose of this paper is to prove the following theorem. Let M be a compact quaternionic-Kähler manifold with positive (nonnegative) quaternionic bisectional curvature. Then M is \(HP^ n\) with the standard metric (a quaternionic symmetric space). This generalizes a result of M. Berger, who considered a manifold M with positive sectional curvature. Since Goldberg-Kobayashi and Mok-Zhong stated that a compact Kähler- Einstein manifold with positive (nonnegative) bisectional curvature is isometric to \(CP^ n\) this theorem is analogous to the mentioned results. To prove this theorem the authors study the twistor space \({\mathfrak Z}\), which is the space of all almost complex structures on M compatible with the quaternionic structure.