It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold ( M , g ) of positive sectional curvature admits a harmonic 2-form of constant length, then M has definite intersection form and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to $$\mathbb {CP}_{2}$$ CP 2 with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on $$\mathbb {CP}_{2}$$ CP 2 .

中文翻译：

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具有等长调和 2-形式的四流形

Seaman 证明，如果一个紧凑的、连通的、有向的、正截面曲率的黎曼 4-流形 ( M , g ) 允许一个等长的调和 2-形式，那么 M 有确定的交形式，并且这种调和形式是唯一的直至常数倍。在本文中，我们证明了这种流形与 $$\mathbb {CP}_{2}$$ CP 2 微分，曲率假设稍弱，并且在 Fubini-Study 附近存在此类度量的无限维模空间$$\mathbb {CP}_{2}$$ CP 2 上的度量。