In this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ , provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

中文翻译：

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具有正曲率的四歧管

在这篇笔记中，我们证明了一个四维紧致半共形平面黎曼流形米4是拓扑的$\mathbb{S}^{4}$要么$\mathbb{C}\mathbb{P}^{2}$, 假设截面曲率都在区间内$\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$此外，我们使用双正交（截面）曲率的概念来获得收缩条件，该条件保证四维紧凑流形同胚于复投影平面或 4 球面的副本的连通和。