On a compact n -dimensional manifold M , it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, we prove the Besse conjecture for compact manifolds with pinched Weyl curvature. Moreover, we shall conclude that such a conjecture is true if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

中文翻译：

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具有收缩曲率的紧凑流形的贝塞猜想

在紧凑的 n 维流形 M 上，已经推测总标量曲率的临界点，限制在单位体积标量曲率不变的度量空间中，是爱因斯坦。在本文中，我们证明了具有收缩 Weyl 曲率的紧凑流形的 Besse 猜想。此外，我们将得出结论，如果其 Weyl 曲率张量和 Ricci 曲率的 Kulkarni-Nomizu 积是正交的，则该猜想为真。