In this paper we study on gradient quasi-Einstein solitons with a fourth-order vanishing condition on the Weyl tensor. More precisely, we show that for n ≥ 4, the Cotton tensor of any n -dimensional gradient quasi-Einstein soliton with fourth order f -divergence free Weyl tensor is flat, if the manifold is compact, or noncompact but the potential function satisfies some growth condition. As corollaries, some local characterization results for the quasi-Einstein metrics are derived.

中文翻译：

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梯度准爱因斯坦孤子上棉花张量的消失结果

在本文中，我们研究了外尔张量上具有四阶消失条件的梯度拟爱因斯坦孤子。更准确地说，我们证明，对于 n ≥ 4，任何具有四阶 f 散度自由外尔张量的 n 维梯度拟爱因斯坦孤子的 Cotton 张量是平坦的，如果流形是紧凑的，或非紧凑但势函数满足一些生长条件。作为推论，导出了准爱因斯坦度量的一些局部特征结果。