We prove that an $n$-dimensional $(n \geq 4)$ gradient shrinking Ricci soliton with fourth-order divergence free Riemannian curvature tensor (i.e. $\mathit{\operatorname{div}}^4 Rm = 0)$ is rigid. In particular, such a soliton in dimension $4$ is either Einstein, or a finite quotient of $\mathbb{R}^4$, $\mathbb{R}^2 \times \mathbb{S}^2$, or $\mathbb{R} \times \mathbb{S}^3$. Under the condition of $\mathit{\operatorname{div}}^3 W (\nabla f) = 0$, we have the same results.

中文翻译：

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梯度收缩Ricci孤子的刚度

我们证明，具有四阶无散度黎曼曲率张量（即$ \ mathit {\ operatorname {div}} ^ 4 Rm = 0）$的，$ n $维的$（n \ geq 4）$梯度收缩Ricci孤子是死板的。特别是，这样一个维数为$ 4 $的孤子可以是爱因斯坦，也可以是$ \ mathbb {R} ^ 4 $，$ \ mathbb {R} ^ 2 \ times \ mathbb {S} ^ 2 $或$ \ mathbb {R} \ times \ mathbb {S} ^ 3 $。在$ \ mathit {\ operatorname {div}} ^ 3 W（\ nabla f）= 0 $的条件下，我们得到相同的结果。