In this paper, we prove an n-dimensional radially flat gradient shrinking Ricci solitons with \(div^2W(\nabla f,\nabla f)=0\) is rigid. Moreover, we show that a four-dimensional radially flat gradient shrinking Ricci soliton with \(\text {div}^2W^\pm (\nabla f,\nabla f)=0\) is either Einstein or a finite quotient of \({\mathbb {R}}^4\), \({\mathbb {S}}^2\times {\mathbb {R}}^2\) or \({\mathbb {S}}^3\times {\mathbb {R}}\).

中文翻译：

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径向条件下梯度收缩Ricci孤子

在本文中，我们证明了具有\（div ^ 2W（\ nabla f，\ nabla f）= 0 \）的n维径向平坦梯度收缩Ricci孤子是刚性的。此外，我们证明具有\（\ text {div} ^ 2W ^ \ pm（\ nabla f，\ nabla f）= 0 \）的四维径向扁平梯度收缩Ricci孤子是爱因斯坦或\的有限商（{\ mathbb {R}} ^ 4 \），\（{\ mathbb {S}} ^ 2 \ times {\ mathbb {R}} ^ 2 \）或\（{\ mathbb {S}} ^ 3 \次{\ mathbb {R}} \）。