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We classify complete gradient Ricci solitons satisfying a fourth-ordervanishing condition on the Weyl tensor, improving previously known results.More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinkingRicci soliton with fourth order divergence-free Weyl tensor is either Einstein,or a finite quotient of $N^{n-k}\times \mathbb{R}^k$, $(k > 0)$, the product ofa Einstein manifold $N^{n-k}$ with the Gaussian shrinking soliton$\mathbb{R}^k$. The technique applies also to the steady and expanding cases inall dimensions. In particular, we prove that a three dimensional gradientsteady soliton with third order divergence-free Cotton tensor, i.e. withvanishing double divergence of the Bach tensor, is either flat or isometric tothe Bryant soliton.

Gradient Ricci solitons with vanishing conditions on Weyl