Complete gradient Ricci solitons with zero radial Weyl curvature

In this paper, we study the complete gradient Ricci solitons $(M^n, g,f)$ with zero radial Weyl curvature, which means that the interior product of $\nabla f$ with the Weyl tensor $W$ is zero, i.e., $i_{\nabla f}W=0$.

We classify completely the complete gradient Ricci solitons with zero radial Weyl curvature for the dimension $n\geq 4$.