Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n,g)$ be a compact gradient shrinking Ricci soliton satisfying ${\mathrm{R}\mathrm{i}\mathrm{c}}_g+Ddf=\rho g$ with $\rho >0$ constant. We show that if $(M,g)$ satisfies $\delta \mathcal W (\cdot, \cdot, \nabla f) = 0$,then$(M, g)$isEinstein\mathrm{.}Here$\mathcal W$denotestheWeylcurvaturetensor\mathrm{.}Inthecaseofnoncompact,if$M$ is complete and satisfies the same condition, then $M$ is rigid in the sense that $M$ is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in \cite{l-r}, \cite{m-s} and \cite{p-w3}.