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 ${\rm Ric}_g + Ddf = \rho g$ with $\rho >0$ constant. We show that if $(M,g)$ satisfies $\delta \mathcal W (\cdot, \cdot, \nabla f) = 0$, then $(M, g)$ is Einstein. Here $\mathcal W$ denotes the Weyl curvature tensor. In the case of noncompact, 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}.