Rigidity of gradient shrinking Ricci solitons

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton R⁴, R²² or the round cylinder R³. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.