Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let (Mn,g,f) be a compact gradient shrinking Ricci soliton satisfying Ric g+Ddf=I g with I >0 constant. We show that if (M,g) satisfies I´W(·,·,∇f)=0, then (M,g) is Einstein. Here 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 and . Finally, we prove that if (Mn,g,f) is a complete noncompact gradient steady Ricci soliton satisfying I´W(·,·,∇f)=0, and if the scalar curvature attains its maximum at some point in the interior of M, then either (M,g) is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in .