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 ( M n , g , f ) be a compact gradient shrinking Ricci soliton satisfyingRic g + Dd f = ρ g with ρ > 0 constant. We show that if ( M , g ) satisfies δ 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 [10][M. Fernández‐López, 2011], [14][O. Munteanu, 2013] and [19][P. Petersen, 2010]. Finally, we prove that if ( M n , g , f ) is a complete noncompact gradient steady Ricci soliton satisfying δ 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 [3][H.‐D. Cao, 2012].