Ridigity of Ricci solitons with weakly harmonic Weyl tensors

Seungsu Hwang and Gabjin Yun

Mathematische Nachrichten, 2018, vol. 291, issue 5-6, 897-907

Abstract: 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=Ï g with Ï >0 constant. We show that if (M,g) satisfies δW(·,·,∇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 δW(·,·,∇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 .

Date: 2018
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