Ridigity of Ricci solitons with weakly harmonic Weyl tensors. Issue 5 (23rd January 2018)

• Record Type: Journal Article
• Authors: Hwang, Seungsu; Yun, Gabjin
• Is Part Of: Mathematische Nachrichten. Volume 291:Issue 5/6(2018)
• Page Start: 897
• Page End: 907
• Publication Date: 2018-01-23
• Subjects: Einstein metric -- gradient Ricci soliton -- harmonic Weyl curvature tensor -- scalar curvature -- weakly harmonic Weyl curvature tensor -- 53C25; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201600285
• Languages: English
• ISSNs: 0025-584X

Abstract

ABSTRACT: 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 satisfying Ric 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 in10 14 and19 . 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 in3 .