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Ridigity of Ricci solitons with weakly harmonic Weyl tensors
Author(s) -
Hwang Seungsu,
Yun Gabjin
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600285
Subject(s) - mathematics , scalar curvature , weyl tensor , ricci curvature , ricci decomposition , mathematical analysis , ricci flow , mathematical physics , rigidity (electromagnetism) , einstein , manifold (fluid mechanics) , einstein manifold , euclidean space , pure mathematics , riemann curvature tensor , curvature , geometry , physics , quantum mechanics , mechanical engineering , engineering
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].