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Four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature
Author(s) -
Kim Jongsu
Publication year - 2021
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.202000126
Subject(s) - ricci curvature , mathematics , curvature of riemannian manifolds , mathematical analysis , scalar curvature , ricci decomposition , riemann curvature tensor , riemannian manifold , curvature , manifold (fluid mechanics) , ricci flat manifold , constant curvature , pure mathematics , harmonic , mathematical physics , sectional curvature , geometry , physics , quantum mechanics , mechanical engineering , engineering
In this article we make a classification of four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature. We prove first that any four‐dimensional (not necessarily complete) gradient almost Ricci soliton ( M , g , f , λ ) with harmonic Weyl curvature has less than four distinct Ricci‐eigenvalues at each point. If it has three distinct Ricci‐eigenvalues at each point, then ( M , g ) is locally a warped product with 2‐dimensional base in explicit form, and if g is complete in addition, the underlying smooth manifold isR 2 × M k 2orR 2 − { ( 0 , 0 ) } × M k 2 . Here M k 2 is a smooth surface admitting a complete Riemannian metric with constant curvature k . If ( M , g ) has less than three distinct Ricci‐eigenvalues at each point, it is either locally conformally flat or locally isometric to the Riemannian productR 2 × N λ 2 , λ ≠ 0 , where R 2 has the Euclidean metric and N λ 2 is a 2‐dimensional Riemannian manifold with constant curvature λ. We also make a complete description of four‐dimensional gradient almost Ricci solitons with harmonic curvature.