Four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature

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.