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 is R2×Mk2 or R2−{(0,0)}×Mk2. Here Mk2 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 product R2×Nλ2, λ≠0, where R2 has the Euclidean metric and Nλ2 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.