Four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature

Abstract

AbstractIn 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 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 is locally a warped product with 2‐dimensional base in explicit form, and if g is complete in addition, the underlying smooth manifold is or . Here is a smooth surface admitting a complete Riemannian metric with constant curvature k. If has less than three distinct Ricci‐eigenvalues at each point, it is either locally conformally flat or locally isometric to the Riemannian product , , where has the Euclidean metric and 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.