Four-dimensional gradient almost Ricci solitons with harmonic Weyl curvature

초록

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,lambda) 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 R2xMk2 or R2-{(0,0)}xMk2. 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 R2xN lambda 2, lambda not equal 0, where R2 has the Euclidean metric and N lambda 2 is a 2-dimensional Riemannian manifold with constant curvature lambda. We also make a complete description of four-dimensional gradient almost Ricci solitons with harmonic curvature.

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