Homogeneous four-manifolds with half-harmonic Weyl curvature tensor (Q2093088)

We say that a Riemannian manifold has half-harmonic Weyl curvature if \(\delta W^+=0\) or \(\delta W^-=0\), where \(\delta\) is the divergence operator and \(W^{+}\) (resp., \(W^-\)) is the positive (resp., negative) part of the Weyl curvature tensor. The purpose of the authors is to state that a four-dimensional homogeneous manifold with half-harmonic Weyl curvature tensor is symmetric or homothetic either to the only nonsymmetric anti-self-dual homogeneous manifold or to the three-symmetric space.