Left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups with zero-divergence Weyl tensor (Q610570)

A Riemannian manifold \((M,g)\) of dimension \(n\geq 4\) is called a \(C\)-space if div\(\,W=0\) where \(W\) denotes the Weyl tensor. Due to the homogeneity property, the study of the divergence of the Weyl tensor on Lie groups reduces to a study at the Lie algebra level. The aim of this work is to determine the four-dimensional nonunimodular Lie groups which are \(C\)-spaces. The key is to find a convenient basis, an idea based on previous works. The resulting Lie algebras are presented in a table. Even though several references are given, the corresponding proofs are not written out explicitly, which makes the understanding of the used tools very difficult.