About the Schouten-Weyl tensor on 3-dimensional Lorenzian Lie groups

The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric Lie groups with almost harmonic (i.e. with zero curl and divergence) Schouten-Weyl tensor are investigated. In addition, the question about the harmonicity of contraction of the Schouten-Weyl tensor is considered.