Four-dimensional locally homogeneous pseudo-Riemannian manifolds with a nontrivial isotropy subgroup and an isotropic Schouten–Weil tensor

The isotropic Schouten–Weyl tensor was previously studied in the case of three-dimensional Lie groups with a left-invariant Lorentzian metric. In the case of locally homogeneous pseudo-Riemannian spaces with a nontrivial isotropy subgroup, manifolds with an isotropic Weyl tensor were classified. In this paper, we obtain a classification of four-dimensional, locally homogeneous pseudo-Riemannian manifolds with an isotropic Schouten—Weyl tensor. Some results on the curvature tensors of similar manifolds are obtained.