Investigation of four-dimensional locally homogeneous (Pseudo)Riemannian Manifolds with an isotropic Schouten - Weyl Tensor
Locally homogeneous (pseudo)Riemannian manifolds were studied by many mathematicians. Their generalization is a locally conformally homogeneous (pseudo)Riemannian manifolds on which a conformal transformations act transitively. Such manifolds were previously studied both in the Riemannian case and in the pseudo-Riemannian case. In the paper of E.D. Rodionov, V.V. Slavsky and L.N. Chibrikova it was proved that a locally homogeneous manifold could be obtained from a locally conformally homogeneous (pseudo)Rieman-nian manifolds by a conformal deformation if the Weyl tensor (or the Schouten - Weyl tensor in the three-dimensional case) has a nonzero squared length. Thus, the problem arises of studying (pseudo)Riemannian locally homogeneous and locally conformally homogeneous manifolds, the Schouten - Weyl tensor of which has zero squared length, and itself is not equal to zero. In this paper, we present an algorithm that can solve the classification problem of four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a nontrivial isotropy subgroup and an isotropic Schouten - Weyl tensor.