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.
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