The Isotropic Weyl Tensor on Four-Dimensional Locally Homogeneous Pseudo-Riemannian Manifolds

Abstract

 The locally homogeneous (pseudo)Riemannian manifolds are studied by many mathematicians. The more general cases of these manifolds are spaces where conformal transformations act transitively. They are also called the locally conformally homogeneous (pseudo)Riemannian manifolds. It is worth noting that such manifolds were investigated both for the case of Riemannian metric, and for the case of pseudo-Riemannian metric. The work of E.D. Ro dionov, V.V. Slavskii, and L.N. Chibrikova claims that if there is a Weyl tensor with the non-zero squared length for a manifold of dimension n ≥ 4, then a locally homogeneous space can be obtained from a locally conformally homogeneous (pseudo)Riemannian space by means of a conformal deformation. Hence, there should be a naturally arisen problem to investigate such (pseudo)Riemannian locally conformally homogeneous and locally homogeneous manifolds for which the squared length of the Weyl tensor equals to zero, but the tensor itself is not equal to zero (such a Weyl tensor is also called isotropic). In this paper, we describe a step-by-step algorithm to solve the problem of classification of four-dimensional locally homogeneous pseudo-Riemannian manifolds with an isotropic Weyl tensor and a non-trivial isotropy subgroup.