Locally Homogeneous Pseudo-Riemannian 4-Manifolds with Isotropic Weyl Tensor

Abstract

Papers of many mathematicians are devoted to studying of conformally flat (i.e., with the trivial Weyl tensor) (pseudo)Riemannian manifolds. Moreover, one can consider manifolds with Weyl tensors having zero squared length while itself being non zero. Also, such manifolds are called manifolds with isotropic Weyl tensors.In the case of Riemannian metric, the squared length of the tensor in some orthonormal basis is the sum of the squares of all components. The squared length equals to zero if the tensor itself is trivial. Therefore, it is natural to consider only the pseudo-Riemannian metric case. For the dimension of 3, the Weyl tensor is trivial, and the Schouten-Weyl tensor (also known as the Cotton tensor) is identical with the Weyl tensor. In the paper of Rodionov E.D., Slavskii V.V., Chibrikova L.N., the Schouten-Weyl tensor was investigated for a left-invariant Lorentzian metric on three-dimensional Lie groups, including the problem of its isotropy.In this paper, results of the study of fourdimensional locally homogeneous spaces with nontrivial isotropy subgroup and with invariant pseudo-Riemannian metric and an isotropic Weyl tensor are presented.DOI 10.14258/izvasu(2018)1-17