On four-dimensional locally homogeneous pseudo-Riemannian manifolds with isotropic Weyl tensor

Abstract

The papers of many mathematicians are devoted to studying of (pseudo)Riemannian manifolds with zero Weyl tensor (i.e. conformally flat manifolds). Moreover, one can consider manifolds whose Weyl tensor has zero squared length, and itself is~not~zero. Also such manifolds are called manifolds with isotropic Weyl tensor. In the case of a Riemannian metric, the squared length of a tensor is the sum of the squares of all components insome orthonormal basis, and it is zero iff the tensor itself is trivial. Therefore, it is natural to consider only the case of a pseudo-Riemannian metric. In the case of dimension 3, the Weyl tensor is trivial, and the Schouten-Weyl tensor (also known as the Cotton tensor) is the analogue of the Weyl tensor. The Schouten-Weyl tensor was investigated for a left-invariant Lorentzian metric on three-dimensional Lie groups, including the problem of its iso\-tropy, in the work of E.D. Rodionov, V.V. Slavskii, L.N. Chibrikova. In this paper results on the investigation of four-dimensional locally homogeneous spaces with nontrivial isotropy subgroup and with invariant pseudo-Riemannian metric and an isotropic Weyl tensor are presented.