Einstein structures on four-dimensional nutral Lie groups

Abstract

When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to explain it. For this purpose, he obtained a combination of Ricci tensor and Ricci scalar through Bianchi identity, which its covariant derivative is zero and is known as Einstein tensor. The main purpose of this paper is to classify Einstein left-invariant metrics of signature (2,2) on four-dimensional Lie groups. Recently, four-dimensional Lie groups equipped with left-invariant metric of neutral signature has been investigated extensively and a complete list of them has been presented. Now we study Einstein structures on these Lie groups. Then some geometric properties of these spaces, such as Ricci-flatness will be considered.