Résumé
The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group4dimensionalG₀as a Lorentzian and flat affine manifold. As the groupG₀is naturally equipped with a bi-invariant Hessian metrick⁺ , relative to a bi-invariant flat affine structure∇ , we examine both structures and the relationships between them. Both structures are defined using the Lie algebra𝔤 , the first one through the tracek(u,v):=\mathrm{trace}{(}{u}∘ v)and the second by the composition∇_(u⁺)v⁺:=(u∘ v)⁺ , whereu,v∈𝔤 . The curvatures, tidal force, and Jacobi vector fields of(G₀, k⁺)are determined in Section 1. Section 2 discusses the causal structure ofk⁺ , while Section 3 focuses on the developed map relative to∇in the sense of C. Ehresmann.