Introduction to Lorentzian and Flat Affine Geometry of GL(2,R)

The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group $4$ dimensional G₀ as a Lorentzian and flat affine manifold. As the group G₀ is naturally equipped with a bi-invariant Hessian metric k⁺, relative to a bi-invariant flat affine structure ∇, we examine both structures and the relationships between them. Both structures are defined using the Lie algebra g, the first one through the trace k(u,v):=trace(u∘ v) and the second by the composition ∇u⁺v⁺:=(u∘ v)⁺, where u,v. The curvatures, tidal force, and Jacobi vector fields of (G₀, k⁺) are determined in Section 1. Section 2 discusses the causal structure of k⁺, while Section 3 focuses on the developed map relative to ∇ in the sense of C. Ehresmann.