Introduction to Lorentzian and Flat Affine Geometry of $\mathsf{GL}(2,\mathbb{R})$

Abstract

The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group $4$ dimensional $G_0$ as a Lorentzian and flat affine manifold. As the group $G_0$ is naturally equipped with a bi-invariant Hessian metric $k^+$, relative to a bi-invariant flat affine structure $\nabla$, we examine both structures and the relationships between them. Both structures are defined using the Lie algebra $\mathfrak{g}$, the first one through the trace $k(u,v):=\mathrm{trace}(u\circ v)$ and the second by the composition $\nabla_{u^+}v^+:=(u\circ v)^+$, where $u,v\in\mathfrak{g}$. The curvatures, tidal force, and Jacobi vector fields of $(G_0, k^+)$ are determined in Section 1. Section 2 discusses the causal structure of $k^+$, while Section 3 focuses on the developed map relative to $\nabla$ in the sense of C. Ehresmann.