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

TL;DR

This paper explores the Lorentzian and flat affine geometries of the 4-dimensional Lie group component of GL(2,R), analyzing its metrics, curvature, causal structure, and affine development to deepen understanding of its geometric properties.

Contribution

It provides a detailed analysis of the bi-invariant Hessian metric and flat affine structure on the group, including curvature, causal structure, and affine development, highlighting their interrelations.

Findings

Determined the curvature, tidal force, and Jacobi fields of the group with the Hessian metric.

Analyzed the causal structure of the Lorentzian metric.

Explored the developed map relative to the flat affine structure.

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.