Introduction to Lorentzian and Flat Affine Geometry of GL(2,R) - Institut de Mathématiques et de Modélisation de Montpellier
Pré-Publication, Document De Travail Année : 2024

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

Introduction aux Geometries Lorentzienne et Affine Plate de GL(2,R)

Résumé

The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group four 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 the natural bi-invariant flat affine structure $\nabla$ (see \cite{AuMe}), we examine these structures and the relationships between them. The curvatures, tidal force, and Jacobi vector fields of $(G_0, k^+)$ are determined in Section 1. Section 2 discusses the causal structure of $(G_0,k^+)$, while Section 3 focuses on the developed map relative to $\nabla$ in the sense of C. Ehresmann.
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Dates et versions

hal-04561292 , version 1 (26-04-2024)
hal-04561292 , version 2 (18-05-2024)
hal-04561292 , version 3 (14-09-2024)

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  • HAL Id : hal-04561292 , version 3

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Alberto Medina, Andrés Villabon. Introduction to Lorentzian and Flat Affine Geometry of GL(2,R). 2024. ⟨hal-04561292v3⟩
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