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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2405.07053 |
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| _version_ | 1866910452487815168 |
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| author | Medina, Alberto Villabon, Andres |
| author_facet | Medina, Alberto Villabon, Andres |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07053 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Introduction to Lorentzian and Flat Affine Geometry of $\mathsf{GL}(2,\mathbb{R})$ Medina, Alberto Villabon, Andres Differential Geometry 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. |
| title | Introduction to Lorentzian and Flat Affine Geometry of $\mathsf{GL}(2,\mathbb{R})$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2405.07053 |