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Hyperbolic Geometry according to Klein is the pair \(({{\mathcal H}}^{n+1},\mathrm {GM}({{\mathcal H}}^{n+1}))\) , where \(\displaystyle {{\mathcal H}}^{n+1}=\{{\mathbf {x}}=(x_1,\dots ,x_{n+1})\in {{\mathbb R}}^{n+1}\:|\; x_{n+1}>0\}, \) and \(\mathrm {GM}({{\mathcal H}}^{n+1})\) is the subgroup of \(\mathrm {GM}(n+1)\) that leaves \({{\mathcal H}}^{n+1}\) invariant. The description of Hyperbolic Geometry in this way corresponds to the upper half-space model which is due to Poincaré.

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Hyperbolic Geometry

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  • 发表日期：

    2026年

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  • Ioannis D. Platis

摘要

Hyperbolic Geometry according to Klein is the pair \(({{\mathcal H}}^{n+1},\mathrm {GM}({{\mathcal H}}^{n+1}))\) , where \(\displaystyle {{\mathcal H}}^{n+1}=\{{\mathbf {x}}=(x_1,\dots ,x_{n+1})\in {{\mathbb R}}^{n+1}\:|\; x_{n+1}>0\}, \) and \(\mathrm {GM}({{\mathcal H}}^{n+1})\) is the subgroup of \(\mathrm {GM}(n+1)\) that leaves \({{\mathcal H}}^{n+1}\) invariant. The description of Hyperbolic Geometry in this way corresponds to the upper half-space model which is due to Poincaré.