AI 中文总结
本文推导了常曲率几何中正多胞形（立方体、正单纯形、正交叉多胞形）的内外角闭式公式，并利用庞加莱关系得到偶数维时的黎曼体积公式，所有结果以标准正态分布函数表示。
AI 中文摘要
我们推导了常截面曲率 $\kappa \in \mathbb R$ 的几何中 $d$ 维立方体、正单纯形和正交叉多胞形的内角和外角的闭式表达式。更一般地，我们确定了任意维度 $d$ 中矩形盒、锐角垂心单纯形、矩形垂心单纯形和不对称交叉多胞形在任意面处的内角和外角。我们还刻画了这些多胞形的黎曼切锥和法锥（在等距意义下）。结合内角公式与庞加莱关系，我们推导了当维度 $d$ 为偶数时这些多胞形的黎曼体积公式。所有公式均以标准正态分布函数 $\Phi(x)$ 及其虚数版本 $\Phi({\rm{i}} x)$ 表示。例如，若 $d\geq 2$ 为偶数，则曲率 $\kappa = -1$ 的 $d$ 维双曲空间中理想正则单纯形的双曲体积为 $$ \frac{\pi^{d/2}} {\sqrt{2}\\, {\rm{i}}^{d}\\, \Gamma\left(\frac{d+1}{2}\right)} \int_{-\infty}^{\infty} \left[ \Phi\left(\frac{{\rm i} y}{\sqrt d}\right)^{d+1} + \Phi\left(-\frac{{\rm i} y}{\sqrt d}\right)^{d+1} \right] {\rm e}^{-y^2/2} {\rm d} y. $$
英文摘要
We derive closed-form expressions for the internal and external angles of $d$-dimensional cubes, regular simplices and regular crosspolytopes in geometries of constant sectional curvature $κ\in \mathbb R$. More generally, we determine internal and external angles at arbitrary faces of rectangular boxes, acute orthocentric simplices, rectangular orthocentric simplices and asymmetric crosspolytopes in arbitrary dimension $d$. We also characterize Riemannian tangent and normal cones of these polytopes, up to isometry. Combining internal angle formulas with the Poincaré relation, we derive formulas for the Riemannian volume of these polytopes if the dimension $d$ is even. All formulas are stated in terms of the standard normal distribution function $Φ(x)$ and its imaginary version $Φ({\rm{i}} x)$. For example, if $d\geq 2$ is even, then the hyperbolic volume of the ideal regular simplex in the $d$-dimensional hyperbolic space of curvature $κ= -1$ is $$ \frac{π^{d/2}} {\sqrt{2}\, {\rm{i}}^{d}\, Γ\left(\frac{d+1}{2}\right)} \int_{-\infty}^{\infty} \left[ Φ\left(\frac{{\rm i} y}{\sqrt d}\right)^{d+1} + Φ\left(-\frac{{\rm i} y}{\sqrt d}\right)^{d+1} \right] {\rm e}^{-y^2/2} {\rm d} y. $$
Comments45 pages
