Synthetic geometry in hyperbolic simplices (Q2692992)

This paper aims to unfurl synthetic geometricc formulas for simplices with constant curvature, which, together with [\textit{B. Minemyer}, Lond. Math. Soc. Lect. Note Ser. 451, 135--145 (2018; Zbl 1430.52012)], establishes a formulation on which further research may be more easily performed due to the formulas' relative simplicity. The paper focuses primarily on hyperbolic simplices with due regard to their mathematical importance. The principal results go as follows: \begin{itemize} \item[1.] \S 3 develops a simple criterion determining whether a set of positive edge lengths for an \(n\)-simplex determines a legitimate hyperbolic simplex (Theorem 3.2). \item[2.] Given a hyperbolic simplex \((\tau,g_{\mathbb{H}})\) and two points \(x,y\in\tau\), \S 5 determines an easy procedure to find \(d_{\mathbb{H}}(x,y)\) using only the edge lengths of \(\tau \) and the barycentric coordinates of \(x\) and \(y\). \item[3.] Given a Euclidean -simplex \((\tau,g_{\mathbb{E}})\) with \(\tau=(v_{1},v_{2},\dots,v_{n},v_{n+1})\), \S 6 unfurls a formula for \(\mathrm{proj}_{\tau_{n+1}}(v_{n+1})\), the orthogonal projection of \(v_{n+1}\) onto the \((n-1)\)-face opposite of it (Theorem 6.2). \item[4.] Given a hyperbolic \(n\)-simplex \((\tau,g_{\mathbb{H}})\) with \(\tau=(v_{1},v_{2},\dots,v_{n},v_{n+1})\), \S 7 develops a formula for \(\mathrm{proj}_{\tau_{1}}^{\mathbb{H}}(v_{1})\) (Theorem 7.1). \end{itemize}