A Schläfli differential formula for simplices in semi-Riemannian hyperquadrics, Gauss-Bonnet formulas for simplices in the de Sitter sphere and the dual volume of a hyperbolic simplex
We refer to the paper \textit{The Schläfli differential equality} by \textit{J. Milnor} [In: Collected papers, Vol. I, Publish Perish (1994; Zbl 0857.01015)] as a reference introducing to this problem, and the paper under review for its recent history. In the present paper, the author proves the Schläfli differential formula for the volume of simplices in central unit hyperquadrics of \(\mathbb R^{n+1}_q\) (\(=\) the semi-Euclidean space of index \(q\)). Next she applies this result to obtain the Gauss-Bonnet formula for simplices with Riemannian faces in the de Sitter sphere. And she also generalizes the formula of L. Santalò relating the volume of a hyperbolic simplex with the measure of the set of hyperbolic hyperplanes intersecting it.
scientific article; zbMATH DE number 1537817
scientific article