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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

In this paper we prove a Schläfli differential formula for the volume of simplices in central unit hyperquadrics of semiEuclidean space R q . Then we apply this result to obtain Gauss-Bonnet formulas for simplices with riemannian faces in the de Sitter sphere, and to generalize a formula of L. Santaló relating the volume of a hyperbolic simplex with the measure of the set of hyperbolic hyperplanes intersecting it.