Metrics on hyperbolic convex polyhedra. (Métriques sur les polyèdres hyperboliques convexes.) (French) Zbl 0912.52008

Schlenker, Jean-Marc

J. Differ. Geom. 48, No. 2, 323-405 (1998).

There is a traditional way of defining a metric for the projective model of hyperbolic space: For a unit ball \(C \subset R^n\) given two interior points \(a,b\) one considers the usual line through \(a,b\) and finds the intersections \(x,y\) of this line with the boundary of \(C\). The Hilbert distance between \(a\) and \(b\) is simply \(-\frac{1}{2} \log(\frac{xa}{xb} \frac{by}{ay})\). Thinking of \(\partial C\) as a quadric not as a convex set one can generalize the above formula for points in \(R^n -C\). The “distance” becomes then a certain complex number. One can for example speak of the distance between two points outside \(C\). The new space with this “distance” is called the Sitter hyperbolic space, denoted \(HS^n\) and it has natural notions of geodesic and polyhedron. Sitter spaces have also a notion of duality between points and hyperplanes.
The main goal of this article is to show that several well-known results about polyhedra in the classical hyperbolic space extend to Sitter spaces. The author presents generalizations of results by Alexandrov, Andreev, Pogorelov and Rivin.

Reviewer: Jesus De Loera (Minneapolis)

MSC:

52A55	Spherical and hyperbolic convexity
51M10	Hyperbolic and elliptic geometries (general) and generalizations
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
52B12	Special polytopes (linear programming, centrally symmetric, etc.)

Keywords:

Sitter spaces; convexity in hyperbolic space; hyperbolic polyhedra; Hilbert metric; compact convex polyhedra; hyperbolic 3-space