About the rigidity of tridimensional hyperbolic polyhedra: finite volume case, hyperideal case, Fuchsian case. (Sur la rigidité de polyèdres hyperboliques en dimension \(3\): cas de volume fini, cas hyperidéal, cas fuchsien.) (French) Zbl 1061.52007

Rousset, Mathias

Bull. Soc. Math. Fr. 132, No. 2, 233-261 (2004).

The article contains the following principal results as well as their generalisation to the case of Fuchsian polyhedra:
(1) A characterisation of the hyperbolic convex polyhedra with \(n\) semi-ideal faces (with vertices at infinity) in terms of their dual metric. As particular cases, that characterisation comprises the case of compact polyhedra previously considered by E. M. Andreev [Math. USSR, Sb. 10, 413–440 (1970; Zbl 0217.46801)] and the case of ideal polyhedra previously considered by I. Rivin and C. D. Hodgson [Invent. Math. 111, No. 1, 77–111 (1993; Zbl 0784.52013)] and by I. Rivin [Ann. Math. (2) 143, No. 1, 51–70 (1996; Zbl 0874.52006)].
(2) A new proof of the characterisation of the hyperbolic hyperideal convex polyhedra of a prescribed combinatorial type (the vertices are at infinity or behind the absolute) by their dihedral angles.

Reviewer: Victor Alexandrov (Novosibirsk)

MSC:

52B10	Three-dimensional polytopes
52A55	Spherical and hyperbolic convexity
51M10	Hyperbolic and elliptic geometries (general) and generalizations
53C45	Global surface theory (convex surfaces à la A. D. Aleksandrov)

Keywords:

hyperbolic geometry; Lobachevskij geometry; polyhedron; hyperideal polyhedron; Fuchsian manifold; rigidity; de Sitter space; Pogorelov’s transformation

Citations:

Zbl 0217.46801; Zbl 0784.52013; Zbl 0874.52006