## A characterization of compact convex polyhedra in hyperbolic 3-space.(English)Zbl 0784.52013

This paper is a comprehensive survey with detailed proofs based on the Ph. D. dissertation of the second author. A. D. Alexandrov, Convex Polyhedra (in Russian), Moscow, GITTL, 1950, (German translation: Berlin, Akademie Verlag 1958) has given a complete characterization of the shapes of compact convex polyhedra in the hyperbolic 3-space $$H^ 3$$ in terms of the intrinsic hyperbolic metric on the boundary. The authors give an analogous extrinsic description in terms of their dihedral angles. The Gauss map of a compact convex polyhedron on the Euclidean 3-space $$E^ 3$$ has been extended onto $$H^ 3$$ by using the hyperboloid model of $$H^ 3$$. The hyperbolic polar image $$G(P)$$ of $$P$$ has a realization as a “convex polyhedron” in the “de Sitter sphere” $$S^ 2_ 1$$. The main result is
Theorem 1.1. A metric space $$(M,g)$$ homeomorphic to $$S^ 2$$ can arise as the Gaussian image $$G(P)$$ of a compact convex polyhedron $$P$$ in $$H^ 3$$ iff the following conditions hold:
(a) The metric $$g$$ has constant curvature 1 away from a finite collection of cone points $$c_ i$$.
(b) The cone angle at each $$c_ i$$ is greater than $$2\pi$$.
(c) The lengths of closed geodesics of $$(M,g)$$ are strictly greater than $$2\pi$$.
These results have interesting consequences, e.g. Andreev’s characterization of compact convex polyhedra in $$H^ 3$$ with dihedral angles not greater than $$\pi/2$$ follows very easily [see the first author, Ohio State Univ. Math. Res. Inst. Publ. 1, 185-193 (1992; Zbl 0765.52013)]. Another result states that the face angles of a convex polyhedron $$P$$ in $$H^ 3$$ determine it up to congruence.
The second author applied the current approach to characterize convex polyhedra of finite volume just in the following review: [I. Rivin: On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32, No. 1, 87-92 (1993; see the paper below)]. In the projective Cayley-Klein model of $$H^ 3$$, an ideal convex hyperbolic polyhedron is represented by convex Euclidean polyhedron inscribed in the 2-sphere. So the result also characterizes the graphs which can be inscribed in the 2-sphere as the 1-skeleton of a convex Euclidean polyhedron. The question of the existence of such a description has been open since 1832, when it was raised by Jakob Steiner. As evidence that the above characterization is the desired one, Warren Smith has shown that Theorem 1 in the previous paper leads to a polynomial time algorithm for deciding whether a polyhedral graph can be inscribed in the sphere.

### MSC:

 52B10 Three-dimensional polytopes 52A55 Spherical and hyperbolic convexity 51M10 Hyperbolic and elliptic geometries (general) and generalizations

### Keywords:

survey; compact convex polyhedra; hyperbolic 3-space

### Citations:

Zbl 0784.52014; Zbl 0765.52013
Full Text:

### References:

 [1] Aleksandrov, A.D.: Convex Polyhedra (in Russian) Moscow: GITTL 1950; (German translation): Berlin: Akademie Verlag 1958) [2] Aleksandrov, A.D., Berestovskii, V.N., Nikolaev, I.G.: Generalized Riemannian spaces. Russ. Math. Surv.41(3), 1-54 (1986) · Zbl 0625.53059 [3] Andreev, E.M.: On convex polyhedra in Lobachevskii space. Math. USSR, Sb.12, 413-440 (1970) · Zbl 0217.46801 [4] Beardon, A.F.: The geometry of discrete groups. Berlin Heidelberg New York: Springer 1983 · Zbl 0528.30001 [5] Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry. (Pure Appl., Math. vol. 67) New York: Dekker 1981 · Zbl 0462.53001 [6] Berger, M.: Géométrie, vols. 1-5. Paris: Cedic/Fernand Nathan 1977; (English translation: Geometry I and II. Berlin Heidelberg New York: Springer 1987) [7] Bowditch, B.H.: Notes on locally CAT(1) spaces. University of Aberdeen (Preprint 1992) · Zbl 0865.53035 [8] Cauchy, A.L.: Sur les polygones et polyèdres, 2nd memoir. J. Éc Polytechnique19, 87-98 (1813) [9] Cheeger, J. Ebin. D.G.: Comparison Theorems in Riemannian Geometry. (North-Holland Math. Libr., vol. 9) Amsterdam Oxford: North-Holland 1975 · Zbl 0309.53035 [10] Colin de Verdière, Y., Marin, A.: Triangulations presque Équilatérales des surfeces. J. Differ. Geom.32, 199-207 (1990) · Zbl 0705.53021 [11] Gromov, M.L.: Hyperbolic groups. In: Gersten S.M. (ed.) Essays in group theory, pp. 75-263 (Pub., Math. Sci. Res. Inst., vol 8) Berlin Heidelberg New York: Springer 1987 [12] Gromov, M.L.: Structures Métriques pour les Variétés Riemanniennes. Paris: Cedic/Fernand Nathan 1982. [13] Hodgson, C.D.: Deduction of Andreev’s Theorem from Rivin’s Characterization of Convex Hyperbolic Polyhedra. In: Topology 90. Proceedings of the Research Semester in Low Dimensional Topology at O.S.U Berlin New York: de Gruyter (to appear) · Zbl 0765.52013 [14] Klingenberg, W.: Global Riemannian Geometry. Berlin New York: de Gruyter 1982. · Zbl 0495.53036 [15] O’Neill, B.: Semi-Riemannian Geometry; with applications to relativity. New York: Academic Press 1983 [16] Pogorelov, A.V.: Extrinsic geometry of Convex Surfaces. (Transl. Math. Monogr., vol.35) Providence: Am. Math. Soc. 1973 · Zbl 0311.53067 [17] Rivin, I.: On geometry of convex ideal polyhdra in hyperbolic 3-space. Topology (to appear) · Zbl 0784.52014 [18] Rivin, I.: A characterization of ideal polyhedra in hyperbolic 3-space. (Preprint 1992) · Zbl 0874.52006 [19] Rivin, I.: On geometry of convex polyhedra in hyperbolic 3-space. Ph. D. thesis, Princeton (June 1986) · Zbl 0784.52014 [20] Rivin, I.: Some applications of the hyperbolic volume formula of Lobachevsky and Milnor (submitted 1992) [21] Royden, H.L.: Real Analysis. New York. Macmillan 1968 · Zbl 0197.03501 [22] Spivak, M.: A Comprehensive Introduction to Differential Geometry, 5 vol. Berkeley: Publish or Perish 1979 · Zbl 0439.53003 [23] Stoker, J.J.: Geometric problems concerning polyhedra in the large. Commun. Pure Applied Math.21, 119-168 (1968) · Zbl 0159.24301 [24] Thurston, W.P.: Geometry and Topology of 3-manifolds. Revised Lectere Notes, Princeton Univ. Math. Dept. (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.