On geometry of convex ideal polyhedra in hyperbolic 3-space (Q2367216)

The author extends a result of \textit{E. M. Andreev} [Math. USSR, Sb., 12(1970), 255-259 (1971; Zbl 0252.52005)]. He characterizes the convex polyhedron \(P\) in the Bolyai-Lobachevskian hyperbolic 3-space \(H^ 3\), such that all the vertices of \(P\) are lying on the sphere at infinity. For this he introduces the Poincaré dual \(P^*\) of \(P\) and assigns the weight \(w(e^*)\) of any edge \(e^*\) of \(P^*\) by the exterior dihedral angle at the corresponding edge of \(P\). The main theorem is Theorem 1. The dual \(P^*\) of a convex ideal polyhedron \(P\) satisfies the following conditions: 1) \(0<w(e^*)<\pi\) for each edge \(e^*\) of \(P^*\) (at Andreev, \(\pi/2<w(e^*)<\pi\) was an additional assumption) 2) If the edges \(e^*_ 1,\ldots,e^*_ k\) form the boundary of a face of \(P^*\), then \(w(e^*_ 1)+ \cdots+w(e^*_ k)=2 \pi\). 3) If \(e^*_ 1,\ldots,e^*_ k\) form a simple circuit which does not bound a face of \(P^*\), then \(w(e^*_ 1)+\cdots+w(e^*_ k)>2\pi\). The author remarks that he has succeded in showing the conditions of Theorem 1 to be sufficient. Moreover, two combinatorially equivalent ideal polyhedra \(P_ 1\) and \(P_ 2\) with pair-wise equal dihedral angles are congruent. In Theorem 2 of the present paper this congruence is proved under the additional assumption that the corresponding faces of \(P_ 1\) and \(P_ 2\) are congruent.