J.J. Stoker, Geometrical problems concerning polyhedra in the large, Comm. Pure Appl. Math. 21 (1958), 119--168.  Home/Search   Document Not in Database   Summary   Related Articles

....P# to a neighborhood of #(P) in K#. 4. The Infinitesimal Rigidity Lemma This section is devoted to the proof of the Infinitesimal Rigidity Lemma of Proposition 10. The proof follows the general lines of the famous arguments of Cauchy [Ca] on the deformations of convex polyhedra in E 3 (see also [St][Cr] as adapted to hyperbolic polyhedra by Andreev [An1] and Rivin [RiH] For polyhedra which are strictly hyperideal, namely whose vertices are all outside of the closure of H 3 , this proof is essentially contained in [RiH, 4] if we use the truncated polyhedron introduced in 1. Recall that ....

....In other words, there cannot be 0 or 2 sign changes in the boundary of f, unless all the edges of f are labelled by 0. We begin with faces of Type (iii) contained in horospheres. The argument uses the following euclidean geometry lemma, which is an infinitesimal and simpler version of Lemma M3 of [St] (a result attributed there to A.D. Aleksandrov) Lemma 13. In the euclidean plane R 2 , let Qt, t # ] #, #[ be a di#erentiable family of strictly convex polygons with straight sides. Suppose that, at t = 0, the derivative of the angle of Qt at each of its vertices is equal to 0, and that the ....

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J.J. Stoker, Geometrical problems concerning polyhedra in the large, Comm. Pure Appl. Math. 21 (1958), 119--168.

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