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Continuous deformations of polyhedra that do not alter the dihedral angles. (English) Zbl 1301.52037
The author proves that in hyperbolic and spherical 3-spaces there are compact non-convex polyhedral surfaces homeomorphic to the sphere that can be non-trivially continuously deformed in such a way that all the dihedral angles are preserved. Here a non-trivial deformation means that it changes the surface area, the total mean curvature, and the Gaussian curvature of some vertex.
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
52B70 Polyhedral manifolds
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
51M20 Polyhedra and polytopes; regular figures, division of spaces
51K05 General theory of distance geometry
Full Text: DOI arXiv
[1] Andreev, E.M.: On convex polyhedra in Lobachevskij spaces (in Russian). Mat. Sb., N. Ser. 81, 445-478 (1970). An English translation in Math. USSR, Sb. 10, 413-440 (1970) · Zbl 0194.23202
[2] Berger, M.: Geometry. I, II. Corrected 4th Printing. Universitext. Springer, Berlin (2009) · Zbl 1065.52015
[3] Connelly, R, A counterexample to the rigidity conjecture for polyhedra, Publ. Math. IHES., 47, 333-338, (1977) · Zbl 0375.53034
[4] Dolbilin, N; Frettlöh, D, Properties of Böröczky tilings in high-dimensional hyperbolic spaces, Eur. J. Comb., 31, 1181-1195, (2010) · Zbl 1198.52015
[5] Dupont, J.L., Sah, C.-H.: Three questions about simplices in spherical and hyperbolic 3-space. In: The Gelfand Mathematical Seminars, 1996-1999, pp. 49-76. Birkhäuser, Boston (2000) · Zbl 1009.52027
[6] Dupont, J.L.: What is ...a Scissors Congruence? Notices Am. Math. Soc. 59(9), 1242-1244 (2012) · Zbl 1284.52001
[7] Mazzeo, R; Montcouquiol, G, Infinitesimal rigidity of cone-manifolds and the stoker problem for hyperbolic and Euclidean polyhedra, J. Differ. Geom., 87, 525-576, (2011) · Zbl 1234.53014
[8] Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. 2d ed. Graduate Texts in Mathematics, 149. Springer, New York (2006) · Zbl 1106.51009
[9] Sabitov, I.Kh.: Algebraic methods for the solution of polyhedra (in Russian). Uspekhi Mat. Nauk 66(3), 3-66 (2011). An English translation in. Russian Math. Surveys 66(3), 445-505 (2011) · Zbl 0159.24301
[10] Souam, R, The schläfli formula for polyhedra and piecewise smooth hypersurfaces, Differ. Geom. Appl., 20, 31-45, (2004) · Zbl 1065.52015
[11] Stoker, JJ, Geometrical problems concerning polyhedra in the large, Commun. Pure Appl. Math., 21, 119-168, (1968) · Zbl 0159.24301
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