Continuous deformations of polyhedra that do not alter the dihedral angles

Abstract

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume of the domain, bounded by such a polyhedral surface, is necessarily constant during such a deformation while, for some families of polyhedral surfaces, the surface area, the total mean curvature, and the Gauss curvature of some vertices are nonconstant during deformations that preserve the dihedral angles. Moreover, we prove that, in the both spaces, there exist tilings that possess nontrivial deformations preserving the dihedral angles of every tile in the course of deformation.