  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.
