Abstracts
for the Seminar

Fall 2016

**Speaker: **Victor Alexandrov, Sobolev Institute of Mathematics, Novosibirsk

**Title: **Continuous deformations of polyhedra that do not alter the dihedral
angles

**Time:** 2:30 PM, Monday, September 19, 2016

**Place:** Malott 206

**Abstract:** We prove that, both in Lobachevskij 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 (1) the volume of the domain,
bounded by such a polyhedral surface, is necessarily constant during such
a deformation; (2) 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; (3)
in the both spaces, there exist tilings that possess nontrivial
deformations preserving the dihedral angles of every tile in the course of
deformation.

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