POLYHEDRA IN CONSTANT CURVATURE SPACES.

In this two-sessions seminar, some results concerning geometrical properties of three dimensional polyhedra1 in Euclidean, hyperbolic and spherical spaces will be presented. In the first session we will introduce some classical problems and theorems. Some of the main results that will be shown are: Dehn’s solution to Hilbert’s third problem, Cauchy’s rigidity theorem and Lobachevsky’s formula for the volume of an ideal hyperbolic tetra-hedron. More details can be found in [R] and [T]. In the second session we will be devoted essentially to non-Euclidean polyhedra, relating them with important concepts in geometry (namely Mostow’s rigidity theorem and hyperbolic orbifolds). By means of a simple and very interesting theorem by Schläfli we will find explicit answers to several problems concerning volume calculation of compact polyhedra. More details can be found in [V]. In particular, at the end of this session there will be presented the main result of [AGM]