| In this thesis, we discuss the proof that all convex polyhedral metrics can be realized in euclidean and hyperbolic 3-space. This result is accredited to A. D. Alexandrov and is fundamental in modern synthetic differential geometry. Nevertheless, gaps appear in currently acknowledged proofs: (1) It is necessary to prove that strictly convex metrics with 4 real vertices can be realized. (2) It must be shown that, within manifolds of convex polyhedra in E;In this thesis we prove these statements.;The remainder of the thesis is devoted to general hyperbolic geometry with emphasis on the synthetic point of view. We first construct horocyclic coordinates and use these to derive the Poincare model for the hyperbolic plane. Then we compute useful formulas for the curvature of a surface, and use these formulas to study C... |
