Problems in hyperbolic geometry

Thesis (Ph. D.)--University of Hawaii at Manoa, 1993.Includes bibliographical references (leaf 124)Microfiche.vii, 124 leaves, bound ill. 29 cmIn 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 E3 or H3, there exist submanifolds of degenerate polyhedra which are "thin" when mapped into manifolds of (abstract) strictly convex metrics. 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 C2 surfaces in H3, infinitesimal deformations of the horosphere, and curves of constant curvature in H2. Finally, we also prove that certain surfaces of rotation in E3 isometrically imbed in H3. These results, some of which are new, provide a background for synthetic methods underlying the theorem of Alexandrov