Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces

Details Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces

2010-11-13

arxiv.org/abs/1011.3123v1

Mathematics

Differential Geometry

Survey paper. No proof. 10 pages

Scientific paper

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover such a polyhedron is unique, up to global isometries, among convex polyhedra invariant under isometries acting on a totally umbilical surface. This general statement falls apart into 10 different cases. The cases when $S$ is the sphere are classical.

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