# Alexandrov's embedding theorem and the meaning of realizable

1. Jan 27, 2014

### IBphysicsy

I am a graduate student working on an algorithm for isometric embeddings of simplicial complexes homeomorphic to $\mathcal{S}^2$with positive Gaussian curvature at each vertex(i.e. deficit angles > 0). Given the constraint on Gaussian curvature, this complex can be regarded as a convex polyhedral metric. Alexandrov's embedding theorem states,

Every convex polyhedral metric can be realized (and in one way only) by a convex polyhedron.

Alexandrov's presentation of his proof is long and cumbersome making it difficult to understand how he draws his conclusion. Because of this, I'm not sure what he means by realizable in this context. Does it mean the polyhedral metrics are identical, or does realizable allow for changes in connections between vertices thus giving a polyhedron that represents the polyhedral metric but is not identical to it? By identical I mean the connections between vertices and edge lengths are the same. If there is anyone who has studied this theorem extensively who can give a precise definition of realizable or give me references concisely summarizing the theorem that would be great. My references are,

author: A.D. Alexandrov title: A.D. Alexandrov Selected Works pt. II Intrinsic Geometry of Convex Surfaces section: Chapter 6 and 7

author: A.V. Pogorelov title: Extrinsic Geometry of Convex Surfaces section: Chapter 1 pg. 20

Last edited: Jan 27, 2014