Isometric Embeddings of Locally Euclidean Metrics in R~3 as Conical Surfaces

摘要

It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane R~2 with the standard metric, then it can be isometrically embedded in R~3 as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.