Isometric embeddings of locally Euclidean metrics in ℝ3 as conical surfaces
Сергей Николаевич Михалeв, I. Kh. Sabitov · Mathematical Notes · 2014
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.