Isometric Embeddings of Polyhedra into Euclidean Space

Abstract

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that each $n$-dimensional simplex of $\mathcal{T}$ is affinely isometric to a simplex in $\mathbb{E}^n$. We prove that any 1-Lipschitz map from an $n$-dimensional Euclidean polyhedron $\mathcal{P}$ into $\mathbb{E}^{3n}$ is $\epsilon$-close to a pl isometric embedding for any $\epsilon > 0$. If we remove the condition that the map be pl then any 1-Lipschitz map into $\mathbb{E}^{2n + 1}$ can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash-Kuiper $C^1$ isometric embedding theorem. Finally, we discuss how these results extend to various other types of polyhedra.