On the isometric embedding problem for length metric spaces

Abstract

We prove that every proper $n$-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space ${\mathbb{R}}^{3n+6,1}$. By an "approximate isometric embedding" we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.

40 pages, 10 figures