On the isometric embedding problem for length metric spaces

TL;DR

This paper demonstrates that any proper n-dimensional length metric space can be approximately embedded into Lorentzian space while preserving geodesic energy, advancing understanding of metric space embeddings.

Contribution

It introduces the concept of approximate isometric embeddings into Lorentzian space for proper length metric spaces, providing a new approach to embedding problems.

Findings

Every proper n-dimensional length metric space admits an approximate isometric embedding into n+6,1.

The embedding preserves the energy functional on a dense set of geodesics.

This result extends the theory of metric space embeddings into Lorentzian geometry.

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.