# Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds

## TL;DR

This paper proves that any non-positively curved metric on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a 2+1 dimensional flat Lorentzian spacetime, using polyhedral approximation.

## Contribution

It establishes a new embedding theorem for non-positively curved surfaces into flat Lorentzian manifolds, expanding understanding of spacetime geometry.

## Key findings

- Any Alexandrov non-positively curved metric on a compact surface can be embedded in flat spacetime.
- Embedding is isometric and convex, preserving the metric.
- Proof utilizes polyhedral approximation techniques.

## Abstract

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.