A Correspondence Between Distances and Embeddings for Manifolds: New Techniques for Applications of the Abstract Boundary

TL;DR

This paper establishes a one-to-one correspondence between manifold embeddings and certain distances, enabling the use of the Abstract Boundary to analyze manifold edges and singularities in General Relativity without external structures.

Contribution

It introduces a novel correspondence linking embeddings and distances, facilitating boundary analysis within manifolds for studying singularities in physics.

Findings

Correspondence between embeddings and distances on manifolds.

Application to describing manifold edges and singularities.

Provides alternative methods for analyzing physical singularities.

Abstract

We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use the Abstract Boundary to describe the structure of the `edge' of our manifold without resorting to structures external to the manifold itself. This is particularly important in the study of singularities within General Relativity where singularities lie on this `edge'. The ability to talk about the same objects, e.g., singularities, via different structures provides alternative routes for investigation which can be invaluable in the pursuit of physically motivated problems where certain types of information are unavailable or difficult to use.