(In)equality distance patterns and embeddability into Hilbert spaces

TL;DR

This paper investigates the limitations of embedding compact Riemannian manifolds into Euclidean spaces, showing they cannot contain certain rigid structures and proposing a concept of loose embeddings that preserve distance relations.

Contribution

It introduces the concept of loose embeddings for Riemannian metric spaces and establishes a local-to-global principle for their embeddability into Euclidean spaces.

Findings

Compact Riemannian manifolds cannot contain arbitrarily large regular simplices.

They cannot contain arbitrarily long sequences of equidistant points.

A local-to-global principle for loose embeddability is proven.

Abstract

We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddability: if every finite subspace thereof is loosely embeddable into a common $\mathbb{R}^N$, then the metric space as a whole is loosely embeddable into $\mathbb{R}^N$ in a weakened sense.