Quantitative Bi-Lipschitz embeddings of bounded curvature manifolds and orbifolds

TL;DR

This paper develops methods to embed bounded curvature manifolds and orbifolds into Euclidean space with controlled distortion and dimension, based on analyzing their structure at multiple scales.

Contribution

It introduces a new approach using collapsing theory to construct bi-Lipschitz embeddings for manifolds and orbifolds with bounded curvature and diameter.

Findings

Embeddings have distortion and dimension bounded by curvature, diameter, and dimension.

Results apply to bounded subsets of complete Riemannian manifolds and certain orbifolds.

Method leverages multi-scale analysis of manifold structure using collapsing theory.

Abstract

We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also apply for bounded subsets of complete Riemannian manifolds, and complete flat and elliptic orbifolds. Our approach is based on analysing the structure of a bounded curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces.