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Tubed embeddings

Anton Petrunin

TL;DR

The paper characterizes when a complete Riemannian manifold with bounded geometry admits an isometric tubed embedding into a higher-dimensional ambient space. It combines a graph-approximation strategy with a Krauthgamer–Lee style large-scale embedding, a locally bi-Lipschitz construction, and a Nash-type uniform immersion to produce a tubed, uniformly smooth isometric embedding into a Euclidean space; the construction yields bounded normal curvatures and a thick tubular neighborhood. A universal space theorem shows there is no single ambient space that accommodates all manifolds with bounded geometry, while the main result provides a constructive, dimension-controlled embedding under growth assumptions. The work also develops the notion of uniformly regular metrics, linking bounded geometry to uniform regularity, and discusses generalizations, limitations, and open questions about tubed embeddings into hyperbolic spaces and products of hyperbolic planes.

Abstract

We consider the following question: When does a Riemannian manifold admit an embedding with a uniformly thick tubular neighborhood in another Riemannian manifold of large dimension?

Tubed embeddings

TL;DR

The paper characterizes when a complete Riemannian manifold with bounded geometry admits an isometric tubed embedding into a higher-dimensional ambient space. It combines a graph-approximation strategy with a Krauthgamer–Lee style large-scale embedding, a locally bi-Lipschitz construction, and a Nash-type uniform immersion to produce a tubed, uniformly smooth isometric embedding into a Euclidean space; the construction yields bounded normal curvatures and a thick tubular neighborhood. A universal space theorem shows there is no single ambient space that accommodates all manifolds with bounded geometry, while the main result provides a constructive, dimension-controlled embedding under growth assumptions. The work also develops the notion of uniformly regular metrics, linking bounded geometry to uniform regularity, and discusses generalizations, limitations, and open questions about tubed embeddings into hyperbolic spaces and products of hyperbolic planes.

Abstract

We consider the following question: When does a Riemannian manifold admit an embedding with a uniformly thick tubular neighborhood in another Riemannian manifold of large dimension?
Paper Structure (12 sections, 19 equations, 3 figures)

This paper contains 12 sections, 19 equations, 3 figures.

Table of Contents

  1. Other ambient spaces.
  2. About the proofs.
  3. Lemma.
  4. Intersection graph.
  5. Proposition.
  6. Uniformly regular metrics.
  7. Proposition.
  8. Proof of the main theorem (§\ref{['par:main']}).
  9. Theorem.
  10. Proof of the proposition in §\ref{['par:other-intro']}.
  11. Question
  12. Acknowledgments.

Figures (3)

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