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An Isometric Embedding of a Bounded Set in a Euclidean Space into the Gromov-Hausdorff Space

Takuma Byakuno

Abstract

We construct an isometric embedding of a bounded set in a Euclidean space into the Gromov-Hausdorff space. In particular, we can embed a bounded and connected Riemannian manifold into the Gromov-Hausdorff space by a bilipschitz map.

An Isometric Embedding of a Bounded Set in a Euclidean Space into the Gromov-Hausdorff Space

Abstract

We construct an isometric embedding of a bounded set in a Euclidean space into the Gromov-Hausdorff space. In particular, we can embed a bounded and connected Riemannian manifold into the Gromov-Hausdorff space by a bilipschitz map.

Paper Structure

This paper contains 4 sections, 5 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Main Theorem
  4. Acknowledgments

Key Result

Proposition 2.1

For $X,Y\in\mathcal{M}$ and $\varepsilon>0$ with $d_{\mathrm{GH}}(X,Y)<\varepsilon$, there exist a positive number $\delta<\varepsilon$ and a $2\delta$-isometry $F:X\rightarrow Y$.

Figures (2)

  • Figure 1: The definition of $\mathcal{K}_x$.
  • Figure 2: In case of $N=2$, $M=2$, $x=(1,2)$

Theorems & Definitions (9)

  • Proposition 2.1
  • proof : Proof
  • Lemma 3.1
  • Remark
  • Corollary 3.2
  • proof
  • Proposition 3.3: Nash embedding theorem
  • Corollary 3.4
  • proof