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

TL;DR

This paper presents a method to embed bounded sets, including Riemannian manifolds, from Euclidean space into the Gromov-Hausdorff space using isometric and bilipschitz maps, facilitating geometric analysis.

Contribution

It introduces an isometric embedding technique for bounded Euclidean sets and Riemannian manifolds into the Gromov-Hausdorff space, expanding tools for geometric topology.

Findings

Successfully embeds bounded Euclidean sets into Gromov-Hausdorff space.

Embeds bounded connected Riemannian manifolds via bilipschitz maps.

Provides a framework for geometric analysis in 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.