A direct proof of one Gromov's theorem

TL;DR

This paper presents a new, direct proof of Gromov's theorem, establishing a quantitative relationship between Gromov--Hausdorff and Lipschitz distances for Riemannian manifolds with bounded curvature and injectivity radius.

Contribution

The authors provide a novel, direct proof of Gromov's theorem, clarifying the relationship between geometric distances for manifolds with controlled curvature and injectivity radius.

Findings

Lipschitz distance can be bounded by a function of Gromov--Hausdorff distance

The function $ ext{Delta}_{C,n}$ tends to zero as $ ext{delta}$ approaches zero

The proof applies to complete Riemannian $n$-manifolds with bounded curvature and injectivity radius

Abstract

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $\Delta_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $\delta$, absolute values of their sectional curvatures $|K_{\sigma}|\leq C$, and their injectivity radii $\geq 1/C$, then the Lipschitz distance between $V$ and $W$ is less than $\Delta_{C,n}(\delta)$ and $\Delta_{C,n}\to 0$ as $\delta\to 0$.