The Distortion of the Reeb Quotient Map on Riemannian Manifolds

TL;DR

This paper investigates how the Reeb quotient map distorts Riemannian manifolds when approximating them with metric graphs, providing bounds involving topological invariants and a new concept called thickness.

Contribution

It introduces bounds on the distortion of the Reeb quotient map for Riemannian manifolds using the first Betti number and a new invariant called thickness.

Findings

Bounds on the metric distortion involving Betti number and thickness

Reeb quotient maps can approximate Riemannian manifolds with controlled distortion

Introduction of the thickness invariant for analyzing Reeb maps

Abstract

Given a metric space $X$ and a function $f: X \to \mathbb{R}$, the Reeb construction gives metric a space $X_f$ together with a quotient map $X \to X_f$. Under suitable conditions $X_f$ becomes a metric graph and can therefore be used as a graph approximation to $X$. The Gromov-Hausdorff distance from $X_f$ to $X$ is bounded by the half of the metric distortion of the quotient map. In this paper we consider the case where $X$ is a compact Riemannian manifold and $f$ is an excellent Morse function. In this case we provide bounds on the distortion of the quotient map which involve the first Betti number of the original space and a novel invariant which we call thickness.