The Distortion of the Reeb Quotient Map on Riemannian Manifolds

2018/01/04 by Facundo Mémoli, Mémoli, Facundo, Osman Berat Okutan +1
Computer Science · Mathematics · #51F99 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1801.01562

openalex publication_date 2018/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a metric space X and a function f: X → ℝ, the Reeb construction gives metric a space X_f together with a quotient map X → 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.

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