The Distortion of the Reeb Quotient Map on Riemannian Manifolds

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Given a metric space X and a function f: X → R, the Reeb construction gives metric a space Xf together with a quotient map X → Xf. Under suitable conditions Xf becomes a metric graph and can therefore be used as a graph approximation to X. The Gromov-Hausdorff distance from Xf 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.