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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Approximating topological metrics by Riemannian metrics
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by Steven C. Ferry and Boris L. Okun PDF
Proc. Amer. Math. Soc. 123 (1995), 1865-1872 Request permission

Abstract:

We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if $f:M \to X$ is a continuous surjection from a smooth manifold to a compact metric space with ${f^{ - 1}}(x)$ connected for every $x \in X$, then there is a metric d on X and a sequence of Riemannian metrics $\{ {\psi _i}\}$ on M so that $(M,{\psi _i})$ converges to (X, d) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function $\rho$ and a sequence of Riemannian manifolds with $\rho$ as contractibility function so that $\lim (M,{\psi _i})$ is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds M and N and a contractibility function $\rho$ so that for every $\varepsilon > 0$ there are Riemannian metrics ${\phi _\varepsilon }$ and ${\psi _\varepsilon }$ on M and N so that $(M,{\phi _\varepsilon })$ and $(N,{\psi _\varepsilon })$ have contractibility function $\rho$ and ${d_{GH}}((M,{\phi _\varepsilon }),(N,{\psi _\varepsilon })) < \varepsilon$.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1865-1872
  • MSC: Primary 53C23; Secondary 57N60, 57R12
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1246524-7
  • MathSciNet review: 1246524