Approximating topological metrics by Riemannian metrics

Authors:
Steven C. Ferry and Boris L. Okun

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1865-1872

MSC:
Primary 53C23; Secondary 57N60, 57R12

MathSciNet review:
1246524

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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 is a continuous surjection from a smooth manifold to a compact metric space with connected for every , then there is a metric *d* on *X* and a sequence of Riemannian metrics on *M* so that converges to (*X, d*) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function and a sequence of Riemannian manifolds with as contractibility function so that is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds *M* and *N* and a contractibility function so that for every there are Riemannian metrics and on *M* and *N* so that and have contractibility function and .

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1246524-7

Keywords:
Riemannian manifold,
length space,
cell-like map

Article copyright:
© Copyright 1995
American Mathematical Society