Proceedings of the American Mathematical Society

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



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 $ 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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Keywords: Riemannian manifold, length space, cell-like map
Article copyright: © Copyright 1995 American Mathematical Society