Approximating topological metrics by Riemannian metrics

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 ${d_{GH}}((M,{\phi _\varepsilon }),(N,{\psi _\varepsilon })) < \varepsilon$.