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
→
X
f:M \to X
is a continuous surjection from a smooth manifold to a compact metric space with
f
−
1
(
x
)
{f^{ - 1}}(x)
connected for every
x
∈
X
x \in X
, then there is a metric d on X and a sequence of Riemannian metrics
{
ψ
i
}
\{ {\psi _i}\}
on M so that
(
M
,
ψ
i
)
(M,{\psi _i})
converges to (X, d) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function
ρ
\rho
and a sequence of Riemannian manifolds with
ρ
\rho
as contractibility function so that
lim
(
M
,
ψ
i
)
\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
ρ
\rho
so that for every
ε
>
0
\varepsilon > 0
there are Riemannian metrics
ϕ
ε
{\phi _\varepsilon }
and
ψ
ε
{\psi _\varepsilon }
on M and N so that
(
M
,
ϕ
ε
)
(M,{\phi _\varepsilon })
and
(
N
,
ψ
ε
)
(N,{\psi _\varepsilon })
have contractibility function
ρ
\rho
and
d
G
H
(
(
M
,
ϕ
ε
)
,
(
N
,
ψ
ε
)
)
>
ε
{d_{GH}}((M,{\phi _\varepsilon }),(N,{\psi _\varepsilon })) > \varepsilon
.