Cheeger's Finiteness Theorem states that For each positive numbers $D,v,n$, the number of diffeomorphism classes of Riemannian manifolds $M$ with $Diameter(M)\le D$, $Vol(M)\ge v$, and $|K(M)|\le 1$ is finite. Where $K(M)$ denotes the sectional curvatures of $M$.

Clearly, when a sequence of manifolds $\{N^n_k\}$, in the class mentioned in Cheegers theorem, converges to $M^n$ in the Gromov-Hausdorff distance, then for $k$ large, $N^n_k$ is diffeomorphic to $M^n$. Is there any estimate of the derivatives of the diffeomorphism constructed in Cheeger's paper? Or is the diffeomorphism converges to idendity in $C^1$ norm?