By a theorem of Cheeger-Colding if $N_i\to M$, where $N_i$ and $M$ are of the same dimension, compact and smooth Riemannian, as $i\to\infty$. If Ricci curvature of $N_i$ is bounded from below by $k\in \mathbb R$, then for $i$ large, $N_i$ is diffeomorphic to $M$.

Question is, what if $N_i$'s are open manifolds? I really care about the smooth structures, so let's assume all $N_i$ are all homeomorphic.

By a theorem of Cheeger-Colding if $N_i\to M$, where $\to$ means Gromov-Haussdorff convergence, $N_i$ and $M$ are of the same dimension, compact and smooth Riemannian manifold with finite volume, as $i\to\infty$. If Ricci curvature of $N_i$ is bounded from below by $k\in \mathbb R$, then for $i$ large, $N_i$ is diffeomorphic to $M$.

Question is, what if $N_i$'s are open manifolds? I really care about the smooth structures, so let's assume all $N_i$ are all homeomorphic.