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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.

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    $\begingroup$ What do you mean by $N_i\to M$ for open manifolds? If you mean pointed GH convergence, then let $N_i$ be the fixed metric on $N$ rescaled by $i$, so that the limit is a Euclidean space, which need not homeomorphic to $N$. $\endgroup$
    – Igor Belegradek
    Commented Jan 14, 2016 at 18:06
  • $\begingroup$ @IgorBelegradek, thanks, the $\to$ means GH convergence. Yes, you are right, I need to exclude the tangent cone here, so what if all $N_i$ and $M$ are of finite volume? $\endgroup$
    – J. GE
    Commented Jan 14, 2016 at 23:41
  • $\begingroup$ The question is still unclear: did you mean pointed GH convergence or unpointed GH convergence? $\endgroup$
    – Igor Belegradek
    Commented Jan 15, 2016 at 0:59
  • $\begingroup$ @IgorBelegradek, I mean global, unpointed. But for pointed version, you have counterexample? moving the center point to infinity can prevent manifold from collapsing in finite volume case? $\endgroup$
    – J. GE
    Commented Jan 15, 2016 at 1:12
  • $\begingroup$ In the pointed case there are counterexamples, coming from e.g. cusp opening of hyperbolic manifolds of dimension 2 and 3: closed hyperbolic manifolds can converge to noncompact hyperbolic manifolds of finite volume. I do not undertand what you said about moving the basepoint and preventing collapse. $\endgroup$
    – Igor Belegradek
    Commented Jan 15, 2016 at 1:29

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