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I am looking for a reference to the following statement. (It should be known --- I saw it before, don't remember where; search by keywords did not help.)

Let $f\colon M\to N$ be a homeomorphism between two smooth $d$-dimensional manifolds. Suppose that $M$ and $N$ admit Riemannian metrics such that the $f$ is $e^{\mp\varepsilon}$-bi-Lipschitz for some $\varepsilon=\varepsilon(d)$. Then $M$ is diffeomorphic to $N$.

(It can be proved by taking a locally finite covering of manifolds by almost isometric charts and applying a smoothing in the charts recursively.)

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  • $\begingroup$ Is this what you want? sciencedirect.com/science/article/pii/S0166864199001455 $\endgroup$
    – John Samples
    Commented Feb 6, 2021 at 21:21
  • $\begingroup$ Oh sorry, the contents of this paper were apparently mischaracterized in the thread I found it in. $\endgroup$
    – John Samples
    Commented Feb 6, 2021 at 21:33
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    $\begingroup$ Mathoverflow asks "Know someone who can answer?". Yes, Anton Petrunin should know the answer to this question. $\endgroup$
    – markvs
    Commented Feb 6, 2021 at 22:17
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    $\begingroup$ @DeaneYang I do not think so, but it could be mentioned there. $\endgroup$
    – Anton Petrunin
    Commented Feb 7, 2021 at 3:10
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    $\begingroup$ @DeaneYang, It is essentially differential topology, metric here only simplifies formulation. Likely quasiconformal-mapping people know the answer. $\endgroup$
    – Anton Petrunin
    Commented Feb 7, 2021 at 4:15

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