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In a paper of P. Michor, it was shown that Emb(M,N) is a smooth principal diff(M)-bundle, M and N are smooth locally compact manifolds provided dim M < dim N. My question is why there is a restriction on the dimensions. Does anyone know a reference for the result when dim M = dim N ?

Thanks in advance

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    $\begingroup$ The link is broken for me. $\endgroup$
    – ಠ_ಠ
    Commented Jan 21, 2016 at 8:07
  • $\begingroup$ Sorry, It's fixed now $\endgroup$
    – s k
    Commented Jan 21, 2016 at 8:28

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Then an open subset of $Emb(M,N)$ is $Diff(M)$. Namely, if $M$ is compact, the image of $M$ under an embedding is open and closed, so you have a diffeomorphism onto a connected component. If $M$ is not compact, there are no smooth movements in $Emb(M,N)$ tangential to the image near infinity. If $M$ is compact with boundary, see

  • MR3263203 Reviewed Gay-Balmaz, François; Vizman, Cornelia Principal bundles of embeddings and nonlinear Grassmannians. Ann. Global Anal. Geom. 46 (2014), no. 3, 293–312.
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  • $\begingroup$ I'm so grateful to have an answer from you personally, thanks $\endgroup$
    – s k
    Commented Jan 22, 2016 at 5:36

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