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.