The action of the diffeomorphism group on the space of immersions
  We study the action of the diffeomorphism group $\Diff(M)$ on the space of proper immersions $\Imm_{\text{prop}}(M,N)$ by composition from the right. We show that smooth transversal slices exist through each orbit, that the quotient space is Hausdorff and is stratified into smooth manifolds, one for each conjugacy class of isotropy groups. 
Vincente Cervera, Francisca Mascar\'o, Peter W. Michor