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We consider two principal bundles of embeddings with total space $Emb(M,N),$with structure groups $Diff(M)$ and $Diff_+(M),$ where $Diff_+(M)$ is thegroups of orientation preserving diffeomorphisms. The aim of this paper is todescribe the structure group of the tangent bundle of the two base manifolds:$$ B(M,N) = Emb(M,N)/Diff(M) \hbox{ and } B_+(M,N)= Emb(M,N)/Diff_+(M).$$ Fromthe various properties described, an adequate group seems to be a group ofFourier integral operators, which is carefully studied. This is the main goalof this paper to analyze this group, which is a central extension of a group ofdiffeomorphisms by a group of pseudo-differential operators which is slightlydifferent from the one developped in \cite{OMYK4}. We show that these groupsare regular, and develop the necessary properties for applications to thegeometry of $ B(M,N) .$ A case of particular interest is $M=S^1,$ whereconnected components of $B_+(S^1,N)$ are deeply linked with homotopy classes oforiented knots. In this example, the structure group of the tangent space$TB_+(S^1,N)$ is a subgroup of some group $GL_{res},$ following the classicalnotations of \cite{PS}. These constructions suggest some approaches in thespirit of \cite{Ma2006} that could lead to knot invariants through a theory ofChern-Weil forms.

On Diff(M)-pseudo-differential operators and the geometry of non linear grassmannians