On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians

We consider two principal bundles of embeddings with total space Emb(M, N), with structure groups Di f f (M) and Di f f+(M), where Di f f+(M) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B(M, N) = Emb(M, N)/Di f f (M) and B+(M, N) = Emb(M, N)/Di f f+(M) from the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. It is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in the mathematical litterature e.g. by H. Omori and by T. Ratiu. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B(M, N). A case of particular interest is M = S1, where connected components of B+(S, N) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space TB+(S, N) is a subgroup of some group GLres, following the classical notations of (Pressley, A., 1988). These constructions suggest some approaches in the spirit of one of our previous works on Chern-Weil theory that could lead to knot invariants through a theory of Chern-Weil forms.