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

Abstract

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 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: $$ B(M,N) = Emb(M,N)/Diff(M) \hbox{ and } B_+(M,N)= Emb(M,N)/Diff_+(M).$$ From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This 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 \cite{OMYK4}. 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 $M=S^1,$ where connected components of $B_+(S^1,N)$ are deeply linked with homotopy classes of oriented 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 classical notations of \cite{PS}. These constructions suggest some approaches in the spirit of \cite{Ma2006} that could lead to knot invariants through a theory of Chern-Weil forms.