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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 isdoi:10.3390/math4010001 fatcat:dsumydzds5gsvlh5lat5hzrwla