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In this article, we examine the geometry of a group of Fourier-integraloperators, which is the central extension of $Diff(S^1)$ with a group ofclassical pseudo-differential operators of any order. Several subgroups areconsidered, and the corresponding groups with formal pseudodifferentialoperators are defined. We investigate the relationship of this group with therestricted general linear group $GL_{res},$ we define a right-invariantpseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannianmetric by the use of renormalized traces of pseudo-differential operators, andwe describe classes of remarkable connections.

On the geometry of $Diff(S^1)-$pseudodifferential operators based on renormalized traces