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

In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of Dif f (S 1) with a group of classical pseudo-differential operators of any order. Several subgroups are considered , and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group GLres, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections. MSC (2010) : 22E66, 47G30, 58B20, 58J40