The Virasoro group and the fourth geometry of Poincar\'e

We investigate, in some details, symplectic equivalence between several conformal classes of Lorentz metrics on the hyperboloid of one sheet $H^{1,1} \cong T \times T - \Delta$ and affine coadjoint orbits of the group $Diff_+(\Delta)$ of orientation preserving diffeomorphisms of $\Delta \cong T$ with its natural projective structure. This will allow for generalizations, namely, to the case of arbitrary projective structures on null infinity.