Differential forms canonically associated to even-dimensional compact conformal manifolds

Abstract

On a 6-dimensional, conformal, oriented, compact manifold $M$ without boundary, we compute a whole family of differential forms $\Omega_6(f,h)$ of order 6, with $f,h \in C^\infty(M).$ Each of these forms will be symmetric on $f,$ and $h,$ conformally invariant, and such that $\int_M f_0 \Omega_6(f_1,f_2)$ defines a Hochschild 2-cocycle over the algebra $C^\infty(M).$ In the particular 6-dimensional conformally flat case, we compute the unique one satisfying $\Wres(f_0[F,f][F,h]) = \int_M f_0\Omega_6(f,h)$ for $(\cH,F)$ the Fredholm module associated by A. Connes \cite{Con1} to the manifold $M,$ and $\Wres$ the Wodzicki residue.