\ell(f^{-1})^m_{sj}\, \ell(f^{-1})^s_{km}+
\ell(f^{-1})_m \, \ell(f^{-1})^m_{jk},\nonumber\end{aligned}$$ for all $\phi\in {{\mathcal{F}}}_\l$ and for all $f\in {\mathrm{Diff}}(M),$ where $R_{ij}$ are the Ricci tensor components.\

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The 1-cocycle ${\cal A},$ ${\cal B}$ and the conformally invariant quantization {#new}
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We will show in this section that the quantization procedure is not invariant if one consider two metrics conformally equivalent. The obstruction of the invariance is given by the 1-coycles ${\cal A}$ and ${\cal B}.$

Given two conformally equivalent metrics $\g,$ $\tilde \g.$ Denote by $\n$, $\cal A$ and $\cal B,$ the covariant derivative, the 1-cocycles (\[MultiSchwar1\]) and (\[MultiSchwar2\]) written with the metric $\g,$ respectively. We have

The quantization map ${\cal Q}_{\l,\mu}^{\g}:$ ${\cal D}^1_{\l,\mu}(M)\rightarrow {\cal S}^1_\d(M)\oplus {\cal S}^0_\d (M)$ defined in (\[yah1\]) depend only on the conformal class of the metric $\g.$

[**Proof.**]{} Let $\tilde \g$ be another metric conformally equivalent to $\g. $ That means that there exists a diffeomorphism $\psi : (M,\tilde \g )\rightarrow (M,\g)$ and a non-zero positive function $F$ such that (locally) $$\tilde \g=F^{-1}\cdot \psi^* \g.$$ The Levi-Civita of the two connections are related by $${\Gamma^k_{ij}}^{ \g}= {\Gamma^k_{ij}}^{\tilde \g}+
\frac{1}{2 F} \left (\partial_i F\, 
\d_j^k 
+\partial_j F\, \d_i^k-\partial_s F\, \tilde \g^{sk}\tilde \g_{ij}\right ) -
\ell(\psi^{-1})^k_{ij},$$ where $\partial_i F=F_i$ and $\ell(\psi^{-1})^k_{ij}$ are 