\end{array}$$ and has the following properties:\
(i) It does not depend on the rescaling of the metric $\g$.\
(ii) If $M={\mathbb{R}}^n$ is endowed with a flat conformal structure, this map is unique, equivariant with respect to the action of the group ${\mathrm{O}}(p+1,q+1) \subset {\mathrm{Diff}}({\mathbb{R}}^n)$.

Before to give the formula of the conformal equivariant map in the case of surfaces, let us recall an interesting approach for the multi-dimensional Schwarzian derivative for conformal mapping [@os] (see also [@ca]). First, recall that all surfaces are conformally flat. This means that every metric can be express (locally) as $$\g=F^{-1}\psi^* \g_0,$$ where $\psi$ is a conformal diffeomorphism of $M$, $F$ is a non-zero positive function and $\g_0$ is a metric of constant curvature. The Schwarzian derivative of $\psi$ is defined in [@os] as the following tensor field $$\label{osg}
S(\psi )=\frac{1}{2F}\nabla dF-\frac{3}{4F^2}\,dF\otimes dF+\frac{1}{8F^2}
\g^{-1}(dF,dF)\, \g \cdot$$

Now we are in position to give the quantization map for the case of surfaces.

For $\d \not=1,2,\frac{3}{2},\frac{5}{2},$ and for each $P=P^{ij}\xi_i\xi_j +P^i\xi_i +P_0\in {\cal S}_{\d,2}(M)$ one associates a linear differential operator given by $$\begin{aligned}
Q(P)&=&
P^{ij}\n_i\n_j \nonumber\\[2mm]
&&+(\a_1 \n_i P^{ij}+\a_2\,\g^{ij}\, \g_{kl}\n_i P^{kl})
\n_j \label{surf}\\[2mm]
&&+\a_3\n_i\n_j P^{ij}+\a_4\, \g^{st}\, \g_{ij}\n_s\n_t P^{ij}
\nonumber\\[2mm]
&& +\frac{4\l(\mu-1)}{2\d -3}\left( S(\psi)_{ij}P^{ij}+
\frac{1}{8(\d-1)} R \,\g_{ij}\,P^{ij}\right )+P_0, 
