\nonumber\end{aligned}$$ where $S(\psi)$ is the tensor (\[osg\]), $R$ is the scalar curvature and the coefficients $\a_1,\ldots,\a_4$ are given as above.

[ The projectively equivariant quantization map was given in [@lo1] (see also [@b3] for the non-flat case). The multi-dimensional projective Schwarzian derivative is defined as an obstruction to extend this isomorphism to the full group ${\mathrm{Diff}}(M).$ We will show in the next section that the conformal Schwarzian derivatives defined in this paper appear as obstructions to extend the isomorphisms (\[Tensor1\]), (\[surf\]) to the full group ${\mathrm{Diff}}(M).$ ]{}

Deformation of the space of symbols ${\cal S}_{2,\d}$ {#va}
-----------------------------------------------------

The goal of this section is to explicate the relation between the 1-cocycles (\[MultiSchwar1\]), (\[MultiSchwar2\]) and the space of second-order linear differential operators ${\cal D}_{\l,\mu}^2(M).$ Since the space ${\cal D}_{\l,\mu}^2(M)$ is a non-trivial deformation of the space of the corresponding space of symbols ${\cal S}_{\d,2}(M),$ where $\d=\mu-\l$, it is interesting to give explicitly this deformation in term of the 1-cocycles (\[MultiSchwar1\]), (\[MultiSchwar2\]). Namely, we are looking for the operator $\bar f_\d={\cal Q}^{-1}_{\l,\mu} \circ f_{\l,\mu} \circ 
{\cal Q}_{\l,\mu}$ such that the diagram below is commutative $$\nonumber
\begin{CD}
{\cal S}_{\d,2}(M)
@> \bar f_\d >>
{\cal S}_{\d,2}(M)
\strut\\ 
@V{\cal Q}_{\l,\mu}VV
@VV{\cal Q}_{\l,\mu}V \strut\\
{{\mathcal{D}}}_{\l,\mu}^2 (M) @>  