is clear that a projective connection on $M$ leads to the following 1-cocycle on ${\mathrm{Diff}}(M)$: $${\cal C}(f^{-1})=
\left(
(f^{-1})^*\Pi_{ij}^k-\Pi_{ij}^k
\right)
dx^i\otimes dx^j\otimes\frac{\partial}{\partial x^k}
\label{ellp}$$ This formula is independent on the choice of the coordinate system.

By definition, the tensor field (\[ellp\]) depends only on the projective class of the connection $M.$ In particular if $\Pi \equiv 0,$ this tensor field vanishes on the projective group ${\mathrm{PSL}}_{n+1}({\mathbb{R}}).$

One can define a 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}^{2}_\d(M), {{\mathcal{S}}}^{1}_\d(M))$ by contracting any symmetric contravariant tensor field with the tensor (\[ellp\]). Therefore, the operator (\[MultiSchwar1\]) can be viewed as the conformal analogue of the tensor field (\[ellp\]). In the same spirit, the operator (\[MultiSchwar2\]) can be viewed as the conformal analogue of the “projective” multi-dimensional Schwarzian derivative introduced in [@b2; @bo2].

Relation to the modules of differential operators {#SecComput}
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Conformally equivariant quantization {#tokyo}
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The quantization procedure explained in this paper was first introduced in [@do; @lo1]. By an equivariant quantization we mean an identification between the space of linear differential operators and the corresponding space of symbols, equivariant with respect to the action of a (finite dimension) sub-group $G\subset {\mathrm{Diff}}({\mathbb{R}}^n).$ Recall that in the one-dimensional case the equivariant quantization process 