\n_{s}\ell(f)_{ij}^s \right )+(\d-1)
 \left ({f^{-1}}^*(R\,\g_{ij})-R\,\g_{ij}\right ),
\nonumber
\label{MultiSchwar3}\end{aligned}$$ where $S(\psi)$ is the derivative (\[osg\]), $\ell(f)$ is the tensor (\[ell\]), $R$ is the scalar curvature of the metric $\g,$ is a differential operator from from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^0(M).$

Theorem (\[mainp\]) remains true for this operator.

[*Acknowledgments*]{}. It is a pleasure to acknowledge numerous fruitful discussions with Prof. V. Ovsienko. I am grateful to, the referee for his pertinent remarks, Prof. Y. Maeda and Keio University for their hospitality.

[99]{} A. L. Besse, Einstein manifolds, Springer-Verlag, Berlin Heidelberg, 1987.

S. Bouarroudj, Multi-dimensional Schwarzian derivative revisited, math. DG/0101056

S. Bouarroudj, Projectively equivariant quantization map, Lett. Math. Phy. [**51:**]{} (4) (2000), 265-274.

S. Bouarroudj & V. Ovsienko, Schwarzian derivative related to the space of linear differential operators on a locally projective manifold, Banach Cent. Pub., vol 51, 2000.

S. Bouarroudj & V. Ovsienko, Three cocycles on ${\mathrm{Diff}}(S^1)$ generalizing the Schwarzian derivative, Internat. Math. Res. Notices 1998, No.1, 25–39.

C. Caratheodory, Theory of functions, Chelsa Pub. Co. N.Y. 1960.

K. Carne, The Schwarzian derivative for conformal maps, J. Reine Angrew. Math. [**408**]{} (1990), 10- 33.

P. Cohen, Yu. Manin and D. Zagier, Automorphic pseudo-differential operators, in Algebraic Aspects of Integrable Systems, Prog. Nonlinear Differential Equations 