-1)(n\d -2)}{(n-1)(n-2)}\cdot 
\nonumber 
\label{ga}
\end{array}$$

\[mainp\] [(i)]{} For all $\d \not= \frac{n+2}{2n},$ the map $f\mapsto {\cal B}(f^{-1})$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}^0_\d(M))$.

[(ii)]{} The operator (\[MultiSchwar2\]) does not depend on the rescaling of the metric $\g.$ In the flat case, this operator vanishes on the conformal group ${\mathrm{O}}(p+1,q+1).$

[**Proof.**]{} To prove that the map $f\mapsto {\cal B}_{ij}(f^{-1})$ is a 1-cocycle, one has to verify the 1-cocycle condition $${\cal B}(f\circ h)=
{h^*}^{-1}{\cal B}(f)+{\cal B}(h), \quad \mbox{for all }f,h\in {\mathrm{Diff}}(M),$$ where $h^*$ is the natural action on ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}_\d^0(M)).$ To do this, we use the formul[æ]{} $$\begin{aligned}
\label{comp}
\n_i\, f^*_{\d} P^{kl}&=&f^*_{\d}\n_i P^{kl}-\mathrm{Sym}_{k,l} \left( 
\ell(f^{-1})_{it}^k\,  f^*_{\d} \, P^{tl}\right) +\d \,\ell(f^{-1})_i\, 
f^*_{\d} P^{kl},\\[2mm]
\n_u h^* \ell(f)_{ij}^k&=&h^*\n_u \ell(f)_{ij}^k - h^*\ell(f)_{ij}^t\, 
\ell(h^{-1})_{ut}^k + \mathrm{Sym}_{i,j}\left( h^*\ell(f)_{it}^k \,
\ell(h^{-1})_{ju}^t \right), \nonumber \end{aligned}$$ for all $f,h \in {\mathrm{Diff}}(M)$ and for all $P^{kl}\xi_k\xi_l\in {{\mathcal{S}}}_{\d}^2(M).$\
Let us prove that this 1-cocycle is not trivial. Suppose that there exists an operator $B_{ij}:= u_{ij}^{st}\nabla_s \nabla_t+ v_{ij}^s\nabla_s +t_{ij}$ such that $${\cal B}(f)={f^*}^{-1} B -B.$$ It is easy to see that ${f^*}^{-1} v_{ij}^s- v_{ij}^s=
(2+n(1-2\delta))\left( \ell(f)_{ij}^s -\frac{1}{n}\, 
Sym_{i,j}\,\delta_{i}^s \,\ell(f)_{j} \right).$ The right-hand side of this relation depends on the second jet of $f,$ while the the left-hand 