(\[mou\]), acts from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^0(M).$

The following two propositions follow by straightforward computation.

[(i)]{} The map $X\mapsto {\mathfrak a}^k_{ij}(X)$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_{\d}^2(M), {{\mathcal{S}}}_{\d}^1(M)).$

[(ii)]{} The operator $\mathfrak a$ does not depend on the rescaling of the metric. In the flat case, it vanishes on the Lie algebra $\mathrm{o}(p+1,q+1),$ where $p+q=n.$

[(i)]{} The map $X\mapsto {\mathfrak b}_{ij}(X)$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_{\d}^2(M), {{\mathcal{S}}}_{\d}^0(M)).$

[(ii)]{} The operator $\mathfrak b$ does not depend on the rescaling of the metric. In the flat case, it vanishes on the Lie algebra $\mathrm{o}(p+1,q+1),$ where $p+q=n.$

In section (\[va\]), we will show that the space ${{\mathcal{D}}}_{\l,\mu}^2(M)$ can be viewed as a non-trivial deformation of the module ${\cal S}_{2,\d}(M)$ in the sense of Neijenhuis and Richardson’s theory of deformation (see also [@do; @lo1]). According to the theory of deformation, the problem of “infinitesimal” deformation is related to the cohomology group $$\label{hic}
\mathrm H^1 ({\mathrm{Vect}}(M), {\mathrm{End}}({\cal S}_{2,\d}(M))\cdot$$ To compute the cohomology group (\[hic\]) we restrict the coefficients to the space of linear differential operators on ${\cal S}_{2,\d}(M),$ denoted by ${{\mathcal{D}}}({\cal S}_{2,\d}(M)).$ This space is decomposed, as a ${\mathrm{Vect}}(M)$-module, into the direct sum $${{\mathcal{D}}}({\cal S}_{2,\d}(M))=\bigoplus_{k,m=0}^{2}{{\mathcal{D}}}({\cal S}_{\d}^k(M), 
{\cal S}_{\d}^m(M)),$$ where 