derivative, the tensor (\[ell\]), and the scalar curvature associated with the metric $\tilde \g,$ respectively.

Cohomology of ${\mathrm{Vect}}(M)$ and Schwarzian derivatives
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We will give here the infinitesimal 1-cocycle associated with the 1-cocycles ${\cal A}$ and ${\cal B}$. First, let us recall the notion of a Lie derivative of a connection. For each $X \in  {\mathrm{Vect}}(M),$ the Lie derivative $$\label{mou}
L_X \nabla :=(Y,Z)\mapsto [X,\n_YZ]-\n_{[X,Y]} Z- \n_{Y} [X,Z]$$ of $\n$ is a well-known symmetric $(2,1)$-tensor field. The map $$X\mapsto L_X {\nabla}$$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in the space of symmetric $(2,1)$-tensor fields on $M.$\
The linear differential operator ${\mathfrak a}$ defined by $${\mathfrak a}^k_{ij}(X):=L_X \left 
(\g^{sk}\, \g_{ij}\, \nabla_s\right) 
 +c \left ((L_X \nabla)^k_{ij} 
-\frac{1}{n}\mathrm{Sym}_{i,j}\delta_j^k \, (L_X \nabla)_i\right),$$ where the constant $c$ is as in (\[qua\]) and $L_X \n$ is the tensor (\[mou\]), acts from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^1(M).$ The linear differential operator ${\mathfrak b}$ defined by $$\begin{aligned}
\label{chr}
{\mathfrak b}_{ij}(X)&:=&L_X \left (\g^{st}\, \g_{ij}\, \nabla_s\nabla_t \right) 
 +c_1  \left ((L_X \nabla)^k_{ij} 
-\frac{1}{n}\mathrm{Sym}_{i,j}\delta_j^k \, (L_X \nabla_i)\right)
\nabla_k\nonumber\\
&& +c_2 \nabla_k \left ( (L_X \nabla)^k_{ij} 
-\frac{1}{n}\mathrm{Sym}_{i,j}\delta_j^k \, (L_X \nabla)_i \right )
+c_6 L_X \left ( R \, \g_{ij}\right ),\end{aligned}$$ where the constants $c_1,c_2$ and $c_6$ are as in (\[ga\]) and $L_X(\n)$ is the tensor 