$\Gamma$ the Levi-Civita connection associated with the metric $\g.$

Space of linear differential operators as a module
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We denote the space of tensor densities on $M$ by ${{\mathcal{F}}}_{\l}(M),$ or ${{\mathcal{F}}}_{\l}$ for simplify. This space is nothing but the space of sections of the line bundle $(\wedge^n T^* M)^{\otimes \l}.$ One can define in a natural way a ${\mathrm{Diff}}(M)$-module structure on it: for $f\in {\mathrm{Diff}}(M)$ and $\phi \in{\cal F}_{\l},$ in a local coordinates $(x^i)$, the action is given by $$\label{den}
f^*\phi=\phi\circ f^{-1}\cdot ( {J_{f^{-1}}})^{\lambda},$$ where $J_f=\left |Df/Dx \right |$ is the Jacobian of $f$.\
By differentiating this action, one can obtain the action of the Lie algebra of vector fields ${\mathrm{Vect}}(M).$

[ ${{\mathcal{F}}}_0= C^{\infty}(M), \, {{\mathcal{F}}}_1= \Omega^{n}(M)$ (space of differential $n$-forms). ]{}

Let us recall the definition of a covariant derivative on densities. If $\phi \in {{\mathcal{F}}}_\l,$ then $\nabla \phi \in \Omega^1 (M)\otimes {{\mathcal{F}}}_\l$ given in a local coordinates by $$\nabla_i \phi= \partial_i \phi-\l \Gamma_i \phi,$$ with $\Gamma_i= \Gamma_{ti}^t.$ (Here and bellow summation is understood over repeated indices).

Consider now ${\cal D}_{\l,\mu}(M),$ the space of linear differential operators acting on tensor densities $$A:{{\mathcal{F}}}_\l\to{{\mathcal{F}}}_\m.\nonumber
\label{Conv}$$ The action of ${\mathrm{Diff}}(M)$ on ${{\mathcal{D}}}_{\l,\mu}(M)$ depends on the two parameters $\l$ and $\m$. This action 