by $$\begin{aligned}
\label{actsym}
f_{\d}(P)&=& f^*P\cdot (J_{f^{-1}})^{\d},\end{aligned}$$ where $J_f=|Df/Dx|$ is the Jacobian of $f,$ and $f^*$ is the natural action of ${\mathrm{Diff}}(M)$ on ${\cal S}(M).$

We then have a graduation of ${\mathrm{Diff}}(M)$-modules given by $${\cal S}_\d(M)=\bigoplus_{k=0}^\infty {\cal S}_\d^k(M),$$ where $ S_\d^k(M)$ is the space of contravariant tensor fields of degree $k$ endowed with the ${\mathrm{Diff}}(M)$-module structure (\[actsym\]).

We want to study the space of contravariant tensor fields of degree less than two, denoted by ${\cal S}_{\d,2}(M)$ (i.e. ${\cal S}_{\d,2}(M):={\cal S}_{\d}^2(M)\oplus 
{\cal S}_{\d}^1(M)\oplus {\cal S}_{\d}^0(M)$).

Conformal Schwarzian derivatives
================================

Let $(M,\g)$ be a pseudo-Riemannian manifold. Denote by $\Gamma$ the Levi-Civita connection associated with the metric $\g.$

Main definition {#enfant}
---------------

It is well known that the difference between two connections is a well-defined tensor field of type $(2,1).$ It follows therefore that the difference $$\label{ell}
\ell(f):=f^*\Gamma-\Gamma,$$ where $f\in {\mathrm{Diff}}(M),$ is a well-defined $(2,1)$-tensor field on $M$.

It is easy to see that the map $$f\mapsto \ell(f^{-1})$$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in the space of tensor fields on $M$ of type $(2,1).$

Our first main definition is the linear differential operator ${\cal A}(f)$ acting from ${\cal S}_{\d}^2(M)$ to ${\cal S}_\d^1(M)$ defined by $${\cal A} (f)_{ij}^k:=
{f^*}^{-1} \left( \g^{sk}\,\g_{ij}\nabla_s \right) 
