Schwarzian derivative is a cohomology class in the cohomology group above for $k-m=2.$ This class is nothing but the operator ${\mathfrak b}$ defined in (\[chr\]).

Comparison with the projective case  {#connex}
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Let $M$ be a manifold of dimension $n.$ Fix a symmetric affine connection $\Gamma$ on $M$ (here $\Gamma$ is any connection not necessarily a Levi-Civita one). Let us recall the notion of projective connection (see [@kn]).

A projective connection is an equivalent class of symmetric affine connections giving the same unparameterized geodesics.

Following [@kn], the symbol of the projective connection is given by the expression $$\label{connection}
\Pi_{ij}^k=\G_{ij}^k-\frac{1}{n+1}\left (\delta_i^k \G_{j}+\delta_j^k \G_{i}
\right ),$$ where $\Gamma_{ij}^k$ are the Christoffel symbols of the connection $\Gamma$ and $\Gamma_i=\Gamma_{ij}^j.$\
Two affine connection $\Gamma$ and $\tilde \Gamma$ are projectively equivalent if the corresponding symbols (\[connection\]) coincide.\
A projective connection on $M$ is called *flat* if in a neighborhood of each point there exists a local coordinate system $(x^1,\ldots,x^n)$ such that the symbols $\Pi_{ij}^k$ are identically zero (see [@kn] for a geometric definition). Every flat projective connection defines a projective structure on $M$.\
Let $\Pi$ and $\tilde \Pi$ be two projective connections on $M.$ The difference $\Pi-\tilde \Pi$ is a well-defined $(2,1)$-tensor field. Therefore, it 