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abstract: 'Let $(M,\g)$ be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of $M$ related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric $\g.$ In particular, if the manifold $(M,\g)$ is conformally flat, these derivatives vanish on the conformal group ${\mathrm{O}}(p+1,q+1),$ where $\mathrm{dim} (M)=p+q.$ This work is a continuation of [@b2; @bo2] where the Schwarzian derivative was defined on a manifold endowed with a projective connection.'
author:
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    Sofiane BOUARROUDJ[^1]\
    [ Department of Mathematics, Keio University, Faculty of Science & Technology]{}\
    [3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan. ]{}\
    [e-mail: sofbou@math.keio.ac.jp]{}
title: Conformal Schwarzian derivatives and conformally invariant quantization
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.1truein \[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Example]{} \[thm\][Remark]{} \[thm\][Definition]{}

Introduction
============

Let $S^1$ be the circle identified with the projective line ${\mathbb{RP}}^1.$ For any diffeomorphism $f$ of $S^1,$ the expression $$\label{clas}
S(f):= \frac{f'''(x)}{f'(x)}-\frac{3}{2} \left (\frac{f''(x)}{f'(x)}\right )^2,$$ where $x$ is an affine parameter on $S^1,$ is called Schwarzian derivative (see [@cara]).\
The Schwarzian derivative has the following properties:

\(i) It defines a 1-cocycle on the group of diffeomorphisms ${\mathrm{Diff}}(S^1)$ with values in differential 