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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:
- |
    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
---

ł

.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 quadratics (cf. [@ki; @s]).

\(ii) Its kernel is the group of projective transformations ${\mathrm{PSL}}_2({\mathbb{R}})$.\
The aim of this paper is to propose a new approach for constructing the multi-dimensional conformal Schwarzian derivative. This approach was recently used in [@b2; @bo2] to introduce the multi-dimensional “projective” Schwarzian derivative. The starting point of our approach is the relation between the Schwarzian derivative (\[clas\]) and the space of Sturm-Liouville operators (see, e.g., [@wi]). The space of Sturm-Liouville operators is not isomorphic as a ${\mathrm{Diff}}(S^1)$-module to the space of differential quadratics. More precisely, the space of Sturm-Liouville operators is a non-trivial deformation of the space of differential quadratics in the sense of Neijenhuis and Richardson’s theory of deformation (see [@nr]), generated by the 1-cocycle (\[clas\]) (see [@ga] for more details). From this point of view, the multi-dimensional Schwarzian derivative is closely related to the modules of linear differential operators. To set out our approach, let us introduce some notation.\
Let $M$ be a smooth manifold. We consider the space of linear differential operators with arguments that are $\l$-densities on $M$ and values that are $\mu$-densities on $M.$ We have, therefore, a two parameter family of ${\mathrm{Diff}}(M)$-modules denoted by ${{\mathcal{D}}}_{\l,\mu}(M).$ The corresponding space of symbols is the space ${{\mathcal{S}}}_{\d}(M)$ of fiberwise polynomials on $T^*M$ with values in $\d$-densities, where $\d=\mu-\l.$ In general, the space ${{\mathcal{D}}}_{\l,\mu}(M)$ is not isomorphic as a ${\mathrm{Diff}}(M)$-module to the space ${{\mathcal{S}}}_{\d}(M)$ (cf. [@do; @lo2]). However, we are interested in the following two cases:\
(i) If $M:={\mathbb{R}}^n$ is endowed with a flat projective structure (i.e. local action of the group ${\mathrm{SL}}_{n+1}({\mathbb{R}})$ by linear fractional transformations) there exists an isomorphism between ${{\mathcal{D}}}_{\l,\mu}({\mathbb{R}}^n)$ and ${{\mathcal{S}}}_{\d}({\mathbb{R}}^n),$ for $\d$ generic, intertwining the action of ${\mathrm{SL}}_{n+1}({\mathbb{R}})$ (cf. [@lo2]). The multi-dimensional “projective” Schwarzian derivative was defined in [@b2; @bo2] as an obstruction to extend this isomorphism to the full group ${\mathrm{Diff}}({\mathbb{R}}^n).$\
(ii) If $M:={\mathbb{R}}^n$ is endowed with a flat conformal structure (i.e. local action of the conformal group ${\mathrm{O}}(p+1,q+1),$ where $p+q=n$), there exists an isomorphism between ${{\mathcal{D}}}_{\l,\mu}({\mathbb{R}}^n)$ and ${{\mathcal{S}}}_{\d}({\mathbb{R}}^n),$ for $\d$ generic, intertwining the action of ${\mathrm{O}}(p+1,q+1)$ (cf. [@dlo; @do]). In this paper we introduce the multi-dimensional “conformal” Schwarzian derivative in this context. Recall that in the one-dimensional case these two notions coincide in the sense that the conformal Lie algebra ${\mathrm o}(2,1)$ is isomorphic to the projective Lie algebra ${\mathrm{sl}}_2({\mathbb{R}}).$

Differential operators and symbols
==================================

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

Space of linear differential operators as a module
--------------------------------------------------

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 is given by the equation $$f_{\l,\m}(A)=f^*\circ A\circ {f^*}^{-1},
\label{Opaction}$$ where $f^*$ is the action (\[den\]) of ${\mathrm{Diff}}(M)$ on ${{\mathcal{F}}}_\l$.

By differentiating this action, one can obtain the action of the Lie algebra ${\mathrm{Vect}}(M).$

The formul[æ]{} (\[den\]) and (\[Opaction\]) do not depend on the choice of the system of coordinates.

Denote by ${\cal D}^2_{\l,\mu}(M)$ the space of second-order linear differential operators with the ${\mathrm{Diff}}(M)$-module structure given by (\[Opaction\]). The space ${\cal D}_{\l,\mu}^2(M)$ is in fact a ${\mathrm{Diff}}(M)$-submodule of ${{\mathcal{D}}}_{\l,\m}(M).$

[The space of Sturm-Liouville operators $\frac{d^2}{dx^2}+u(x): {{\mathcal{F}}}_{-1/2}\rightarrow {{\mathcal{F}}}_{3/2}$ on $S^1,$ where $u(x)\in 
{{\mathcal{F}}}_{2}$ is the potential, is a submodule of ${{\mathcal{D}}}_{-\frac{1}{2},\frac{3}{2}}^2(S^1)$ (see [@wi]). ]{}

The module of symbols
---------------------

The space of symbols, ${\mathrm{Pol}}(T^*M),$ is the space of functions on the cotangent bundle $T^*M$ that are polynomials on the fibers. This space is naturally isomorphic to the space ${\cal S}(M)$ of symmetric contravariant tensor fields on $M.$ In local coordinates $(x^i,\xi_i)$, one can write $P\in {\cal S}(M)$ in the form $$P=\sum_{l\geq 0}P^{i_1, \ldots,i_l}\xi_{i_1}\cdots \xi_{i_l},$$ with $P^{i_1, \ldots,i_l}(x)\in C^{\infty}(M).$\
We define a one parameter family of ${\mathrm{Diff}}(M)-$module on the space of symbols by $${\cal S}_{\d}(M):={\cal S}(M)\otimes {{\mathcal{F}}}_{\d}.$$ For $f\in {\mathrm{Diff}}(M)$ and $P\in {\cal S}_{\d}(M),$ in a local coordinate $(x^i)$, the action is defined 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 $${\cal A} (f)_{ij}^k:=
{f^*}^{-1} \left( \g^{sk}\,\g_{ij}\nabla_s \right) 
-\g^{sk}\,\g_{ij}\nabla_s + c\,\left(\ell(f)^k_{ij} -\frac{1}{n} 
\, \mathrm{Sym}_{i,j}\,\delta_{i}^k \,\ell(f)_{j}\right), 
\label{MultiSchwar1}$$ where $$\label{qua} 
c=2-\delta n,$$ and $\ell(f)_{ij}^k$ are the components of the tensor (\[ell\])

\[main\] [(i)]{} For all $\d \not= 2/n,$ the map $f\mapsto {\cal A}(f^{-1})$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({\cal S}_{\d}^2(M), {\cal S}_\d^1(M)).$\
[(ii)]{} The operator (\[MultiSchwar1\]) does not depend on the rescaling of the metric $\g.$ In particular, if $M:={\mathbb{R}}^n$ and $\g$ is the flat metric of signature $p-q$, this operator vanishes on the conformal group ${\mathrm{O}}(p+1,q+1).$

[**Proof.**]{} To prove (i) we have to verify the 1-cocycle condition $${\cal A}(f\circ h)={h^*}^{-1} {\cal A}(f)+ {\cal A}(h), 
\quad \mbox{for all } f,h\in {\mathrm{Diff}}(M),$$ where $h^*$ is the natural action on ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}_\d^1(M)).$ This condition holds because the first part of the operator (\[MultiSchwar1\]) is a coboundary and the second part is a 1-cocycle.

Let us proof that this 1-cocycle is not trivial for $\d\not=2/n$. Suppose that there is a first-order differential operator $A^k_{ij}=u^{sk}_{ij}\nabla_s+v^k_{ij}$ such that $$\label{cn}
{\cal A}(f)={f^*}^{-1} A-A.$$ From (\[cn\]), it is easy to see that ${f^*}^{-1} v^k_{ij}-v^k_{ij}=
(2-\delta n)\left(\ell(f)^k_{ij} -\frac{1}{n} 
\, \mathrm{Sym}_{i,j}\,\delta_{i}^k \,\ell(f)_{j}\right)$. The right-hand side of this equation depends on the second jet of the diffeomorphism $f,$ while the left-hand side depends on the first jet of $f,$ which is absurd.

For $\d=2/n,$ one can easily see that the 1-cocycle (\[MultiSchwar1\]) is a coboundary.

Let us prove (ii). Consider a metric $\tilde \g= F\cdot\g,$ where $F$ is a non-zero positive function. Denote by $\tilde {\cal A}(f)$ the operator (\[MultiSchwar1\]) written with the metric $\tilde \g.$ We have to prove that $\tilde {\cal A}(f)={\cal A}(f).$ The Levi-Civita connections associated with the metrics $\g$ and $\tilde \g$ are related by $$\label{lien}
\tilde \Gamma^k_{ij}=\Gamma^k_{ij}+\frac{1}{2F}\left (F_i \d^k_j +F_j \d^k_i
-F_t \,\g^{tk} \g_{ij}\right),$$ where $F_i=\partial_i F.$\
We need some formul[æ]{}: denote by $\ell(f)$ the tensor (\[ell\]) written with the connection $\tilde \g,$ then we have $$\begin{aligned}
\label{bebe}
\tilde \nabla_k P^{ij}&=&\nabla_k P^{ij}+\frac{1}{2F}\left(  \mathrm{Sym}_{i,j} 
P^{mi} \left(F_m \delta^j_k-F_t \,\g^{tj}\g_{km}\right) +(2-n\delta )\,
P^{ij}F_k\right),\\
\tilde \ell(f)^k_{ij}&=&\ell(f)^k_{ij}+\frac{1}{2 F\circ f}
\left(\mathrm{Sym}_{i,j} \stackrel{*}{F_i}\delta^k_j -\stackrel{*}{F}_t\, 
\stackrel{*}{\g}^{tk}\stackrel{*}{\g}_{ij}\right)
-\frac{1}{2F}\left( \mathrm{Sym}_{i,j} \,F_i\, \delta^k_j -F_t\, 
\g^{tk}\g_{ij}\right), 
\nonumber\end{aligned}$$ where $\stackrel{*}{F}_i={f^{*}}^{-1}F_i$ and $\stackrel{*}{\g}_{ij}=
{f^{*}}^{-1}\g_{ij}$ for all $P^{ij}\xi_i \xi_j\in {\cal S}_{\d}^2 (M).$\
By substituting the formul[æ]{} (\[bebe\]) into (\[MultiSchwar1\]) we get $${\cal A} (f)_{ij}^k=\tilde {\cal A} (f)_{ij}^k+
\frac{1}{2 F} (\delta n +c -2 ) \,{\g}^{sk} \,\g_{ij}\, F_s\, 
+  
\frac{1} {2 F\circ f}(2-c -\delta  n) 
\stackrel{*}{\g}^{tk} \stackrel{*}{\g}_{ij}\, \stackrel{*}{F}_t \, \,
\cdot$$ We see that ${\cal A} (f)=\tilde {\cal A} (f)$ if and only if $c=2-\delta n .$

Let us prove that the operator (\[MultiSchwar1\]) vanishes on the conformal group ${\mathrm{O}}(p+1,q+1)$ in the case when $M:={\mathbb{R}}^n$ is endowed with the flat metric $\g_0:=\mathrm{diag}(1,\ldots,1,-1,\ldots,-1)$ whose trace is $p-q.$ Any $f\in {\mathrm{O}}(p+1,q+1)$ satisfies ${{f^*}}^{-1} {\g_0}=F\cdot{\g_0},$ where $F$ is a non-zero positive function. This relation implies $$\begin{aligned}
2 \ell(f)^k_{ij} +\mathrm{Sym}_{i,j}\,\ell(f)^s_{it}\,\g_0^{tk}\,
{\g_0}_{sj}&=&\frac{1}{F}\mathrm{Sym}_{i,j}\, \partial_i F\, \delta^k_j, 
\nonumber \\ 
\mathrm{Sym}_{i,j}\, \ell(f)^s_{li} \, {\g_0}_{sj}&=&\frac{\partial_l F}{F}
{\g_0}_{ij},\nonumber \\
\ell(f)_t&=& \frac{n}{2}\frac{\partial_t F}{F}\cdot  
\nonumber\end{aligned}$$ Sibstitute these formul[æ]{} into (\[MultiSchwar1\]). Then we obtain by straightforward computation that ${\cal A}(f)\equiv 0.$\

------------------------------------------------------------------------

Suppose now that ${\rm dim }(M)>2.$

Our second main definition is the linear differential operator ${\cal B}(f)$ acting from ${\cal S}^2_{\d}(M)$ to ${\cal S}^0_\d(M)$ defined by $$\begin{aligned}
{\cal B}(f)_{ij}&=&
{f^*}^{-1} \left(\g^{st}\, \g_{ij}\nabla_s\nabla_t \right ) -\g^{st}\,\g_{ij} 
\nabla_s \nabla_t
+c_1\,\left( \ell(f)_{ij}^s -\frac{1}{n}\, 
\mathrm{Sym}_{i,j}\,\delta_{i}^s \,\ell(f)_{j} \right)\nabla_s \nonumber \\
&&+\,c_2 \, \ell(f)_i\,\ell(f)_j  +c_3 \n_{s} 
\left ( \ell(f)_{ij}^s-\frac{1}{n}
\,\mathrm{Sym}_{i,j}\, \d_{i}^s \ell_{j}(f)\right) +c_4\, \ell(f)_{ij}^s\ell(f)_s 
\nonumber  \\
&& +\,c_5\, \ell(f)_{si}^u\ell(f)_{uj}^s + c_6\left( 
 {f^{-1}}^*(R\,\g_{ij})-R\,\g_{ij}\right ),
\label{MultiSchwar2}\end{aligned}$$ where $\ell(f)$ is the tensor (\[ell\]), $R$ is the scalar curvature of the metric $\g,$ and the constants $c_1,\ldots,c_6,$ are given by $$\begin{array}{ll}
c_1=2+n(1-2\delta ), & \displaystyle c_2=\frac{(2+n(1-2\delta ))(\d- 1)}{n}, 
\nonumber \\[3mm]
c_3=\displaystyle \frac{(2+n(1-2\delta ))(\d n-2)}{n-2} , &c_4=
\displaystyle \frac{(2+n(1-2\delta ))(2\d-2)}{n-2}, \\[3mm]
c_5= \displaystyle \frac{(2+n(1-2\delta ))(1-\d)n}{n-2},& c_6=
\displaystyle \frac{n(\d -1)(n\d -2)}{(n-1)(n-2)}\cdot 
\nonumber 
\label{ga}
\end{array}$$

\[mainp\] [(i)]{} For all $\d \not= \frac{n+2}{2n},$ the map $f\mapsto {\cal B}(f^{-1})$ defines a non-trivial 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}^0_\d(M))$.

[(ii)]{} The operator (\[MultiSchwar2\]) does not depend on the rescaling of the metric $\g.$ In the flat case, this operator vanishes on the conformal group ${\mathrm{O}}(p+1,q+1).$

[**Proof.**]{} To prove that the map $f\mapsto {\cal B}_{ij}(f^{-1})$ is a 1-cocycle, one has to verify the 1-cocycle condition $${\cal B}(f\circ h)=
{h^*}^{-1}{\cal B}(f)+{\cal B}(h), \quad \mbox{for all }f,h\in {\mathrm{Diff}}(M),$$ where $h^*$ is the natural action on ${{\mathcal{D}}}({\cal S}^2_{\d}(M),{\cal S}_\d^0(M)).$ To do this, we use the formul[æ]{} $$\begin{aligned}
\label{comp}
\n_i\, f^*_{\d} P^{kl}&=&f^*_{\d}\n_i P^{kl}-\mathrm{Sym}_{k,l} \left( 
\ell(f^{-1})_{it}^k\,  f^*_{\d} \, P^{tl}\right) +\d \,\ell(f^{-1})_i\, 
f^*_{\d} P^{kl},\\[2mm]
\n_u h^* \ell(f)_{ij}^k&=&h^*\n_u \ell(f)_{ij}^k - h^*\ell(f)_{ij}^t\, 
\ell(h^{-1})_{ut}^k + \mathrm{Sym}_{i,j}\left( h^*\ell(f)_{it}^k \,
\ell(h^{-1})_{ju}^t \right), \nonumber \end{aligned}$$ for all $f,h \in {\mathrm{Diff}}(M)$ and for all $P^{kl}\xi_k\xi_l\in {{\mathcal{S}}}_{\d}^2(M).$\
Let us prove that this 1-cocycle is not trivial. Suppose that there exists an operator $B_{ij}:= u_{ij}^{st}\nabla_s \nabla_t+ v_{ij}^s\nabla_s +t_{ij}$ such that $${\cal B}(f)={f^*}^{-1} B -B.$$ It is easy to see that ${f^*}^{-1} v_{ij}^s- v_{ij}^s=
(2+n(1-2\delta))\left( \ell(f)_{ij}^s -\frac{1}{n}\, 
Sym_{i,j}\,\delta_{i}^s \,\ell(f)_{j} \right).$ The right-hand side of this relation depends on the second jet of $f,$ while the the left-hand side depends on the first jet of $f,$ which is absurd.

For $\d= \frac{n+2}{2n},$ the 1-cocycle (\[MultiSchwar2\]) is trivial: ${\cal B}(f)_{ij}={f^*}^{-1} (\g_{ij}B) -B\,\g_{ij},$ where $B:= \g^{st}\nabla_s\nabla_t -\frac{1}{4}\frac{n-2}{n-1}R$ is the so-called Yamabe-Laplace operator (see, e.g., [@besse]).\
Let us prove (ii). Consider a metric $\tilde \g:=F\cdot \g,$ where $F$ is a non-zero positive function. Denote by $\tilde {\cal B}(f)$ the operator (\[MultiSchwar2\]) written with the metric $\tilde \g.$ We have to prove that $\tilde {\cal B}(f)={\cal B}(f).$\
The proof is similar to the proof of part (ii) of Theorem (\[main\]), by means of the equation (\[lien\]), (\[bebe\]) and $$\begin{aligned}
\tilde \nabla_l \tilde \nabla_k P^{ij}&=&\nabla_l \tilde \nabla_k P^{ij}+
\frac{1}{2F} \left( (1-\d n) F_l\,\tilde\nabla_k P^{ij}- F_k\, 
\tilde \nabla_l P^{ij}+\g^{st}\, \g_{lk}\, F_s \tilde 
\nabla_t P^{ij}\right ) \nonumber \\
&& +\frac{1}{2F} \mathrm{Sym}_{i,j} \tilde \nabla_k P^{mi} \left (F_m \, \delta^j_l -
{\g}^{sj}{\g}_{ml}\, F_s\right),\nonumber\\
\tilde \nabla_l \, \tilde \ell(f)^k_{ij}&=&\nabla_l \,\tilde \ell(f)^k_{ij} - 
\frac{1}{2F}\, F_l\,\,\tilde \ell(f)^k_{ij}+\frac{1}{2F} \left( F_t \, 
\delta^k_l -\g^{sk}\, \g_{tl}\, F_s \right ) \tilde \ell(f)^t_{ij}
\nonumber \\
&&-\frac{1}{2F} \mathrm{Sym}_{i,j} \left (F_i \delta^s_l-F_m \,{\g}^{ms}
{\g}_{il}\right)\tilde \ell(f)^k_{js}\nonumber\\
\tilde R&=& \frac{1}{F} \left ( R-(n-1)\frac{1}{F} 
\left( \g^{ij}\nabla_i F_j+ (n-6)\frac{1}{4 \, F}\g^{ij}F_i 
F_j \right) \right)  \end{aligned}$$ for all $P^{ij}\xi_i\xi_j\in {\cal S}_\d^2(M),$ where $\tilde \nabla,$ $\tilde \ell(f)$ and $\tilde R$ are, the covariant derivative, the tensor (\[ell\]), and the scalar curvature associated with the metric $\tilde \g,$ respectively.

Cohomology of ${\mathrm{Vect}}(M)$ and Schwarzian derivatives
-------------------------------------------------------------

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 (\[mou\]), acts from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^0(M).$

The following two propositions follow by straightforward computation.

[(i)]{} The map $X\mapsto {\mathfrak a}^k_{ij}(X)$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_{\d}^2(M), {{\mathcal{S}}}_{\d}^1(M)).$

[(ii)]{} The operator $\mathfrak a$ does not depend on the rescaling of the metric. In the flat case, it vanishes on the Lie algebra $\mathrm{o}(p+1,q+1),$ where $p+q=n.$

[(i)]{} The map $X\mapsto {\mathfrak b}_{ij}(X)$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_{\d}^2(M), {{\mathcal{S}}}_{\d}^0(M)).$

[(ii)]{} The operator $\mathfrak b$ does not depend on the rescaling of the metric. In the flat case, it vanishes on the Lie algebra $\mathrm{o}(p+1,q+1),$ where $p+q=n.$

In section (\[va\]), we will show that the space ${{\mathcal{D}}}_{\l,\mu}^2(M)$ can be viewed as a non-trivial deformation of the module ${\cal S}_{2,\d}(M)$ in the sense of Neijenhuis and Richardson’s theory of deformation (see also [@do; @lo1]). According to the theory of deformation, the problem of “infinitesimal” deformation is related to the cohomology group $$\label{hic}
\mathrm H^1 ({\mathrm{Vect}}(M), {\mathrm{End}}({\cal S}_{2,\d}(M))\cdot$$ To compute the cohomology group (\[hic\]) we restrict the coefficients to the space of linear differential operators on ${\cal S}_{2,\d}(M),$ denoted by ${{\mathcal{D}}}({\cal S}_{2,\d}(M)).$ This space is decomposed, as a ${\mathrm{Vect}}(M)$-module, into the direct sum $${{\mathcal{D}}}({\cal S}_{2,\d}(M))=\bigoplus_{k,m=0}^{2}{{\mathcal{D}}}({\cal S}_{\d}^k(M), 
{\cal S}_{\d}^m(M)),$$ where ${{\mathcal{D}}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M))\subset 
{\mathrm{Hom}}({\cal S}_{\d}^k(M), {\cal S}_{\d}^m(M)).$

The relation between the Schwarzian derivative (\[clas\]) and the cohomology group above is as follows: recall that in the one dimensional case the space ${{\mathcal{S}}}^k_\d (S^1)$ is nothing but ${{\mathcal{F}}}_{\d-k}.$ In this case, the problem of deformation with respect to the Lie algebra ${\mathrm{sl}}_2({\mathbb{R}})$ is related to the cohomology group $$\label{ser}
{\mathrm H}^1({\mathrm{Vect}}(S^1), {\mathrm{sl}}_2 ({\mathbb{R}}); {{\mathcal{D}}}({{\mathcal{F}}}_{\d-k}, {{\mathcal{F}}}_{\d-l})),$$ where $k,l=0,1,2.$ The cohomology group (\[ser\]) was calculated in [@bo1], it is one dimension for $k=2,\, l=0,$ and zero otherwise. The (unique) non-trivial class, for $k=2$ and$l=0,$ can be integrated to the group of diffeomorphisms ${\mathrm{Diff}}(S^1);$ it is a zero-order operator given as a multiplication by the Schwarzian derivative (\[clas\]) (see [@bo1] for more details).

In the multi-dimensional case and for $\d=0,$ the first group of differential cohomology of ${\mathrm{Vect}}(M)$, with coefficients in the space ${{\mathcal{D}}}({\cal S}^k(M),{\cal S}^m(M))$ of linear differential operators from ${\cal S}^k(M)$ to ${\cal S}^m(M)$ was calculated in [@lo2]. For $n\geq2$ the result is as follows $$\label{cal}
\mathrm H^1({\mathrm{Vect}}(M),  {{\mathcal{D}}}({\cal S}^k(M),{\cal S}^m(M)))=
\left\{
\begin{array}{ll}
{\mathbb{R}}\oplus \mathrm{H}_{\mathrm{DR}}^1(M), & \mbox{if}\quad  k-m=0,\\
{\mathbb{R}},& \mbox{if} \quad k-m=1,m\not=0,\\
{\mathbb{R}},& \mbox{if} \quad k-m=2,\\
0,&\hbox{otherwise.}
\end{array}
\right.$$ We believe, by analogy for the one-dimensional case, that the “infinitesimal” multi-dimensional 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}
------------------------------------

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 is clear that a projective connection on $M$ leads to the following 1-cocycle on ${\mathrm{Diff}}(M)$: $${\cal C}(f^{-1})=
\left(
(f^{-1})^*\Pi_{ij}^k-\Pi_{ij}^k
\right)
dx^i\otimes dx^j\otimes\frac{\partial}{\partial x^k}
\label{ellp}$$ This formula is independent on the choice of the coordinate system.

By definition, the tensor field (\[ellp\]) depends only on the projective class of the connection $M.$ In particular if $\Pi \equiv 0,$ this tensor field vanishes on the projective group ${\mathrm{PSL}}_{n+1}({\mathbb{R}}).$

One can define a 1-cocycle on ${\mathrm{Diff}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}^{2}_\d(M), {{\mathcal{S}}}^{1}_\d(M))$ by contracting any symmetric contravariant tensor field with the tensor (\[ellp\]). Therefore, the operator (\[MultiSchwar1\]) can be viewed as the conformal analogue of the tensor field (\[ellp\]). In the same spirit, the operator (\[MultiSchwar2\]) can be viewed as the conformal analogue of the “projective” multi-dimensional Schwarzian derivative introduced in [@b2; @bo2].

Relation to the modules of differential operators {#SecComput}
=================================================

Conformally equivariant quantization {#tokyo}
------------------------------------

The quantization procedure explained in this paper was first introduced in [@do; @lo1]. By an equivariant quantization we mean an identification between the space of linear differential operators and the corresponding space of symbols, equivariant with respect to the action of a (finite dimension) sub-group $G\subset {\mathrm{Diff}}({\mathbb{R}}^n).$ Recall that in the one-dimensional case the equivariant quantization process was carried out for $G={\mathrm{SL}}_2({\mathbb{R}})$ in [@cmz] (see also [@ga]).

The following theorems are proven in [@do].

For all $\d \not=1,$ there exists an isomorphism $${\cal Q}_{\l,\mu}: {\cal S}_{\d}^1(M)\oplus {\cal S}_{\d}^0(M)  
\rightarrow {{\mathcal{D}}}_{\l,\mu}^1(M),$$ given as follows: for all $P=P^i\xi_i+P_0\in~{\cal S}_{\d}^1(M)
\oplus {\cal S}_{\d}^0(M),$ one can associate a linear differential operator given by $$\begin{aligned}
\label{yah1}
{\cal Q}_{\l,\mu} (P)&=&
P^{i}\n_i +\a \n_i P^{i}+P_0,\end{aligned}$$ where $$\alpha=\displaystyle \frac{\l}{1-\delta}\cdot$$ This map does not depend on the rescaling of the metric, intertwines the action of ${\mathrm{Diff}}(M).$

For $n>2$ and for all $\d \not=\frac{2}{n}, 
\frac{n+2}{2n}, \frac{n+1}{n},\frac{n+2}{n},$ there exists an isomorphism $${\cal Q}_{\l,\mu}: {\cal S}_{\d}^2(M) \rightarrow {{\mathcal{D}}}_{\l,\mu}^2(M),$$ given as follows: for all $P=P^{ij}\xi_i\xi_j \in~{\cal S}_{\d}^2(M),$ one can associate a linear differential operator given by $$\begin{aligned}
{\cal Q}_{\l,\mu} (P)&=&
P^{ij}\n_i\n_j \nonumber\\[2mm]
&&+(\a_1 \n_i P^{ij}+\a_2\,\g^{ij}\, \g_{kl}\n_i P^{kl})
\n_j 
\label{Tensor1}\\[2mm]
&&+\a_3\n_i\n_j P^{ij}+\a_4\, \g^{st}\, \g_{ij}\n_s\n_t P^{ij}
+\a_5 R_{ij}P^{ij}+\a_6 R \,\g_{ij}\,P^{ij}, 
\nonumber\end{aligned}$$ where $R_{ij}$ (resp. $R$) are the Ricci tensor components (resp. the scalar curvature) of the metric $\g,$ the constants $\a_1,\ldots,\a_6$ are given by $$\begin{array}{ll}
\a_1=\displaystyle 
\frac{2(n\l+1)}
{2+n(1-\d)}, &\a_2= \displaystyle 
\frac{n(\l+\mu-1)}
{(2+n(1-\delta ))
(2-n\d)},\\[3mm]
\a_3= \displaystyle 
\frac{n\l(n\lambda+ 1)}
{(1+n(1-\delta))(2+n(1-\delta ))}, & 
\a_5= \displaystyle
\frac{n^2\l(\mu-1)}
{(n-2)(1+n(1-\delta))} ,\\[3mm]
\a_4= \displaystyle 
\frac{n\l(n^2 \mu (2-\l-\mu)+2(n\l +1)^2-n(n+1))}
{(1+n(1-\delta))(2+n(1-\d))(2+n(1-2\d))(2-n\delta )}, & 
\a_6= \displaystyle
\frac{(n\d-2)}
{(n-1)(2+n(1-2\delta))}\, \a_5 \cdot\\[3mm]
 &\\
\end{array}$$ and has the following properties:\
(i) It does not depend on the rescaling of the metric $\g$.\
(ii) If $M={\mathbb{R}}^n$ is endowed with a flat conformal structure, this map is unique, equivariant with respect to the action of the group ${\mathrm{O}}(p+1,q+1) \subset {\mathrm{Diff}}({\mathbb{R}}^n)$.

Before to give the formula of the conformal equivariant map in the case of surfaces, let us recall an interesting approach for the multi-dimensional Schwarzian derivative for conformal mapping [@os] (see also [@ca]). First, recall that all surfaces are conformally flat. This means that every metric can be express (locally) as $$\g=F^{-1}\psi^* \g_0,$$ where $\psi$ is a conformal diffeomorphism of $M$, $F$ is a non-zero positive function and $\g_0$ is a metric of constant curvature. The Schwarzian derivative of $\psi$ is defined in [@os] as the following tensor field $$\label{osg}
S(\psi )=\frac{1}{2F}\nabla dF-\frac{3}{4F^2}\,dF\otimes dF+\frac{1}{8F^2}
\g^{-1}(dF,dF)\, \g \cdot$$

Now we are in position to give the quantization map for the case of surfaces.

For $\d \not=1,2,\frac{3}{2},\frac{5}{2},$ and for each $P=P^{ij}\xi_i\xi_j +P^i\xi_i +P_0\in {\cal S}_{\d,2}(M)$ one associates a linear differential operator given by $$\begin{aligned}
Q(P)&=&
P^{ij}\n_i\n_j \nonumber\\[2mm]
&&+(\a_1 \n_i P^{ij}+\a_2\,\g^{ij}\, \g_{kl}\n_i P^{kl})
\n_j \label{surf}\\[2mm]
&&+\a_3\n_i\n_j P^{ij}+\a_4\, \g^{st}\, \g_{ij}\n_s\n_t P^{ij}
\nonumber\\[2mm]
&& +\frac{4\l(\mu-1)}{2\d -3}\left( S(\psi)_{ij}P^{ij}+
\frac{1}{8(\d-1)} R \,\g_{ij}\,P^{ij}\right )+P_0, 
\nonumber\end{aligned}$$ where $S(\psi)$ is the tensor (\[osg\]), $R$ is the scalar curvature and the coefficients $\a_1,\ldots,\a_4$ are given as above.

[ The projectively equivariant quantization map was given in [@lo1] (see also [@b3] for the non-flat case). The multi-dimensional projective Schwarzian derivative is defined as an obstruction to extend this isomorphism to the full group ${\mathrm{Diff}}(M).$ We will show in the next section that the conformal Schwarzian derivatives defined in this paper appear as obstructions to extend the isomorphisms (\[Tensor1\]), (\[surf\]) to the full group ${\mathrm{Diff}}(M).$ ]{}

Deformation of the space of symbols ${\cal S}_{2,\d}$ {#va}
-----------------------------------------------------

The goal of this section is to explicate the relation between the 1-cocycles (\[MultiSchwar1\]), (\[MultiSchwar2\]) and the space of second-order linear differential operators ${\cal D}_{\l,\mu}^2(M).$ Since the space ${\cal D}_{\l,\mu}^2(M)$ is a non-trivial deformation of the space of the corresponding space of symbols ${\cal S}_{\d,2}(M),$ where $\d=\mu-\l$, it is interesting to give explicitly this deformation in term of the 1-cocycles (\[MultiSchwar1\]), (\[MultiSchwar2\]). Namely, we are looking for the operator $\bar f_\d={\cal Q}^{-1}_{\l,\mu} \circ f_{\l,\mu} \circ 
{\cal Q}_{\l,\mu}$ such that the diagram below is commutative $$\nonumber
\begin{CD}
{\cal S}_{\d,2}(M)
@> \bar f_\d >>
{\cal S}_{\d,2}(M)
\strut\\ 
@V{\cal Q}_{\l,\mu}VV
@VV{\cal Q}_{\l,\mu}V \strut\\
{{\mathcal{D}}}_{\l,\mu}^2 (M) @>  f_{\l,\mu} >> 
{{\mathcal{D}}}_{\l,\mu}^2 (M) \strut
\end{CD}
\label{TheDiagram}$$

For all $\d \not=\frac{2}{n},\frac{n+2}{2n},1,\frac{n+1}{n}, 
\frac{n+2}{n},$ the deformation of the space of symbols ${\cal S}_{\d,2}(M)$ by the space ${{\mathcal{D}}}_{\l,\mu}^2(M)$ as a ${\mathrm{Diff}}(M)$-module is given as follows: for all $P=P^{ij}\xi_i \xi_j +P^i\xi_i + P_0\in {\cal S}_{\d,2}(M),$ one has $$\bar f_{\d}\cdot (P  )=
T^{ij}\xi_i\xi_j+T^{i}\xi_i+T^{0},$$ where $$\begin{aligned}
T^{ij}&=&
(f_{\d}\,P)^{ij},
\nonumber\\\label{ExplAction}
T^i&=&
(f_{\d}\,P)^i\;+\;
\displaystyle
\frac{n(\mu+\l -1)}{(2+n(1-\d ))(2-n\delta)}\,{\cal A}^i_{kl}(f^{-1})
(f_{\d}\,P)^{kl},\\[2mm]
\displaystyle
T^0&=&
(f_\d\, P)_0
\;-\;
\displaystyle
\frac{n \l(\mu -1)}{(2+n(1-2\d))(1-\d )(1+n(1-\d))}\,{\cal B}_{kl}(f^{-1})
(f_{\d}\, P)^{kl},\nonumber\end{aligned}$$ and $f_\d$ is the action (\[actsym\]). \[MainAct\]

[**Proof.**]{} The proof is a simple computation using (\[comp\]) and the formul[æ]{} $$\begin{aligned}
\n_i\n_j \,{f^*_{\d}}^{-1} P^{kl}&=&{f^*_{\d}}^{-1} \n_i \n_j\, P^{kl} - 
\mathrm{Sym}_{l,k}\left (
{f^*_{\d}}^{-1} \n_i\, P^{tl} \;\ell(f)_{tj}^k \right) + 
{f^*_{\d}}^{-1} \n_u\, P^{kl} \; \ell(f)_{ij}^u \nonumber \\
&& +\d \left ({f^*_{\d}}^{-1} \n_i\, P^{kl} \;\ell(f)_{j}\right) 
-\mathrm{Sym}_{k,l} \left( \n_j \ell(f)^k_{it}\; {f^*_{\d}}^{-1} 
P^{tl}\right) \\
&&+\d \n_j \ell(f)_i \; {f^*_{\d}}^{-1} P^{kl} 
-\mathrm{Sym}_{k,l}\left( \ell(f)_{it}^k \n_j \, {f^*_{\d}}^{-1} 
P^{tl}\right) +\d \ell(f)_i\,\n_j {f^*_{\d}}^{-1}P^{kl}
\nonumber\\[2mm]
\n_{i} \, f^* \phi&=& f^* \n_i \phi +\l \,\ell(f^{-1})_{i}\,  f^* \phi
\nonumber\\[2mm]
\n_j \n_{i} \, f^* \phi&=& f^* \n_j \n_i \phi +\ell(f^{-1})_{ji}^t \, f^* 
\n_t \phi +\l \mathrm{Sym}_{i,j} \,  \ell(f^{-1})_j\, f^* \n_i \phi 
\nonumber\\[2mm]
&& +\left (\l \n_j\, \ell(f^{-1})_{i}
+ \l^2 \ell(f^{-1})_{j}\, \ell(f^{-1})_{i}\right) \, f^* \phi
\nonumber\\[2mm]
f^* R_{jk}-R_{jk}&=& \n_i\, \ell(f^{-1})^i_{jk}-\n_j \, \ell(f^{-1})_k 
- \ell(f^{-1})^m_{sj}\, \ell(f^{-1})^s_{km}+
\ell(f^{-1})_m \, \ell(f^{-1})^m_{jk},\nonumber\end{aligned}$$ for all $\phi\in {{\mathcal{F}}}_\l$ and for all $f\in {\mathrm{Diff}}(M),$ where $R_{ij}$ are the Ricci tensor components.\

------------------------------------------------------------------------

The 1-cocycle ${\cal A},$ ${\cal B}$ and the conformally invariant quantization {#new}
-------------------------------------------------------------------------------

We will show in this section that the quantization procedure is not invariant if one consider two metrics conformally equivalent. The obstruction of the invariance is given by the 1-coycles ${\cal A}$ and ${\cal B}.$

Given two conformally equivalent metrics $\g,$ $\tilde \g.$ Denote by $\n$, $\cal A$ and $\cal B,$ the covariant derivative, the 1-cocycles (\[MultiSchwar1\]) and (\[MultiSchwar2\]) written with the metric $\g,$ respectively. We have

The quantization map ${\cal Q}_{\l,\mu}^{\g}:$ ${\cal D}^1_{\l,\mu}(M)\rightarrow {\cal S}^1_\d(M)\oplus {\cal S}^0_\d (M)$ defined in (\[yah1\]) depend only on the conformal class of the metric $\g.$

[**Proof.**]{} Let $\tilde \g$ be another metric conformally equivalent to $\g. $ That means that there exists a diffeomorphism $\psi : (M,\tilde \g )\rightarrow (M,\g)$ and a non-zero positive function $F$ such that (locally) $$\tilde \g=F^{-1}\cdot \psi^* \g.$$ The Levi-Civita of the two connections are related by $${\Gamma^k_{ij}}^{ \g}= {\Gamma^k_{ij}}^{\tilde \g}+
\frac{1}{2 F} \left (\partial_i F\, 
\d_j^k 
+\partial_j F\, \d_i^k-\partial_s F\, \tilde \g^{sk}\tilde \g_{ij}\right ) -
\ell(\psi^{-1})^k_{ij},$$ where $\partial_i F=F_i$ and $\ell(\psi^{-1})^k_{ij}$ are the components of the tensor (\[ell\]). This equation implies that $$\begin{array}{lcl}
\n^{\g}_i\phi&=&\n^{\tilde \g}_i\phi-\displaystyle \frac{\l n}{2} 
\displaystyle \frac{F_i}{F}+
\l \,\ell(\psi^{-1})_i \, \phi,\\[2mm]
\n^{\g}_i P^i &=&\n^{\tilde \g}_i P^i+\displaystyle 
\frac{n(1-\d)}{2}\frac{F_i P^i}{F}-(1-\d) \ell(\psi^{-1})_{i}P^i,
\end{array}$$ for all $\phi\in {{\mathcal{F}}}_\l$ and for all $P^i\xi_i\in {{\mathcal{S}}}_{\d}^1(M).$\
Substitute these formul[æ]{} into (\[yah1\]) we see that ${\cal Q}^{\tilde \g}_{\l,\mu}={\cal Q}^{\g}_{\l,\mu}.$

The quantization map ${\cal Q}_{\l,\mu}^{\g}:$ ${\cal D}^2_{\l,\mu}(M)\rightarrow {\cal S}^2_\d(M)$ defined in (\[Tensor1\]) has the property $${\cal Q}_{\l,\mu}^{\tilde \g} (P)={\cal Q}_{\l,\mu}^{\g} (P)+d_1 \, 
{\cal A}^{s}(\psi^{-1})
(P )\nabla_s + d_2 \,\nabla_s \left ( {\cal A}^{s}(\psi^{-1})
(P) \right ) +d_3 \, {\cal B}(\psi^{-1})(P),$$ for all $P\in {\cal S}_{\d}^2(M),$ where the constants $d_1,d_2$ and $d_3$ are given by $$\begin{aligned}
d_1&=&\frac{n(\l+\mu-1)}{(2+n(1-\d))(2-n\d))},  \quad 
\displaystyle d_2=\frac{n\l(\l+\mu-1)}{(2+n(1-\d))(2-n\d)(1-\d)}, \nonumber 
\\[1mm]
d_3&=&\displaystyle \frac{n \l(\mu -1)}{(2+n(1-2\d))(\d -1)(1+n(1-\d))} \cdot 
\nonumber \end{aligned}$$

[**Proof.**]{} The proof involves the calculation of $\nabla_i ^{\tilde \g}\nabla_j ^{\tilde \g} \phi,$ $\nabla_i ^{\tilde \g}\nabla_j ^{\tilde \g} P^{kl}$ and $R^{\tilde \g}$ wich is straightforward but quite complicated.

[ The system $d_1=d_2=d_3=0$ admits a unique solution: $(\l,\mu)=(0,1).$ The value of $\d=1$ is called “resonant”. In this case, the quantization map is not unique; there exists a one-parameter family of such isomorphism (see [@do] for more dtails.) ]{}

For all $f\in {\mathrm{Diff}}(M)$ and for all conformal map $\psi: (M,\tilde \g)
\rightarrow (M,\g)$, one has

[(i)]{} ${\cal A}^{\tilde \g}(f)=\psi^* {\cal A}^{\g}
(\psi \circ f\circ \psi^{-1}),$

[(ii)]{} ${\cal B}^{\tilde \g}(f)=\psi^* {\cal B}^{\g} 
(\psi \circ f\circ \psi^{-1}).$

[**Proof.**]{} Straightforward computation.

For all conformal map $\psi (M,\tilde \g)
\rightarrow (M,\g)$ one has

[(i)]{} $\tilde {\cal A}^{\tilde \g}(\psi)=-{\cal A}^{\g}(\psi^{-1} ),$

[(ii)]{} $\tilde {\cal B}^{\tilde \g}(\psi)=-{\cal B}^{\g}(\psi^{-1} )\cdot$

[ The Corollary above shows that for a conformal map $\psi,$ the 1-cocycle $\cal B$ is still a second-order differential operator and then does not coincide with the Schwarzian derivative (\[osg\]) defined by Osgood and Stow. ]{}

Appendix
========

We will give a formula for the Schwarzian derivative for the case of surfaces. As explained in section (\[tokyo\]), all surfaces are conformally flat. That means that every metric can be express (locally) as $$\g=F^{-1}\psi^* \g_0,$$ where $\psi$ is a conformal diffeomorphism of $M$, and $F$ is a non-zero positive function, $\g_0$ is a metric of constant curvature.

The explicit formula of the Schwarzian derivative in the case of surfaces is: the following $$\begin{aligned}
{\cal B}'_2(f)_{ij}&=&
{f^*}^{-1} \left( {\g}^{st}\, {\g}_{ij}\nabla_s\nabla_t \right ) 
-\g^{st}\,\g_{ij}\nabla_s \nabla_t +4(1-\delta)\left( \ell(f)_{ij}^s 
-\frac{1}{2}\,Sym_{i,j}\, 
\delta_{i}^s \,\ell(f)_{j} \right)\nabla_s \nonumber \\[2mm]
&&+4 (1-\d)^2 \ell(f)_s \left ( \ell(f)_{ij}^s -\frac{1}{4}
Sym_{i,j}\, \delta_i^s \, \ell(f)_j\right) + 
2(\d-2)(1-\d) \,Sym_{i,j}\,\n_{j}\ell_{i}(f) \nonumber\\[2mm]
&&+8\,(\d-1)^2 \left( 
 {f^{-1}}^*(S(\psi)_{ij})-S(\psi)_{ij}+\frac{1}{2}
\n_{s}\ell(f)_{ij}^s \right )+(\d-1)
 \left ({f^{-1}}^*(R\,\g_{ij})-R\,\g_{ij}\right ),
\nonumber
\label{MultiSchwar3}\end{aligned}$$ where $S(\psi)$ is the derivative (\[osg\]), $\ell(f)$ is the tensor (\[ell\]), $R$ is the scalar curvature of the metric $\g,$ is a differential operator from from ${{\mathcal{S}}}_{\d}^2(M)$ to ${{\mathcal{S}}}_{\d}^0(M).$

Theorem (\[mainp\]) remains true for this operator.

[*Acknowledgments*]{}. It is a pleasure to acknowledge numerous fruitful discussions with Prof. V. Ovsienko. I am grateful to, the referee for his pertinent remarks, Prof. Y. Maeda and Keio University for their hospitality.

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[^1]: Research supported by the Japan Society for the Promotion of Science.
