POISSON GEOMETRY \ centerlineBANACH CENTER{POISSON PUBLICATIONS GEOMETRY } comma VOLUME 5 1 INSTITUTE OF \noindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 5 1 WARSZAWA 2 0 0 \ centerlineSCHWARZIAN{INSTITUTE OF MATHEMATICS RELATED TO} MODULESPOISSON GEOMETRY OF DIFFERENTIALBANACH CENTER .. OPERATORS PUBLICATIONS , VOLUME 5 1 \ centerlineON A LOCALLY{POLISH PROJECTIVE ACADEMY OF MANIFOLD SCIENCESINSTITUTE} OF MATHEMATICS S period .. B OUARROUDJ .. and V periodPOLISH .. Yu ACADEMY period .. OF OVSIENKO SCIENCES \ centerlineCentre de Physique{WARSZAWA Th acute-e 2 0 0 orique} WARSZAWA 2 0 0 CPT hyphen CNRSSCHWARZIAN comma Luminy Case DERIVATIVE 907 RELATED TO MODULES \ centerlineF hyphen 1{ 3288SCHWARZIAN Marseille Cedex DERIVATIVEOF 9 DIFFERENTIAL comma RELATED France TO MODULES OPERATORS} E hyphen mail : so f-b ouON at cpt A period LOCALLY univ hyphen PROJECTIVE mrs period r-f sub MANIFOLD comma ovsienko at cpt period univ hyphen mrs\ centerline period fr {OF DIFFERENTIALS .\ Bquad OUARROUDJOPERATORS and} V . Yu . OVSIENKO Abstract period We introduce a 1 hyphenCentre cocycle de onPhysique the group Th ofe´ orique diffeomorphisms Diff open parenthesis M closing parenthesis\ centerline of a{ON smooth A LOCALLY PROJECTIVECPT MANIFOLD - CNRS , Luminy} Case 907 manifold M endowed with a projective connectionF - 1 3288 Marseille period This Cedex cocycle 9 , France represents a nontrivial cohomol hyphen \ centerline {S. \quad B OUARROUDJ \quad and V . \quad Yu . \quad OVSIENKO } ogy class of DiffE open - mail parenthesis : so f − b Mou closing@ cpt parenthesis . univ - mrs related.r − f to, ovsienko the Diff open@ cpt parenthesis . univ - mrs M . fr closing parenthesis hyphen modules ofAbstract second order . We linear introduce differential a 1 - cocycle operators on the group of diffeomorphisms Diff (M) of a smooth \ centerlineon M period{manifold InCentre the oneM de hyphenendowed Physique dimensional with a Th projective case $ \acute comma { thise} . cocycle$ This orique cocycle coincides represents} with athe nontrivial Schwarzian cohomol derivative - comma while comma ogy class of Diff (M) related to the Diff (M)− modules of second order linear differential operators \ centerlinein the multi{on hyphenCPTM.−In dimensionalCNRS the one , - Luminy dimensional case comma Case case it ,907 represents this} cocycle its coincides natural with and the new Schwarzian generalization derivative period , while This work is a continuation, of in open the multi square - dimensional bracket 3 closing case , it square represents bracket its natural where and the new same generalization problems have . This been work treated is a in the one hyphen\ centerline dimensional{continuationF − case1 3288 period of [ 3 Marseille ] where the same Cedex problems 9 , have France been treated} in the one - dimensional case . 1 period .. Introduction 1 . Introduction \ centerline1 period 1 period{E − mail .. The1 .: classical 1 so . $ The Schwarzian f− classicalb$ ou derivative Schwarzian $@$ period derivative cpt Consider . univ . Consider the− mrs group the $Diff group . open Diff r− parenthesisf (S1{) of, }$ S to ovsienko the power$@$ of 1 cpt closing .diffeomor univ parenthesis− - mrs of diffeomor. f r } hyphen phisms of thephisms circle preservingof the circle its preserving orientation its period orientation Identifying . Identifying S to theS power1 with ofR 1P with1, fix R an P affine to the pa power - of 1 comma fix\ hspace an affine∗{\ paframeter ihyphen l l } Abstractx on S1 such . We that introduce the natural a PSL 1 − (2,cocycleR)− action on is giventhe group by the linear of diffeomorphisms - fractional Diff $(rameter M x on)$ S to ofasmooth the power of 1 such that the natural PSL open parenthesis 2 comma R closing parenthesis hyphen action is given by the linear hyphen fractional transformations : \noindentLine 1 transformationsmanifold : $ Line M $ 2 x endowed rightax arrow+ b with ax plus a b projective divideda b by cx connection plus d sub comma . This where cocycle Row 1 a b represents Row 2 c d a nontrivial cohomol − x → where ∈ SL(2, R). (1.1) . in SL open parenthesis 2 comma R closingcx + parenthesisd , periodc open d parenthesis 1 period 1 closing parenthesis \noindent ogy class of Diff $( M )$ related to the Diff $( M ) − $ modules of second order linear differential operators The classicalThe Schwarzian classical derivativeSchwarzian is derivative then given is by then : given by : Line 1 S open parenthesis f closing parenthesis = parenleftbigg f to the power of prime prime prime open parenthesis \noindent on $M .$ Intheone − dimensional case , this cocycle coincides with the Schwarzian derivative , while , x closing parenthesis divided by f to the power off 000 prime(x) open3 f 00( parenthesisx) x closing parenthesis minus 3 divided by 2 in the multi − dimensional caseS(f) ,= ( it represents− ( )2)( itsdx natural)2, (1.2) and new generalization . This work is a parenleftbigg f to the power of prime prime openf parenthesis0(x) 2 f 0 x(x closing) parenthesis divided by f to the power of prime opencontinuation parenthesis x of closing [ 3 parenthesis ] where parenrightbiggthe same problems 2 parenrightbigg have been open treatedparenthesis in dx the closing one parenthesis− dimensional to the case . wheref ∈ Diff(S1). power of 2 comma open parenthesis 1 period 2 closing parenthesis Line 2 where f in Diff open parenthesis S to the power of\ centerline 1 closing parenthesis{1 . \quad period1 . 2Introduction . The Schwarzian} derivative as a 1 - cocycle . It is well known that the 1 period 2 periodSchwarzian .. The Schwarzian derivative as a 1 hyphen cocycle period It is well known that the Schwarzian \ hspacederivative∗{\ canfderivative i l l be}1 intrinsically . can1 . be\quad intrinsically definedThe as the classical defined unique as 1 the hyphen Schwarzianunique cocycle 1 - cocycle derivative on Diffon open Diff parenthesis ( .S1 Consider) with values S to the the in power group of Diff 1 closing$ ( parenthesis S ˆ{ 1the} withspace)values $ of quadratic of in diffeomor differentials− on S1, equivariant with respect to the M o¨ b ius group the space of quadratic differentials on S to the power of 1 comma equivariant with respect to the M dieresis-o b ius group\noindent phisms of the circle preserving its orientation . Identifying $ S ˆ{ 1 }$ with $ Rhline P ˆ{ 1 } , $ fix an affine pa − 2000 Mathematics Subject Classification : Primary 1 7 B 56 comma 1 7 B 66 comma 1 3 N 1 0 semicolon Secondary 81\noindent T 70 periodrameter2000 Mathematics $x$ on Subject $Sˆ Classification{ 1 }$: suchPrimary that 1 7 B the 56 , 1 naturalPSL 7 B 66 , 1 3 N 1 0 $ ; Secondary ( 2 , R ) The− paper$ action is81 in T final 70 is . form The given and paper byno is version in the final linear ofform it will and− benof versionpublishedr a c t i o of n it a elsewhere l will be published period elsewhere . open square bracket 1 5 closing square bracket \ [ \ begin { a l i g n e d } transformations : \\ [15] x \rightarrow \ f r a c { ax + b }{ cx + d } { , } where \ l e f t (\ begin { array }{ cc } a & b \\ c & d \end{ array }\ right ) \ in SL(2,R).(1.1 ) \end{ a l i g n e d }\ ]

\noindent The classical Schwarzian derivative is then given by :

\ [ \ begin { a l i g n e d } S ( f ) = ( \ f r a c { f ˆ{\prime \prime \prime } ( x ) }{ f ˆ{\prime } ( x ) } − \ f r a c { 3 }{ 2 } ( \ f r a c { f ˆ{\prime \prime } ( x ) }{ f ˆ{\prime } ( x ) } ) 2 ) ( dx ) ˆ{ 2 } , ( 1 . 2 ) \\ where f \ in Di f f ( S ˆ{ 1 } ). \end{ a l i g n e d }\ ]

\ hspace ∗{\ f i l l }1 . 2 . \quad The Schwarzian derivative as a 1 − cocycle . It is well known that the Schwarzian

\noindent derivative can be intrinsically defined as the unique 1 − cocycle on Diff $ ( S ˆ{ 1 } ) $ with values in the space of quadratic differentials on $ S ˆ{ 1 } , $ equivariant with respect to the M $ \ddot{o} $ b i u s group

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

2000 Mathematics Subject Classification : Primary 1 7 B 56 , 1 7 B 66 , 1 3 N 1 0 ; Secondary 81 T 70 . The paper is in final form and no version of it will be published elsewhere .

\ [ [ 1 5 ] \ ] 1 6 .. S period BOUARROUDJ AND V period YU period OVSIENKO \noindentPSL open parenthesis1 6 \quad 2 commaS . BOUARROUDJ R closing parenthesis AND V subset . YU Diff . OVSIENKO open parenthesis S to the power of 1 closing parenthesis comma cf period .... open square bracket 2 comma 6 closing square bracket period That means comma the map open parenthesis\noindent 1PSL period $ 2 closing ( 2 parenthesis , R satisfies ) the\subset following$ two D i f f $ ( S ˆ{ 1 } ) , $ c f . \ h f i l l [ 2 , 6 ] . That means , the map ( 1 . 2 ) satisfies the following two Line 1 conditions : Line 2 S open parenthesis f circ g closing parenthesis = g to the power of * S open parenthesis f closing\ [ \ begin parenthesis{ a l i1 g 6n e plus d }S . S BOUARROUDJconditions open parenthesis AND : V g .\\ closing YU . OVSIENKO parenthesis comma open parenthesis 1 period 3 closing parenthesis 1 Swhere ( f to f thePSL power\ (2circ, R of) ⊂ *Diff isg the (S ) natural), cf = . Diff [ 2g ,open ˆ 6{ ] . ∗ parenthesis That } S(f)+S(g),( means S , to the the map power ( 1 . of 2 1 )closing satisfies parenthesis the following hyphen action on1 the . space 3 oftwo quadratic ) \end{ differentialsa l i g n e d }\ and] Equation: open parenthesis 1 period 4 closing parenthesis .. S open parenthesis f closing parenthesis = S open paren- thesis g closing parenthesis comma g open parenthesis x closing parenthesisconditions = : af open parenthesis x closing parenthesis ∗ plus\noindent b dividedwhere by cf open $ parenthesis f ˆ{ ∗ }$ x closing isS the( parenthesisf ◦ naturalg) = g S plus(f Diff) + dS sub(g) $ period, ((1.3) Sˆ{ 1 } ) − $ action on the space of quadratic differentials and Moreover commawhere thef ∗ is Schwarzian the natural derivative Diff (S1) is− characterizedaction on the byspace open of quadraticparenthesis differentials 1 period 3 and closing parenthesis and open\ begin parenthesis{ a l i g n ∗} 1 period 4 closing parenthesis period S(f)=S(g),g(x)=1 period 3 period .. Relation to the module of s econd order differential operators\ f r period a c { a The f Schwarzian ( x ) + b }{Morec f to the ( power x of ) derivative + dprecisely} { . sub}\ commatag ∗{$ to ( the power 1af .of(x) appeared + 4b ) consider $} to the power of in the the to \end{ a l i g n ∗} S(f) = S(g), g(x) = (1.4) the power of classical space literature sub of Sturm hyphen sub Liouvillecf(x to) + thed . power of in closed relation with operators sub : A sub u differential = minus 2 parenleftbig d divided by dx operators parenrightbig 2 plus u open parenthesis x \noindent MoreoverMoreover , , the the Schwarzian Schwarzian derivative derivative is characterized is characterized by ( 1 . 3 ) and ( 1 by . 4 ( ) .1 . 3 ) and ( 1 . 4 ) . closing parenthesis sub comma1 . to 3 the . power Relation of period to the module of s econd order differential operators . The where u openSchwarzian parenthesis x closing parenthesis in C to the power of infinity open parenthesis S to the power of 1 closing\ hspace parenthesis∗{\ f i l l } comma1 . 3 the . \ actionquad ofRelation Diff open parenthesis to the module S to the of power s econd of 1 closing order parenthesis differential on this space operators is . The Schwarzian given by f open parenthesis A sub u closing parenthesis = A sub v with \ [ More ˆ{ d e r i vderivative a t i v e } p rappeared e c i s e l y ˆ{ appearedinthe classical} { , } c o nin s i d e r ˆ{ in the } the ˆ{ c l a s s i c a l } . Equation: open parenthesis 1 period 5 closing parenthesis .. v = u circ f to the power of minus 1 times open parenthesis d More precisely, consider the spaceliteratureofSturm−Liouvilleclosedrelationwithoperators:Au differential=−2( operators)2+u(x), fspace to the power literature of minus 1 to{ theo f power Sturm of prime− closing }ˆ{ parenthesisin } { L toi ou the v i powerl l e } of 2closed plus f to the relation power of prime with prime{ o p e r a t o r s } { : dx A { u }} di ff erential∞ 1 { = − 2 ( 1 \ f r a c { d }{ dx }} o p e r a t o r s { ) } 2 + u prime open parenthesiswhere u( x) closing∈ C ( parenthesisS ), the action divided of Diff by ( fS to) theon this power space of prime is given open by f parenthesis(Au) = Av xwith closing parenthesis minus( x 3 divided ) ˆ{ by. 2} parenleftbigg{ , }\ ] f to the power of prime prime open parenthesis x closing parenthesis divided by f to 000 00 the power of prime open parenthesis x closing parenthesis0 parenrightbiggf (x) 3 2 f (x) v = u ◦ f −1 · (f −1 )2 + − ( )2 (1.5) open parenthesis see e period g period open square bracket 1f 60(x closing) 2 squaref 0(x) bracket closing parenthesis period \noindentIt comma thereforewhere comma$ u seems ( x to be ) clear\ thatin theC natural ˆ{\ infty approach} to( understanding S ˆ{ 1 } of) multi , hyphen $ the action of Diff $ (dimensional S ˆ{ 1( analogues see} e)$ . g .of [ onthe 1 6 Schwarzian] this ) . space derivative is givenby should be based $ f on ( therelation A { u with} ) = A { v }$ with modules of differentialIt , therefore operators , seems period to be clear that the natural approach to understanding of multi \ begin1 period{ a l i 4 g periodn-∗} dimensional .. The contents analogues of of this the paper Schwarzian period derivativeIn this paper should we introduce be based ona multi the relation hyphen with dimensional ana hyphenv = u modules\ circ of differentialf ˆ{ − operators1 }\ .cdot ( f ˆ{ − 1 ˆ{\prime }} ) ˆ{ 2 } + \ f r a c { f ˆ{\prime \primelogue of\ theprime Schwarzian1 .} 4 .( derivative The x contents ) related}{ off this ˆ to{\paper theprime Diff . In open} this( parenthesis paper x we ) introduce M} closing −\ af multiparenthesis r a c { -3 dimensional}{ hyphen2 } modules( \ f r a of c { f ˆ{\prime differential\prime } operators(ana x - logue ) }{ of thef ˆ Schwarzian{\prime derivative} ( x related ) } to the) Diff 2 (\Mtag)−∗{modules$ ( of 1 differential . 5 ) $} \endon{ Ma l periodi g n ∗}operators on M. Following open squareFollowing bracket [ 4 ] 4and closing [ 1 0 square] , the module bracket of and differential open square operators bracketDλ,µ 1 0will closing be viewed square as bracket a comma the\noindent module of(seee.g.[16]). differentialdeformation operatorsof the D module sub lambda of symmetric comma mucontravariant will be viewed tensor as fields a on M. This approach deformationleads of the to module Diff (M of)− symmetriccohomology contravariant first evoked tensor in [ 4 fields ] . The on M corresponding period This cohomologyapproach of the Itleads , therefore to DiffLie open , parenthesis seems to M be closing clear parenthesis that the hyphen natural cohomology approach first evoked to understanding in open square bracket of multi 4 closing− squaredimensional bracket periodalgebra analogues The of vector corresponding of fields the Vect Schwarzian cohomology (M) has been of derivative the calculated Lie in [ should 1 0 ] for a be manifold basedM onendowed the relation with with modulesalgebra of of vectora differential flat fields projective Vect openstructure operators parenthesis . We . use M theseclosing results parenthesis to determine has been the calculated projectively in open equivariant square bracket 1 0 closing square bracketcohomology for a ofmanifold Diff (M M) arising endowed in with this context . 1 .a flat 4 . projective\quad NoteThe structure that contents periodmulti - We of dimensional use this these paper analoguesresults . to In determine of this the Schwarzian paper the projectively we derivative introduce equivariant is a subject a multi alr− - dimensional ana − loguecohomology of theeady of Diff Schwarzian considered open parenthesis in derivativethe literature M closing . related We parenthesis will refer to arising [ the 1 , 7in Diff, 1 this 1 , 1 context $ 2 , ( 1 3 period , M 1 5 , ) 1 4 ]− for$ various modules of differential operators onNote $ that M multi .versions $ hyphen of multi dimensional - dimensional analogues Schwarzians of the Schwarzian in projective derivative , conformal is a subject , symplectic alr hyphen and non - eady consideredcommutative in the literature period We will refer open square bracket 1 comma 7 comma 1 1 comma 1 2 comma 1\ hspace 3 comma∗{\ 1 5f commageometryi l l } Following 1 4 . closing [square 4 ] bracket and [ for 1 various0 ] , versions the module of of differential operators $ D {\lambda , multi\mu hyphen}$ willdimensional2 . be viewedProjective Schwarzians as connections a in projective comma . Let conformalM be a smoothcomma symplectic ( or complex and ) non manifold hyphen of commutative geometry perioddimension n. There exists a notion of projective connection on M, due to E . Cartan . Let us \noindent2 period ..deformation Projectiverecall here connections the simplestof the period ( module and Let naive M of be ) way symmetric a smooth to define open a contravariant projective parenthesis connection or complex tensor as closing an fields equivalence parenthesis on $ manifold M . $ ofThis dimension approachclass of standard ( affine ) connections . leadsn period to There Diff exists $( a notion M of ) projective2− . 1$ . connection cohomology Symbols of on projective M first comma connectionsevoked due to E in period [ 4 Cartan ] . The period corresponding Let us recall cohomology of the Lie here the simplestDefinition open parenthesis . A andprojective naive closing connection parenthesison M wayis the to define class ofa projective affine connections connection corre as an - equivalence \noindentclass of standardalgebrasponding open of to parenthesis the vector same expressions affine fields closing Vect parenthesis $ ( connections M ) $ period has been calculated in [ 1 0 ] for a manifold $ M2 period $ endowed 1 period with .. Symbols of projective connections a flat projective structure . We use these results to determine the projectively equivariant Definition period A projective connection on M is the class1 of affine connections corre hyphen cohomology of Diff $ ( M )Π $k = arising Γk − in(δ thiskΓl + contextδkΓl ), . (2.1) sponding to the same expressions ij ij n + 1 i jl j il Equation: open parenthesis 2 period 1 closing parenthesis .. Capital Pi sub ij to the power of k = Capital Gamma subNote ij to that the power multi of− k minusdimensional 1 divided by analogues n plus 1 parenleftbig of the Schwarzian delta sub i to the derivative power of k Capital is a subject Gamma sub alr j l− to theeady power considered of l plus delta in sub the j to literature the power of k . Capital We will Gamma refer sub i [ l to 1 the , 7 power , 1 of 1 l , parenrightbig 1 2 , 1 3 comma , 1 5 , 1 4 ] for various versions of multi − dimensional Schwarzians in projective , conformal , symplectic and non − commutative

\noindent geometry .

2 . \quad Projective connections . Let $ M $ be a smooth ( or complex ) manifold of dimension $ n . $ There exists a notion of projective connection on $M , $ due to E . Cartan . Let us recall here the simplest ( and naive ) way to define a projective connection as an equivalence

\noindent class of standard ( affine ) connections .

\ centerline {2 . 1 . \quad Symbols of projective connections }

Definition . A projective connection on $ M $ is the class of affine connections corre − sponding to the same expressions

\ begin { a l i g n ∗} \Pi ˆ{ k } { i j } = \Gamma ˆ{ k } { i j } − \ f r a c { 1 }{ n + 1 } ( \ delta ˆ{ k } { i } \Gamma ˆ{ l } { j l } + \ delta ˆ{ k } { j }\Gamma ˆ{ l } { i l } ), \ tag ∗{$ ( 2 . 1 ) $} \end{ a l i g n ∗} SCHWARZIAN DERIVATIVE .. 1 7 \ hspacewhere∗{\ Capitalf i l l Gamma}SCHWARZIAN sub ij to DERIVATIVE the power of k\ arequad the1 Christoffel 7 symbols and we have assumed a summation over repeated \noindentindices periodwhere $ \Gamma ˆ{ k } { i j }$ are the Christoffel symbols and we have assumed a summation over repeated The symbols open parenthesis 2 period 1 closing parenthesis naturally appear if one considers projective connections as\noindent a par hypheni n d i c e s . SCHWARZIAN DERIVATIVE 1 7 k ticular casewhere of so hyphen Γij are called the Christoffel Cartan normal symbols connection and we have comma assumed see open a summation square bracket over repeated 8 closing square bracket periodThe symbolsindices ( 2 . . 1 ) naturally appear if one considers projective connections as a par − ticularRemarks case periodThe of .. open sosymbols− parenthesiscalled ( 2 . 1 )aCartan naturally closingparenthesis normal appear ifconnection oneThe considers definition , projective is see correct [ 8 connectionsopen ] . parenthesis as a par i period - e period does not dependticular on the case choice of so of - lo called cal Cartan normal connection , see [ 8 ] . Remarkscoordinates . \quad on MRemarks closing( a ) parenthesis . The( definition a ) The period definition is is correct correct ( ( i . i e . . does e . not does depend not on depend the choice on of the lo choice of lo cal coordinatesopen parenthesiscal on coordinates b $Mclosing parenthesison )M) .. $ .. The formula open parenthesis 2 period 1 closing parenthesis .. defines a natural projection to( b the ) space The formula of trace ( hyphen 2 . 1 ) less defines open parenthesisa natural projection 2 comma to 1 theclosing space parenthesis of trace - hyphen less ( 2 \ hspacetensors∗{\ commaf, i l 1 l ) one} -( hasb ) :\ Capitalquad The Pi sub formula ik to the power( 2 . of 1 k )=\ 0quad perioddefines a natural projection to the space of trace − l e s s ( 2 , 1 ) − k 2 period 2 periodtensors .. , Flat oneprojective has : Πik = connections 0. and projective s tructures period A manifold M is said to \noindentbe lo callytensors projective2 . 2 . ,open one Flat parenthesis has projective $or : connections endowed\Pi ˆ with{ andk a} projectiveflat{ projectivei k } s= tructures s tructure 0 . .A closing $ manifold parenthesisM is said if there exists an atlas to be lo cally projective ( or endowed with a flat projective s tructure ) if there exists an 2 .on 2 M . with\quad linearatlasFlat onhyphenM projectivewith fractional linear - coordinate fractional connections changescoordinate and : changes projective : s tructures . A manifold $ M $ i sEquation: s a i d to open parenthesis 2 period 2 closing parenthesis .. x to the power of i = a sub j to the power of i x to the be lo cally projective ( or endowed with a flat projective s tructure ) if there exists an atlas power of j plus b to the power of i divided by c sub j x to thei powerj i of j plus d sub period on $M$ with linear − fractional coordinateajx + changesb : A projective connection on M is called flat if in a neighborhoodxi = of each point comma there (2.2) c xj + d exists a lo cal coordinate system open parenthesis x to thej power. of 1 comma period period period comma x to the power\ begin of{ na closingl i g n ∗} parenthesis such that the symbols Capital Pi sub ij to the power of k are identically zero A projective connection on M is called flat if in a neighborhood of each point , there xopen ˆ{ i parenthesis} = \ seef r a open c { a square ˆ{ i bracket} { j 8 closing} x ˆ square{ j } bracket+ for b a ˆ{ geometrici }}{ definitionc { j closing} x ˆ parenthesis{ j } + period d } { . }\ tag ∗{$ ( exists a lo cal coordinate system (x1, ..., xn) such that the symbols Πk are identically zero Every2 . flat projective 2 ) $} connection defines a projective ij ( see [ 8 ] for a geometric definition ) . Every flat projective connection defines a projective \endstructure{ a l i g n on∗} M period structure on M. 2 period 3 period .. A .. projectively .. invariant .. 1 hyphen cocycle .. on Diff open parenthesis M closing parenthesis 2 . 3 . A projectively invariant 1 - cocycle on Diff (M). A common Aperiod projective A .. common connection way of producing on $M$ is called flat if in a neighborhood of each point , there way of producing nontrivial cocycles on Diff (M) using affine connections on M is as follows existsnontrivial a cocycleslo cal on coordinate Diff open parenthesis system M $ closing ( x parenthesis ˆ{ 1 } using, affine . connections. . , on M x is ˆ{ asn follows} ) period $ such that the symbols . The map : The$ \Pi mapˆ{ : k } { i j }$ are identically zero (f ∗Γ)k − Γk is a 1 - cocycle on Diff (M) with values in the space of symmetric ( 2 , 1 ) - open parenthesis fij to theij power of * Capital Gamma closing parenthesis sub ij to the power of k minus Capital Gamma tensor sub\noindent ij to the power( see of [k is 8 a ] 1 hyphenfor a cocycle geometric on Diff definition open parenthesis ) . M Every closing flat parenthesis projective with values connection in the space definesof a projective fields . It is , therefore , clear that a projective connection on M leads to the following 1 - symmetric open parenthesis 2 comma 1 closing parenthesis hyphen tensor cocycle on Diff (M): \noindentfields periodstructure It is comma on therefore $M comma . $ .. clear that a projective connection on M leads to the following 1 hyphen cocycle on Diff open parenthesis M closing parenthesis : 2 . 3 . \quad A \quad projectively \quad i n v a r i a n t \quad 1 − c o c y c l e \quad on D i f f $ ( Equation: open parenthesis 2 period 3 closing parenthesis∗ k k ..) l openi parenthesisj ∂ f closing parenthesis = parenleftbig M ) . $ A \quad common way`(f of) = producing ((f Π) − Π dx ⊗ dx ⊗ (2.3) open parenthesis f to the power of * Capital Pi closingij parenthesisij sub ij to∂x thek power of k minus Capital Pi sub ij to thenontrivial power of k to cocycles the power of on parenrightbig Diff $ ( dx to M the power ) $ of using i oslash affine dx to the connections power of j oslash on partialdiff $M$ divided is as by follows . The map : partialdiff x to thevanishing power on of k( lo cally ) projective diffeomorphisms . \noindentvanishing on$ open (Remarks parenthesis f ˆ{ . ∗ } lo( cally \ aGamma ) The closing expression parenthesis) ˆ{ k (} 2 .projective{ 3i ) j is} well diffeomorphisms − defined \Gamma ( doesˆ period{ notk depend} { i onj } the$ i s a 1 − cocycle on Diff $ (Remarks M period )choice $ .. with open of lo parenthesis calvalues coordinates in a closing the ) . space parenthesis This follows of symmetricThe from expression a well ( - open known 2 , parenthesis 1 fact ) − thatt e 2 nthe period s o difference r 3 closing of parenthesis is well defined opentwo ( parenthesis projective) does connections not depend defines on the a ( choice 2 , 1 ) - tensor field . \noindent fields( b ) . Already It is the , formula therefore ( 2 . 3 ,) implies\quad thatclear the map thatf 7→ a` projective(f) is , indeed , connection a 1 - cocycle on $ M $ of lo cal coordinates closing parenthesis period .. This follows∗ from a well hyphen known fact that the difference of twoleads to the, that following is , it satisfies the relation `(f ◦ g) = g `(f) + `(g). 1 open− cocycleonDiff parenthesis( c projective ) It is clear $( closing that M theparenthesis cocycle ) :$` connectionsis nontrivial defines ( cf . a open[ 1 0 ] parenthesis ) , otherwise 2comma it would 1 depend closing parenthesis only on the first jet of the diffeomorphism f. Note that the formula ( 2 . 3 ) looks as a hyphen tensor field period k \ beginopen{ parenthesisa l i g ncoboundary∗} b closing , however parenthesis , the symbolsAlready Πtheij do formula not transform open parenthesis as components 2 period of a 3 ( closing 2 , 1 ) - parenthesis tensor implies that\ e l the l map( ffield mapsto-arrowright f ( but) as = symbols ( l open of ( a parenthesisprojective f ˆ{ ∗ connection } f closing \Pi parenthesis ) .) ˆ{ k is} comma{ i j } indeed − comma \Pi ˆ a{ 1k hyphen} { i cocycle j }ˆ{ ) } commadx ˆ{ i }\otimesExampledx . In ˆ{ thej case}\ ofotimes a smooth manifold\ f r a c {\ endowedpartial with}{\ a flatpartialprojective connectionx ˆ{ k }}\ tag ∗{$ ( 2that . is comma3, ) ( with it $} satisfies symbols the ( 2relation . 1 ) l identically open parenthesis zero ) or f circ , equivalently g closing parenthesis , with a projective = g to structure the power , of * l open parenthesis\end{ a l i g fn closing∗}the cocycle parenthesis ( 2 . 3plus ) obviously l open parenthesis takes the form g closing : parenthesis period open parenthesis c closing parenthesis It is clear that the cocycle l is nontrivial open parenthesis cf period .. open \noindent vanishing on ( lo cally ) projective diffeomorphisms . square bracket 1 0 closing square bracket∂2f l closing∂xk parenthesis1 ∂ log commaJ otherwise∂ log J it would depend∂ `(f, x) = ( − (δk f + δk f ))dxi ⊗ dxj ⊗ (2.4) only on the first jet of the diffeomorphism∂xi∂xj ∂f f periodl n + Note 1 j that∂x thei formulai ∂x openj parenthesis 2 period∂xk 3 closing parenthesis looksRemarks as a . \quad ( a ) The expression ( 2 . 3 ) is well defined ( does not depend on the choice of lo cal coordinates ) . \quad This follows from a well − i known fact that the difference of two coboundary comma however1 n comma the1 symbols Capitaln Pi sub ij to the power∂f of k do not transform as components where f(x , ..., x ) = (f (x), ..., f (x)) and Jf = det ( j ) is the Jacobian . This of( a projectiveopen parenthesis ) 2 connections comma 1 closing defines parenthesis a ( hyphen 2 , 1 tensor ) − tensor field∂x . field open parenthesis but as symbols of a projective connection closing parenthesis period ( bExample ) Already period the In the formula case of a ( smooth 2 . 3 manifold ) implies endowed that with the a flat map projective $ f connection\mapsto comma\ e l l ( f )$open is parenthesis , indeed with , a1symbols− c open o c y c parenthesis l e , 2 period 1 closing parenthesis .. identically zero closing parenthesis orthat comma is equivalently , it satisfies comma with the a projective relation structure $ \ e l comma l ( the f \ circ g ) = g ˆ{ ∗ } \ e l l (cocycle f )open + parenthesis\ e l l 2 period( g 3 closing ) parenthesis . $ obviously takes the form : Equation: open parenthesis 2 period 4 closing parenthesis .. l open parenthesis f comma x closing parenthesis = parenleftbigg( c ) It is partialdiff clear to that the power the cocycle of 2 f to the $ power\ e l l of$ l divided is nontrivial by partialdiff (x to cf the . power\quad of i[ partialdiff 1 0 ] ) x to, otherwisethe it would depend poweronly of on j partialdiff the first x to the jet power of ofthe k divided diffeomorphism by partialdiff f to $ the f power . $ of lNote minus that 1 divided the by formula n plus 1 parenleftbigg ( 2 . 3 ) looks as a delta sub j to the power of k partialdiff log J sub f divided by partialdiff x to the power of i plus delta sub i to the power of\noindent k partialdiffcoboundary log J sub f divided , however by partialdiff , the x symbols to the power $ of\Pi j parenrightbiggˆ{ k } { i parenrightbigg j }$ do not dx transformto the power asof i components of a ( 2 , 1 ) − t e n s o r oslash dx to the power of j oslash partialdiff divided by partialdiff x to the power of k \noindentwhere f openfield parenthesis ( but x to as the symbols power of 1 of comma a projective period period connection period comma ) x to . the power of n closing parenthesis = parenleftbig f to the power of 1 open parenthesis x closing parenthesis comma period period period comma f to the powerExample of n open. In parenthesis the case x closingof a smooth parenthesis manifold parenrightbig endowed .... and with J sub a f = flat .... det projective open parenthesis connection partialdiff f , to the( power with of symbols i divided ( by 2 partialdiff . 1 ) \ xquad to theidentically power of j closing zero parenthesis ) or , .... equivalently is the Jacobian period , with .... a This projective structure , the cocycle ( 2 . 3 ) obviously takes the form :

\ begin { a l i g n ∗} \ e l l ( f , x ) = ( \ f r a c {\ partial ˆ{ 2 } f ˆ{ l }}{\ partial x ˆ{ i } \ partial x ˆ{ j }}\ f r a c {\ partial x ˆ{ k }}{\ partial f ˆ{ l }} − \ f r a c { 1 }{ n + 1 } ( \ delta ˆ{ k } { j }\ f r a c {\ partial \ log J { f }}{\ partial x ˆ{ i }} + \ delta ˆ{ k } { i }\ f r a c {\ partial \ log J { f }}{\ partial x ˆ{ j }} )) dx ˆ{ i }\otimes dx ˆ{ j }\otimes \ f r a c {\ partial }{\ partial x ˆ{ k }}\ tag ∗{$ ( 2 . 4 ) $} \end{ a l i g n ∗}

\noindent where $ f ( x ˆ{ 1 } , . . . , x ˆ{ n } ) = ( f ˆ{ 1 } ( x ) , . . . , f ˆ{ n } ( x ) ) $ \ h f i l l and $ J { f } = $ \ h f i l l det $ ( \ f r a c {\ partial f ˆ{ i }}{\ partial x ˆ{ j }} ) $ \ h f i l l is the Jacobian . \ h f i l l This 1 8 .. S period BOUARROUDJ AND V period YU period OVSIENKO \noindentexpression1 is 8globally\quad definedS . BOUARROUDJ and vanishes if AND f is given V . open YU . parenthesis OVSIENKO in the lo cal coordinates of the projective structure closing parenthesis as a linear hyphen fractional transformation open parenthesis 2 period 2 closing parenthesis\noindent periodexpression is globally defined and vanishes if $ f $ is given ( in the lo cal coordinates of the The cocycle .. open parenthesis 2 period 3 comma 2 period 4 closing parenthesis .. was introduced in .. open square bracket\noindent 1 5 commaprojective1 8 .. 1S .1 BOUARROUDJ closing structure square AND bracket ) V . as YU .. a. OVSIENKOas linear a multi− hyphenfractional dimensional transformation projective ( 2 . 2 ) . analogue ofexpression the Schwarzian is globally derivative defined period and However vanishes comma if f is given in contradistinction ( in the lo cal coordinates with the Schwarzian of the Thederivative c o c y c l e..projective open\quad parenthesis( structure 2 . 3 1)period , as 2 a . linear 2 4 closing ) -\ fractionalquad parenthesiswas transformation introduced comma this ( map 2 in . 2 ..\ )quad .open parenthesis[ 1 5 , \ 2quad period1 4 1 closing ] \quad as a multi − dimensional projective parenthesisanalogue .. ofdepends theThe on Schwarzian cocycle the second ( 2 hyphen . derivative 3 , 2 order . 4 jets) . ofwas However diffeomorphisms introduced , in in contradistinction period [ 1 5 , 1 1 ] as awith multi the - Schwarzian d eMoreover r i v a t i v ecommadimensional\quad in the( 1 projective one . 2 hyphen ) analogue , thisdimensional mapof the\ Schwarziancasequad ....( open 2 derivative . parenthesis 4 ) \ .quad However ndepends = 1 , in closing contradistinction on parenthesis the second comma− order the jets of diffeomorphisms . expression openwith parenthesis the Schwarzian 2 period derivative 3 comma 2 period ( 1 . 2 4 ) closing , this map parenthesis ( 2 . is4 )identically depends on the second - \noindentzero periodMoreoverorder jets of , diffeomorphisms in the one − . dimensional case \ h f i l l $( n = 1 ) ,$ the expression (2.3 ,2.4) is identically 3 period ..Moreover Introducing , in the the Schwarzian one - dimensional derivative case period Assume (n = that 1), the dim expression M greater ( equal 2 . 3 ,2 2 period . 4 ) is Let S to the power\noindent of k openzeroidentically parenthesis . M closing parenthesis open parenthesiszero . or S to the power of k for short closing parenthesis be the space of k hyphen th order symmetric contravariant\ hspace ∗{\ tensorf i l l }3 fields3 . .\ onquad M Introducing periodIntroducing the Schwarzian the Schwarzian derivative derivative . Assume . that Assume dim M that≥ 2. Let dim $ M k \geq3 period2 1 period .S $(M ..) Let Operator $ S symbols ˆ{ k } .. of( a projective M ) $ .. connection period For an arbitrary system of lo cal k coordinates( fix or theS followingfor short linear ) be the differential space of operatork− th order T : symmetricS to the power contravariant of 2 right tensor arrow Cfields to the on powerM. of infinity open\noindent parenthesis( or M3 closing .$ 1 S . ˆparenthesis{ Operatork }$ given forsymbols shortfor every of ) a beprojective the space connection of $ . k For− an$ arbitrary th order system symmetric contravariant tensor fields on $ MLine 1. a $ in Sof to lo the cal power of 2 by T open parenthesis a closing parenthesis = T sub ij open parenthesis a to the power 2 ∞ of ij closing parenthesiscoordinates with fix Line the 2following T sub ij linear = Capital differential Pi sub operator ij to theT power: S → ofC hline(M sub) given partialdiff for every to the power of k \ hspace ∗{\ f i l l }3 . 1 . \quad Operator symbols \quad of a projective \quad connection . For an arbitrary system of lo cal sub x to the power of k to the power of partialdiff minus 2 divided by2 n minus 1 parenleftbiggij partialdiff Capital Pi sub ij to the power of k divided by partialdiff x to the power of k minusa ∈ S nby plusT (a) 1 = dividedTij(a by)with 2 Capital Pi sub i l to the power \noindent coordinates fix the following lineark $ T : S ˆ{ 2 } of k Capital Pi sub kj to the power of l parenrightbiggk ∂ comma2 ∂Π openij parenthesisn + 1 k l 3 period 1 closing parenthesis \rightarrow C ˆ{\ inftyTij}= Π(ij Mxk )− $ given( for− everyΠilΠkj), (3.1) where Capital Pi sub ij to the power∂ of k are then symbols− 1 ∂x ofk a projective2 connection open parenthesis 2 period 1 closing parenthesis on M period where Πk are the symbols of a projective connection ( 2 . 1 ) on M. \ [ \Itbegin is clear{ a lthat i g n e the d } differentialija \ in operatorS ˆ{ open2 } parenthesisby T 3 period( a 1 closing ) = parenthesis T { i is j not} intrinsically( a ˆ{ definedi j } ) with \\ It is clear that the differential operator ( 3 . 1 ) is not intrinsically defined , indeed , comma indeed comma already k already its principal symbol , Π , is not a tensor field . In the same spirit that the difference Tits principal{ i j } symbol= \ commaPi ˆ{ Capitalk } { Pii j sub ˆ{\ij ij tor the u l e power{3em}{ of0.4 comma pt }} to the{\ powerpartial of k is}} notˆ{\ a tensorpartial field period} { x In ˆ{ k }} − \ f r a c { 2 }{ n − 1 } ( \ f r ak c {\kpartial \Pi ˆ{ k } { i j }}{\ partial x ˆ{ k }} the same spiritof that two the projective difference connections of Πe ij − Πij is a well - defined tensor field , we have the following − \ f r a c { n + 1 }{ 2 }\Pi ˆ{ k } { i l }\Pi ˆ{ l k } { kj k} ) , ( 3 . two projectiveTheorem connections 3 . Pi-tildewide1 . Given sub arbitrary ij to the projective power of connections k minus CapitalΠe ij and Pi subΠij, ijthe to differencethe power of k is a well hyphen1 ) defined\end{ a tensor l i g n e field d }\ comma] we have the following Theorem 3 period 1 period .. Given arbitrary projective connections Pi-tildewide sub ij to the power of k and Capital Pi sub ij to the power of comma to the power of k the differenceT = Te − T (3.2) \noindentEquation:where open parenthesis $ \Pi 3ˆ{ periodk } 2{ closingi j }$ parenthesis are the .. T symbols = T-tildewide of a minus projective T connection ( 2 . 1 ) on 2 ∞ $ Mis a linear . $ differentialis a linear operator differential from operator S to the from powerS to of 2C to(M C) towell the defined power ( of glo infinity bally ) open on M parenthesis( i . e . M closing parenthesis well, defined it open parenthesis glo bally closing parenthesis on M open parenthesis i period e period comma it Itdoes is notclear dependdoes that noton the depend choice differential on of the local choice coordinates of operator local coordinates closing ( 3 parenthesis . ) 1 . ) is period not intrinsically defined , indeed , already itsProof principal period To provesymbolProof that . $ theTo , prove expression\Pi thatˆ{ the openk expression} parenthesis{ i j( ˆ{ 3 3., 2 period}} ) is$ , indeed 2 is closing not a well parenthesis a -tensor defined is differential field comma . indeed In the a well same spirit that the difference of hyphentwo projective definedoper differential - connections oper hyphen $ \ widetilde {\Pi} ˆ{ k } { i j } − \Pi ˆ{ k } { i j }$ i s a w e l l − defined tensor field , we have the following 2 ∞ ator from Sator to the from powerS into of 2C into(M C) to, we the need power an of explicit infinity formula open parenthesis of coordinate M transformation closing parenthesis for such comma we need an\ centerline explicit formula{kindTheorem of of coordinate operators 3 . 1 .transformation . \quad Given for arbitrary projective connections $ \ widetilde {\Pi} ˆ{ k } { i j }$ andsuch $ kind\Pi ofˆ operators{ k Lemma} { periodi j 3 ˆ .{ 2 ., }}The$ the coefficients difference of} a first - o rder linear differential operator 2 Lemma 3 periodA : 2 periodS ..→ The .. coefficients .. of a first hyphen o rder linear differential operator A : S to the power \ begin { a l i g n ∗}∞ k ij of 2 right arrowC (M)A(a) = (tij∂k + uij)a transform under coordinate changes as follows : TC to = the power\ widetilde of infinity{T open} − parenthesisT \ tag M∗{ closing$ ( parenthesis 3 . A 2open parenthesis) $} a closing parenthesis = parenleft- \end{ a l i g n ∗} big t sub ij to the power of k partialdiff sub k plus u sub ij parenrightbig a to thea powerb ofk ij transform under coordinate k c ∂x ∂x ∂y changes as follows : tij(y) = tab(x) (3.3) \noindent is a linear differential operator from $ S ˆ{ 2∂y}i$∂yj to∂xc $ C ˆ{\ infty } ( Equation: open parenthesis 3 period 3 closing parenthesisa b .. t sub ij2 tok thea powerb ofl k open parenthesis y closing Mparenthesis )$ = well t sub defined a b to the (power glo of bally c open parenthesis)on∂x $M∂x x closing ($c parenthesis i∂ .y e∂x . partialdiff ,∂x it ∂x x to the power of a divided by uij(y) = uab(x) − 2tab(x) (3.4) partialdiff y to the power of i partialdiff x to the power∂yi of∂y bj divided by∂x partialdiffc∂xl ∂yk ∂y y to(i ∂y thej) power of j partialdiff y to the power\noindent of k divideddoes by not partialdiff depend x to on the the power choice of c Equation: of local open coordinates parenthesis 3 period ) . 4 closing parenthesis .. u sub ij where round b rackets mean symmetrization . open parenthesis y closing parenthesis = u sub a b open parenthesis x closing parenthesis partialdiff x to the power of a Proof of the lemma : straightforward . Consider the following expression : divided\ hspace by∗{\ partialdifff i l l } Proof y to the . power To prove of i partialdiff that the x to the expression power of b divided( 3 . by2 )partialdiff is , indeed y to the apower well of− j minusdefined 2 differential oper − t sub a b to the power of c open parenthesis x closing parenthesis partialdiff to the power of 2 y to the power of k divided T (α, β) = (Πe k − Πk )∂ + α∂ (Πe k − Πk ) + β(Πe k Πe l − Πk Πl ) by\noindent partialdiff xator to the from power $ of S c partialdiff ˆ{ij 2 }$ij x to i n the tij o powerk $ C ofk ˆ{\ l partialdiffij inftyij } x to(li thejk powerMli ) ofjk a divided , $ we by partialdiffneed an y explicit to formula of coordinate transformation for thesuch power kind of k partialdiffof operators x to the . power of b divided by partialdiff y to the power of open parenthesis i partialdiff x to the power of l divided by partialdiff y to the power of j closing parenthesis \ hspacewhere∗{\ roundf i bl l rackets}Lemma mean 3 . symmetrization 2 . \quad The period\quad coefficients \quad o f a f i r s t − o rder linear differential operator $ AProof : of the S lemma ˆ{ 2 }\ : straightforwardrightarrow period$ Consider the following expression : \noindentT open parenthesis$ C ˆ{\ alphainfty comma} beta(M)A(a)=(tˆ closing parenthesis sub ij = open parenthesis to the{ powerk } of{ Pi-tildewidei j }\ partial sub ij { k } +to the u power{ i of j k} minus) Capital a ˆ{ Pii j sub}$ ij to transform the power of under k closing coordinate parenthesis partialdiff changes sub as k follows plus alpha : partialdiff sub k open parenthesis to the power of tildewide-Pi sub ij to the power of k minus Capital Pi sub ij to the power of k closing parenthesis\ begin { a l plus i g n ∗} beta open parenthesis to the power of Pi-tildewide sub l i to the power of k to the power of tildewide-Pi subt jkˆ{ tok the} power{ i j of} l minus( y Capital ) Pi = sub l ti to ˆ{ thec power} { ofa k Capital b } Pi( sub x j k to ) the\ f power r a c {\ of lpartial closing parenthesisx ˆ{ a }}{\ partial y ˆ{ i }}\ f r a c {\ partial x ˆ{ b }}{\ partial y ˆ{ j }}\ f r a c {\ partial y ˆ{ k }}{\ partial x ˆ{ c }}\ tag ∗{$ ( 3 . 3 ) $}\\ u { i j } ( y ) = u { a b } ( x ) \ f r a c {\ partial x ˆ{ a }}{\ partial y ˆ{ i }}\ f r a c {\ partial x ˆ{ b }}{\ partial y ˆ{ j }} − 2 t ˆ{ c } { a b } ( x ) \ f r a c {\ partial ˆ{ 2 } y ˆ{ k }}{\ partial x ˆ{ c }\ partial x ˆ{ l }}\ f r a c {\ partial x ˆ{ a }}{\ partial y ˆ{ k }}\ f r a c {\ partial x ˆ{ b }}{\ partial y ˆ{ ( i }}\ f r a c {\ partial x ˆ{ l }}{\ partial y ˆ{ j ) }}\ tag ∗{$ ( 3 . 4 ) $} \end{ a l i g n ∗}

\noindent where round b rackets mean symmetrization .

\noindent Proof of the lemma : straightforward . Consider the following expression :

\ [T( \alpha , \beta ) { i j } = ( ˆ{\ widetilde {\Pi}}ˆ{ k } { i j } − \Pi ˆ{ k } { i j } ) \ partial { k } + \alpha \ partial { k } ( ˆ{\ widetilde {\Pi}}ˆ{ k } { i j } − \Pi ˆ{ k } { i j } ) + \beta ( ˆ{\ widetilde {\Pi}}ˆ{ k } { l i }ˆ{\ widetilde {\Pi}}ˆ{ l } { jk } − \Pi ˆ{ k } { l i }\Pi ˆ{ l } { j k } ) \ ] SCHWARZIAN DERIVATIVE .. 1 9 \ hspaceFrom the∗{\ definitionf i l l }SCHWARZIAN open parenthesis DERIVATIVE 3 period 1\ commaquad 1 3 period9 2 closing parenthesis for Line 1 alpha = minus 2 divided by n minus 1 sub comma beta = n plus 1 divided by n minus 1 sub comma open parenthesis\ centerline 3 period{From 5 closing the definition parenthesis Line ( 3 2 . 1 gets 1 , T 3 open . 2 parenthesis ) for } alpha comma beta closing parenthesis = T period \ [ \Nowbegin comma{ a l i g it n follows e d }\ immediatelyalpha = from− the \ f fact r a c that{ 2 Pi-tildewide}{ n − sub1 ijSCHWARZIAN to} the{ , power}\ DERIVATIVE ofbeta k minus= Capital\1f 9 r a Pi c { subn ij to + the1 }{ powern of− k is a1 well} { hyphen, } defined( 3 openFrom . parenthesis the 5 definition ) 2\\ comma ( 3 . 1 1 closing , 3 . 2 ) parenthesis for hyphen tensor 1field ong e t M s comma T that ( the\alpha condition, open\ parenthesisbeta ) 3 period = T 3 closing . \end parenthesis{ a l i g n e for d }\ the] principal symbol of T 2 n + 1 open parenthesis alpha comma beta closing parenthesisα = − is satisfiedβ = period(3.5) n − 1 n − 1 The transformation law for the symbols of a projective, connection reads, : \ hspaceCapital∗{\ Pif sub i l l ij}Now to the , power it follows of k open parenthesis immediately y closing1gets from parenthesisT ( theα, β) fact = T =. Capital that Pi $ sub\ widetilde a b to the{\ powerPi} ofˆ c{ openk } { i j } − \Pi ˆ{ k } { i j }$ i s a w e l l − defined ( 2 , 1 ) − t e n s o r parenthesis x closing parenthesis partialdiff x to the power of a dividedk by partialdiffk y to the power of i partialdiff x to Now , it follows immediately from the fact that Πe ij − Πij is a well - defined ( 2 , 1 ) - the power of btensor divided by partialdiff y to the power of j partialdiff y to the power of k divided by partialdiff x to the power\noindent of c plusfield l open on parenthesis $M y , comma $ that x closing the parenthesis condition comma ( 3 . 3 ) for the principal symbol of $ T ( \alphafield on M,,that\ thebeta condition) $ ( 3 is. 3 ) satisfied for the principal . symbol of T (α, β) is satisfied . where l open parenthesisThe transformationy comma x closing law forparenthesis the symbols is given of a byprojective open parenthesis connection 2 reads period : 4 closing parenthesis period Let u open parenthesis alpha comma beta closing parenthesis sub ij be the zero hyphen order term in T open \ centerline {The transformation law for the symbols of a projective connection reads : } parenthesis alpha comma beta closing parenthesis sub ij comma∂xa ∂x oneb ∂yk Πk (y) = Πc (x) + `(y, x), readily gets : ij ab ∂yi ∂yj ∂xc \ [ Line\Pi 1ˆ u{ openk } parenthesis{ i j } ( alpha y comma ) beta = closing\Pi ˆ parenthesis{ c } { a open b parenthesis} ( yx closing ) \ parenthesisf r a c {\ partial sub ij = u where `(y, x) is given by ( 2 . 4 ) . Let u(α, β) be the zero - order term in T (α, β) , one openx ˆ{ parenthesisa }}{\ partial alpha commay beta ˆ{ closingi }}\ parenthesisf r a c {\ partial open parenthesisij x ˆ{ x closingb }}{\ parenthesispartial sub ay bij ˆ partialdiff{ j }}\ xf to r a the c {\ partial readily gets : powery ˆ{ ofk a}}{\ dividedpartial by partialdiffx y ˆ{ toc the}} power+ of i\ partialdiffe l l ( x to y the power , x of b divided ) , by\ ] partialdiff y to the power of j Line 2 minus 2 open parenthesis alpha plus beta closing parenthesis open parenthesis∂xa ∂xb to the power of Pi-tildewide sub u(α, β)(y) = u(α, β)(x) b a b to the power of c open parenthesis x closing parenthesisij minus Capital Pia sub∂yi a∂y bj to the power of c open parenthesis x\noindent closing parenthesiswhere closing $ \ e parenthesis l l ( partialdiffy , to x the )$ power isgivenby(2.4).Let of 2 y to the power of k divided by partialdiff $u x ( to the\alpha , \beta ) { i j }$ be the zero − order term∂ in2yk $T∂xa ∂xb ( ∂xl \alpha , \beta ) { i j } power of c partialdiff x to the power of− l2( partialdiffα + β)(Πe c x( tox) the− Π powerc (x)) of a divided by partialdiff y to the power of k partialdiff ab ab c l k (i j) x, to $ the one power of b divided by partialdiff y to the power of open∂x parenthesis∂x ∂y ∂y i partialdiff∂y x to the power of l divided by readily gets : a b partialdiff y to the power of j closing parenthesis2β LineΠe 3c plus openc parenthesis∂ log Jy alpha∂x ∂x plus 2 beta divided by n plus 1 closing +(α + )( ab(x) − Πab(x)) c i j parenthesis open parenthesis to the power of Pi-tildewiden + 1 sub a b to the∂x power∂y of c∂y open. parenthesis x closing parenthesis minus\ [ \ begin Capital{ a l iPi g n sub e d } a bu to the ( power\alpha of c open, parenthesis\beta x) closing ( parenthesis y ) closing{ i j } parenthesis= u partialdiff ( \alpha log J , \beta The) transformation ( x ) law{ a ( 3} . 4b ) for\uf( rα, a c β{\)ij ispartial satisfied if andx ˆ only{ a if}}{\α andpartialβ are given y ˆ{ i }}\ f r a c {\ partial sub y divided byby partialdiff ( 3 . 5 ) . x Theorem to the power 3 . 1 of is provenc partialdiff . x to the power of a divided by partialdiff y to the power of i partialdiffx ˆ{ b }}{\ x to thepartial power of by divided ˆ{ j by}}\\ partialdiff y to the power of j sub period We call Tij given by ( 3 . 1 ) the operator symbols of a projective connection . This −The transformation2 ( \alpha law open+ parenthesis\beta 3) period ( 4 ˆ{\ closingwidetilde parenthesis{\Pi for}} u openˆ{ c parenthesis} { a alphab } comma( x beta ) − \Pi ˆ{ cnotion} { a b } ( x ) ) \ f r a c {\ partial ˆ{ 2 } y ˆ{ k }}{\ partial x ˆ{ c } closing parenthesisis the sub main ij is tool satisfied of this if and paper only . if alpha and beta are given \ partial x ˆ{ l }}\ f r a c {\ partial x ˆ{ a }}{\ partial y ˆ{ k }}\ f r a c {\ partial k k by open parenthesisk 3 period.The 5 closing parenthesis periodkij Theorem 3 period 1 is proven period Πij =−∂ Πij /∂x + x ˆ{ b }}{\ΠpartialRemark l y ˆ{scalartogethertermwithΠ( i }}\ f r a c {\ partial(characterize3x.1) ˆ looks{ l ∼}}{\thenormalpartialto the symbolsy ˆ{ j ) }}\\ We call T subil ij given byΠ open, which parenthesis 3 period 1 closingof parenthesis the operator symbols of a projectiveCartanprojective connection connection + ( \alpha + kj \ f r a c { 2 \beta }{ n + 1 } ) ( ˆ{\ widetilde {\Pi}}ˆ{ c } { a period This notion( see [ 8 ] ) . We will show that the operator symbols Tij, and not the symbols of the normal b }is the( main x toolprojective ) of this− connection paper \Pi periodˆ{ ,c lead} { toa a natural b } notion( x of multi ) - dimensional) \ f r a c {\ Schwarzianpartial deriva\ log - J { y }}{\ partial x ˆCapital{ c }}\ Pi subftive r a i c .l{\ to thepartial power of kx Remark ˆ{ a }}{\ Capitalpartial Pi sub kj toy the ˆ{ poweri }}\ of commaf r a c {\ to thepartial power of lx which ˆ{ b to}}{\ the partial powery ˆ{ ofj period}} { The. 3}\ . scalarend 2 .{ a together l i g The n e d main term}\ ] definition with Capital . PiConsider to the powera manifold of k fromM endowed ij to of withopen a parenthesis projective characterize 3 period 1 closingconnec parenthesis - tion . looksThe expression thicksim sub the normal to the symbols Cartan projective to the power of Capital Pi sub ij = minus partialdiff sub connection to the power of Capital Pi sub ij to the power of k slash partialdiff x to the \noindent The transformation law ( 3 . 4 ) for $ u ( \alpha , \beta ) { i j }$ power of k to the power of plus S(f) = f ∗(T ) − T, (3.6) isopen satisfied parenthesis if see and open only square if bracket $ \alpha 8 closing$ square and bracket $ \beta closing$ areparenthesis given period We will show that the operator symbolswhere T subT ijis comma the ( lo and cally not defined the symbols ) operator of the ( normal 3 . 1 ) , is a linear differential operator well \noindentprojectiveby connectiondefined ( 3 . comma 5 ) . lead Theorem to a natural 3 . 1notion is provenof multi hyphen . dimensional Schwarzian deriva hyphen tive period \ hspace ∗{\ f i l l }We c a l l $ T { i j }$ given by ( 3 . 1 ) the operator symbols of a projective connection . This notion 3 period 2 period .. The main definition period Consider(globally)on a manifoldM. M endowed with a projective connec hyphen tion period The expression \noindentEquation:is open the parenthesisProposition main tool 3 period3 of . 6 3 this closing . paperThe parenthesis map . ..f S open7→ parenthesisS(f) is f closing a nontrivial parenthesis 1 - cocycle = f to the power of * open parenthesison Diff T (closingM) with parenthesis minus T comma \noindentwhere T is$values the\Pi open inˆ{ parenthesisHomk } (S{2,Ci lo∞( callyM l ))}. definedRemark closing{\ parenthesisPi ˆ{ l } operator{ kj openˆ{ , parenthesis}} which 3 period}ˆ{ . 1 closing The } s c a l a r { t o g e t h e r } term{ with } parenthesis\Pi ˆ{ k comma}ˆ{ iProof j is a} linear{ .o f differential}The( cocycle{ operatorcharacterize property well for definedS(f})3 follows . directly 1 from )$ the looks definition $ \sim ( 3 . 6{ the normal }$ toopen the parenthesis $ symbols) . This globally cocycle{ Cartan closing is not parenthesis a coboundary projective on M . Indeed period}ˆ{\ , everyPi { coboundaryi j } = d B−on Diff \ partial (M) with} valˆ{\ - Pi ˆ{ k } { i j } / Proposition\ partialues 3 period inx the ˆ{ 3 space periodk }} Hom{ .. Theconnection (S2,C ..∞ map(M)) f arrowright-mapsto is}ˆ of{ the+ } form$ B(f)( Sa open) = parenthesisf ∗(B) − B, fwhere closingB parenthesis∈ .. is .. a nontrivial 1 hyphenHom cocycle (S2,C∞ ..(M on)) Diff. Since openS( parenthesisf) is a first M - order closing differential parenthesis operator .. with , the coboundary con - \noindentvalues in Hom(dition see open [S parenthesis 8=d ]B )would . We S to imply will the power that showB of thatis 2 also comma the a first C operator to - order the power differential symbols of infinity operator $ open T { and parenthesisi j so} , d , M $ closing and not the symbols of the normal parenthesis closingB depends parenthesis at most period on the second j et of f. But ,S(f) depends on the third j et of f. This \noindentProof periodprojectivecontradiction .. The cocycle connection proves property that for the S , cocycle open lead parenthesis ( to 3 . a 6 ) natural is f nontrivialclosing notion parenthesis . of .. multi follows− directlydimensional from the definition Schwarzian deriva − ..t open i v e parenthesis . 3 period 6 closing parenthesis period This cocycle is not a coboundary period Indeed comma every coboundary d B on Diff open parenthesis M closing parenthesis3 . 2 . \ withquad valThe hyphen main definition . Consider a manifold $ M $ endowed with a projective connec − tionues in . the The space expression Hom open parenthesis S to the power of 2 comma C to the power of infinity open parenthesis M closing parenthesis closing parenthesis is of the form B open parenthesis f closing parenthesis open parenthesis a closing parenthesis\ begin { a l = i g f n to∗} the power of * open parenthesis B closing parenthesis minus B comma where B in SHom ( open f parenthesis ) = S to f the ˆ{ power ∗ } of(T) 2 comma C to− the powerT, of\ infinitytag ∗{$ open ( parenthesis 3 . M 6 closing ) $ parenthesis} closing\end{ a parenthesis l i g n ∗} period Since S open parenthesis f closing parenthesis is a first hyphen order differential operator comma the coboundary con hyphen \noindentdition S =where d B would $ imply T $ that is Bthe is also ( lo a first cally hyphen defined order differential ) operator operator ( 3 and . 1so comma) , is d a B linear differential operator well defined depends at most on the second j et of f period But comma S open parenthesis f closing parenthesis depends on the third\ begin j et{ ofa l if g period n ∗} This (contradiction globally proves ) that on the cocycle M . open parenthesis 3 period 6 closing parenthesis is nontrivial period \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } Proposition 3 . 3 . \quad The \quad map $ f \mapsto S ( f ) $ \quad i s \quad a nontrivial 1 − c o c y c l e \quad onDiff $( M )$ \quad with

\noindent values inHom $ ( S ˆ{ 2 } , C ˆ{\ infty } ( M ) ) . $

Proof . \quad The cocycle property for $S ( f ) $ \quad follows directly from the definition \quad ( 3 . 6 ) . This cocycle is not a coboundary . Indeed , every coboundary d $ B $ on Diff $ ( M ) $ with val −

\noindent ues in the spaceHom $ ( S ˆ{ 2 } , C ˆ{\ infty } ( M ) )$ isoftheform $B(f)(a)=fˆ{ ∗ } (B) − B , $ where $ B \ in $

\noindent Hom $ ( S ˆ{ 2 } , C ˆ{\ infty } ( M ) ) .$Since$S ( f )$ isafirst − order differential operator , the coboundary con −

\noindent dition $S =$ d $B$ would imply that $B$ is also a first − order differential operator and so , d $ B $ depends at moston the second j et of $ f . $ But $ , S ( f ) $ depends on the third j et of $ f . $ This contradiction proves that the cocycle ( 3 . 6 ) is nontrivial . 20 .. S period BOUARROUDJ AND V period YU period OVSIENKO \noindentThe cocycle20 open\quad parenthesisS . BOUARROUDJ 3 period 6 closing AND V parenthesis . YU . OVSIENKO will be called the projectively equivariant Schwarzian derivative period It \ hspaceis clear∗{\ thatf i the l l } kernelThe cocycle of S is precisely ( 3 . the 6 )subgroup will be of Diff called open parenthesis the projectively M closing parenthesis equivariant preserving Schwarzian the derivative . It projective \noindentconnectionis period20 clearS . BOUARROUDJ that the AND kernel V . YU of . OVSIENKO $ S $ is precisely the subgroup of Diff $ ( M ) $Example preserving periodThe In the the cocycle projectively projective ( 3 . 6 ) flat will case be called comma the Capitalprojectively Pi sub equivariant ij to the Schwarzianpower of k equiv derivative 0 comma. It the cocycle open parenthesisis 3 clear period that 6 theclosing kernel parenthesis of S is precisely takes the the form subgroup : of Diff (M) preserving the projective \noindentEquation:connection openconnection parenthesis . . 3 period 7 closing parenthesis .. S open parenthesis f closing parenthesis sub ij = l open k parenthesis f closingExample parenthesis . In subthe projectively ij to the power flat case of k, partialdiffΠij ≡ 0, the divided cocycle by ( partialdiff 3 . 6 ) takes x tothe the form power : of k minus 2\ centerline divided by n{ minusExample 1 partialdiff . In the divided projectively by partialdiff flat x to the case power $ of , k parenleftbig\Pi ˆ{ k l} open{ i parenthesis j }\equiv f closing0 , $ the cocycle ( 3 . 6 ) takes the form : } parenthesis sub ij to the power of k to the powerk ∂ of parenrightbig2 ∂ plusk ) n plusn + 1 1 dividedk bym n minus 1 l open parenthesis S(f) = `(f) − (`(f) + `(f) `(f) , (3.7) f closing parenthesis sub im to the powerij ofij k∂x l openk n parenthesis− 1 ∂xk f closingij n parenthesis− 1 im subkj kj to the power of comma to the\ begin power{ a of l i mg n ∗} S ( f ) { i jk } = \ e l l ( f ) ˆ{ k } { i j }\ f r a c {\ partial }{\ partial where l openwhere parenthesis`(f)ij are f closing the components parenthesis of the sub cocycle ij to the ( 2 power . 3 ) with of k values are the in components symmetric ( of2 , the 1 ) -cocycle open parenthesisx ˆ{ k }} 2 period −tensor \ f3 r a closing c { 2 }{ parenthesisn − with1 values}\ f r a in c {\ symmetricpartial open}{\ parenthesispartial 2 commax ˆ{ 1k closing}} ( parenthesis\ e l l hyphen( f tensor ) ˆ{fieldsk } . The{ i jcocycle}ˆ{ () 3} . 7+ ) vanishes\ f r a c if{ andn only + if f 1 is}{ a linearn − - fractional1 }\ transformatione l l ( . f ) ˆ{ k } { im } \ e lfields l period( f The ) cocycle ˆ{ m open}It{ is parenthesiskj easy ˆ{ to, compute}}\ 3 periodtag this∗{ 7$ closingexpression ( parenthesis 3 in .lo cal 7 vanishes coordinates ) $} if and : only if f is a linear hyphen fractional\end{ a l i transformation g n ∗} period It is easy to compute this expression∂ in lo∂ cal3f k coordinates∂xl n :+ 3 ∂2J n + 2 ∂J ∂J \noindent whereS(f $) \=e` l( lf)k (+ f ) ˆ{ k }−{ i j }$f areJ −1 the+ componentsf f J −2 of. the(3.8) cocycle ( 2 . 3 ) with values in symmetric ( 2 , 1 ) − t e n s o r Equation: open parenthesisij 3ij period∂xk 8∂x closingi∂xj∂x parenthesisl ∂f k n + .. 1 S∂x openi∂xj parenthesisf n + f 1 closing∂xi ∂x parenthesisj f sub ij = l open parenthesis f closing parenthesis sub ij to the power of k partialdiff divided by partialdiff x to the power of k plus partialdiff to\noindent the power offieldsTo 3 f obtain to the . thisThe power formula cocycle of k divided from ( ( 3 3 by . . partialdiff7 7 ) , ) one vanishes uses x to the the relation powerif and of: onlyi partialdiff if x$ to f the $ power is a of linear j partialdiff− fractional transformation . x to the power of l partialdiff x to the power of l divided by partialdiff f to the power of k minus n plus 3 divided by n ∂3f k ∂xl ∂2f k ∂2f l ∂xm ∂xs ∂2J ∂J ∂J plus\ centerline 1 partialdiff{ It to the is power easy of to 2 J compute sub f divided this by expression partialdiff x to in the lo powerf cal− of1 coordinates i partialdifff f x−2 to : the} power of j J sub i j l k − i m j s l k = i j Jf + i j Jf . f to the power of minus 1∂x plus∂x n∂x plus∂f 2 divided∂x ∂x by∂x n plus∂x 1∂f partialdiff∂f J∂x sub∂x f divided∂x by∂x partialdiff x to the power of i partialdiff\ begin { a J l isub g nWe∗} f divided observe by that partialdiff , in the x one to the - dimensional power of j J case sub f to (n the= power 1), ofthe minus expression 2 period ( 3 . 8 ) STo obtain ( f thisis preci) formula{ - selyi j from} − open=S(f) parenthesis,\ e lwhere l (S 3is period the f classical 7 ) closing ˆ{ k Schwarzian parenthesis} { i j derivative}\ commaf r a c one{\ . usespartial ( Recall the relation that}{\ in partial : x ˆpartialdiff{ k }} to+this the\ case powerf r a c {\ of 3partial f to the powerˆ{ 3 of} k dividedf ˆ{ k by}}{\ partialdiffpartial x to the powerx ˆ{ ofi i}\ partialdiffpartial x to thex power ˆ{ j of } j\ partialdiffpartial x tox the ˆ{ powerl }}\ of lf r partialdiff a c {\ partial x to the powerx ˆ{ of l divided}}{\ bypartial partialdifff to ˆ{ thek power}} − of k \ minusf r a c { partialdiffn + to3 the}{ powern + of 2 1f to}\ thef r power a c {\ ofpartial k dividedˆ by{ partialdiff2 } J x{ tof the}}{\ powerpartial of i partialdiffx ˆ x{ toi the}\ powerpartial of m partialdiffx ˆ{ j }} `(f) ≡ 0.) toJ theˆ{ −power1 of} 2{ f tof the} power+ \ off r la divided c { n by + partialdiff 2 }{ xn to the + power 1 }\ of jf partialdiffr a c {\ partial x to the powerJ of{ sf partialdiff}}{\ partial x x ˆ{ i }}\ f r a c {\ partial J { f }}{\ partial x ˆ{ j }} J ˆ{ − 2 } { f } . \ tag ∗{$ ( to the power of m dividedRemarks by . partialdiff( a ) The f to infinitesimal the power of analogue l partialdiff of the x cocycle to the power ( 3 . 7 of ) has s divided been introduced by partialdiff f to the power3 . of k =8 partialdiff ) $} to the power of 2 J sub f divided by partialdiff x to the power of i partialdiff x to the power of j J \end{ a l i g n ∗}in [ 1 0 ] . sub f to the power( of b minus ) We will 1 plus show partialdiff in Section J sub 4 . 3 f ,divided that the by analogue partialdiff of xthe to operator the power ( 3 of . i6 partialdiff ) in the one J sub- f divided by partialdiff x to the power of j J sub f to the power of minus 2 period \noindent Todimensional obtain this case , isformula , in fact ,from the operator ( 3 . of 7 multiplication ) , one uses by the the Schwarzian relation derivative : . We observe that3 comma. 3 . in A the remark one hyphen on dimensional the projectively case .. open equivariant parenthesis cohomology n = 1 closing . parenthesisConsider comma .. the expression .. open parenthesis 3 period 8 closing parenthesisn .. is preci hyphen \ [ \ f r a c {\ partialthe standardˆ{ 3 sl} (n+1f, ˆR{)−k action}}{\ onpartialR ( by infinitesimalx ˆ{ i projective}\ partial transformationsx ˆ{ )j . The}\ partial sely .. minusfirst S open group parenthesis of differential f closing cohomology parenthesis of Vect comma (Rn) .., vanishing where S is on the the classical subalgebra Schwarzian sl (n + 1 derivative, R), period ..x open ˆ{ l parenthesis}}\ f r a c Recall{\ partial that in thisx case ˆ{ l }}{\ partial f ˆ{ k }} − \ f r a c {\ partial ˆ{ 2 } f ˆ{ k }}{\withpartial coeffi - x ˆ{ i }\ partial x ˆ{ m }}\ f r a c {\ partial ˆ{ 2 } f ˆ{ l }}{\ partial l open parenthesis f closing parenthesisk ` equiv 0 period closing parenthesis k ` x ˆ{ j }\ partialcients in the spacex ˆ{ Ds(S}}\, S )f of r a linear c {\ partial differential operatorsx ˆ{ m from}}{\S partialto S , was calculatedf ˆ{ l in}}\ f r a c {\ partial Remarks period[ 1 0 .. ] . open For n parenthesis≥ 2 the result a closing is as followsparenthesis : The infinitesimal analogue of the cocycle open parenthesis 3x period ˆ{ s 7}}{\ closingpartial parenthesis hasf ˆ{ beenk introduced}} = \ f r a c {\ partial ˆ{ 2 } J { f }}{\ partial x ˆ{ i } \ partial x ˆ{ j }} J ˆ{ − 1 } { f } + \ f r a c {\ partial J { f }}{\ partial x ˆ{ i }}\ f r a c {\ partial in open square bracket 1 0 closing square bracket period R, k − ` = 2, J { f }}{\ partial x ˆ{ j }} J ˆ{ − 2 } { f } . \ ] open parenthesis b closing parenthesis1 n We will show in Sectionk ` 4 period 3 comma that the analogue of the operator H (Vect(R ), sl(n + 1, R); D(S , S )) = R, k − ` = 1, ` 6= 0, open parenthesis 3 period 6 closing parenthesis in the one hyphen  0, otherwise dimensional case comma is comma in fact comma the operator of multiplication by the Schwarzian derivative period \noindent3 period 3We periodThe observe cocycle .. A .. ( that 3 remark . 7 ) , is ..in , in on the fact .. the ,one corresponds projectively− dimensional to .. the equivariant nontrivial case cohomology..\quad cohomology$ class( period inn the Consider = case 1 .. ) the .. , $ \quad the expression \quad ( 3 . 8 ) \quad in s p r e c i − standard k = 2, ` = 0 integrated to the group Diff (R ), while the nontrivial cohomology class in the s esl l y open\quad parenthesiscase$ −k − n` plus=S 1 1 is( comma given f by R the closing ) operator , parenthesis $ of\quad contraction hyphenwhere with action $ the S on $ tensor R to is the field the power ( 2classical . 4of ) n . open Schwarzianparenthesis by derivative . \quad ( Recall that in this case infinitesimal projectiveFor transformations any lo cally projective closing manifold parenthesisM periodit follows The that first the group cocycle of ( 3 . 6 ) generates the \ begindifferential{ a l i g ncohomologyunique∗} nontrivial of Vect class open of parenthesis the cohomology R to ofthe Diff power (M) of with n closing coefficients parenthesis in D(S comma2,C∞(M vanishing)), on the subalgebra\ e l l ( sl openvanishing f parenthesis ) on\equiv the n plus( pseudo 10 comma ) group . R closing ) of ( lo parenthesis cally defined comma ) projective with coeffi transformations hyphen . The \endcients{ a l i in g n the∗}same space fact D openis true parenthesis for the cocycle S to ( the 2 . power 3 ) . of k comma S to the power of l closing parenthesis of linear differential operators4 . from Relation S to the power to the of modules k to S to of the differential power of l comma operators was calculated. Consider , for simplicity Remarksin open . square\quad, a bracket smooth( a 1 oriented) 0 The closing infinitesimal manifold square bracketM. Denote period analogueD( ForM) n the greater of space the equal of cocycle scalar 2 the linear result ( 3 differential is . as 7 follows ) has op : been- introduced inH [to 1 the 0 power ]erators . of 1A open: C∞ parenthesis(M) → C∞ Vect(M). openThere parenthesis exists a two R to- parameter the power family of n closing of Diff parenthesis (M)− module comma sl open parenthesis n plusstructures 1 comma on D R( closingM). To parenthesis define it , one semicolon identifies D the open arguments parenthesis of differential S to the power operators of k comma S to the power\ hspace of l∗{\ closingf i l l parenthesis}( b ) We closing will parenthesis show in = Section Case 1 R 4 comma . 3 k, minus that l the = 2comma analogue Case of 2 R the comma operator k minus (l 3 . 6 ) in the one − = 1 comma l equal-negationslash 0 comma Case 3 0 comma otherwise \noindentThe cocycledimensional open parenthesis case 3 period , is 7 , closing in fact parenthesis , the is operator comma in factof multiplication comma corresponds by to the the nontrivial Schwarzian derivative . cohomology class in the case 3 .k = 3 2 . comma\quad l =A 0\ integratedquad remark to the\ groupquad Diffon open\quad parenthesisthe projectively R to the power of\quad n closingequivariant parenthesis comma\quad whilecohomology . Consider \quad the \quad standard thes l nontrivial $ ( cohomology n + 1 class , in R ) − $ action on $Rˆ{ n } ( $ by infinitesimal projective transformations ) . The first group of differentialthe case k minus cohomology l = 1 is given by of the Vect operator $ ( of contraction R ˆ{ n } with) the tensor , $ field vanishing open parenthesis on the 2 subalgebraperiod 4 closing sl parenthesis$( n period + 1 , R ) ,$ withcoeffi − For any lo cally projective manifold M it follows that the cocycle open parenthesis 3 period 6 closing parenthesis generates\noindent the cients in the space $D ( Sˆ{ k } , S ˆ{\ e l l } ) $ of linear differential operators from $ Sunique ˆ{ k nontrivial}$ to class $ S of ˆ the{\ cohomologye l l } , of $ Diff was open calculated parenthesis M closing parenthesis with coefficients in D open parenthesisin [10] S to the .For power $nof 2 comma\geq C to the2 power$ the of infinity result open is parenthesis as follows M closing : parenthesis closing parenthesis comma \ [vanishing H ˆ{ 1 on} the( open Vect parenthesis ( pseudo R ˆ{ n closing} ),sl(n+1,R);D( parenthesis group of open parenthesis lo cally defined closing paren- thesisS ˆ{ projectivek } , transformations S ˆ{\ e l l } period) The ) same = \ l e f t \{\ begin { a l i g n e d } & R , k − \ e l l =fact 2 is true , \\ for the cocycle open parenthesis 2 period 3 closing parenthesis period &4 period R .. , Relation k to− the \ modulese l l = of differential 1 , operators\ e l l period\ne Consider0 , comma\\ for simplicity comma &a smooth 0 oriented , otherwise manifold M\end period{ a l i Denote g n e d }\ Dright open. parenthesis\ ] M closing parenthesis the space of scalar linear differential op hyphen erators A : C to the power of infinity open parenthesis M closing parenthesis right arrow C to the power of infinity open\noindent parenthesisThe M cocycle closing parenthesis ( 3 . 7 period ) is There , in exists fact a , two corresponds hyphen parameter to the family nontrivial of Diff open cohomology parenthesis M class in the case closing$ k parenthesis = 2 hyphen , module\ e l l = 0 $ integrated to the group Diff $ ( Rˆ{ n } ) , $ whilestructures thenontrivial on D open parenthesis cohomology M closing class parenthesis in period To define it comma one identifies the arguments of differentialthe case operators $ k − \ e l l = 1 $ is given by the operator of contraction with the tensor field ( 2 . 4 ) . \ hspace ∗{\ f i l l }For any lo cally projective manifold $M$ it follows that the cocycle ( 3 . 6 ) generates the

\noindent unique nontrivial class of the cohomology of Diff $ ( M ) $ with coefficients in $ D ( S ˆ{ 2 } , C ˆ{\ infty } ( M ) ) , $ vanishing on the ( pseudo ) group of ( lo cally defined ) projective transformations . The same fact is true for the cocycle ( 2 . 3 ) .

4 . \quad Relation to the modules of differential operators . Consider , for simplicity , a smooth oriented manifold $M . $ Denote $D ( M ) $ the space of scalar linear differential op − erators $A : Cˆ{\ infty } (M) \rightarrow C ˆ{\ infty } (M) . $ There exists a two − parameter family of Diff $ ( M ) − $ module structures on $ D ( M ) . $ To define it , one identifies the arguments of differential operators SCHWARZIAN DERIVATIVE .. 2 1 \ hspacewith tensor∗{\ f densitiesi l l }SCHWARZIAN on M of degree DERIVATIVE lambda and\quad their values2 1 with tensor densities on M of degree mu period \noindent4 period 1with period tensor .. Differential densities operators on acting $M$ on tensor of densities degree period $ \lambda Consider$ the and the space their F sub values lambda with of tensor densities on $ Mt ensor $ o densitiesf on M comma that mean comma of sections of the line bundle open parenthesis Capital Lambda to thedegree power of $ n\ Tmu to the. power $ of * M closing parenthesis to the power of lambdaSCHWARZIAN period It DERIVATIVE is clear 2 1 that F sub lambdawith tensor is naturally densities a Diff on M openof parenthesis degree λ and M their closing values parenthesis with tensor hyphen densities module on periodM of Sincedegree M is oriented comma\ hspace F∗{\ subf lambdaµ. i l l }4 can . 1 be . identified\quad withDifferential operators acting on tensor densities . Consider the the space $ F {\lambda }$ o f C to the power of4 . infinity 1 . open Differential parenthesis operators M closing acting parenthesis on tensor densitiesas a vector . spaceConsider period the The the spaceDiff openFλ parenthesis M closing parenthesisof hyphen module structures are comma however comma different period \noindentDefinitiont periodt ensor ensor We densities densitiesconsideron theM, differential onthat $M mean operators , , of $ sections that acting of mean on the tensor line , bundle of densities sections (Λn commaT ∗M) ofλ. namelyIt the is clear comma line bundle $ ( \Lambda ˆ{ n } T ˆ{ ∗ } M ) ˆ{\lambda } . $ It is clear Equation: openthat parenthesisFλ is naturally 4 period a Diff 1 closing(M)− module parenthesis . Since .. AM : Fis sub oriented lambda, Fλ rightcan arrowbe identified F submu with period The Diff openC∞ parenthesis(M) as a vector M closing space parenthesis . The Diffhyphen (M)− module action onstructures D open are parenthesis , however M , closingdifferent parenthesis . comma depending\noindent onthat two parametersDefinition $ F {\ lambda .lambdaWe and consider} mu$ comma the is differential naturally is defined operators by a the Diff acting $( on tensor M densities ) − $ , namely module . Since $M$usual formula is oriented, : $ , F {\lambda }$ can be identified with Equation: open parenthesis 4 period 2 closing parenthesis .. f lambda comma mu to the power of open parenthesis A closing\noindent parenthesis$ C = ˆ f{\ to theinfty power} of( * minus M 1 )$ circ A asavector circ f to the power space of * comma .TheDiff $( M ) − $ A : F → F . (4.1) modulewhere structuresf to the power areof * is , the however natural Diff , different open parenthesisλ . Mµ closing parenthesis hyphen action on F sub lambda period The Diff (M)− action on D(M), depending on two parameters λ and µ, is defined by the \ hspace ∗{\ f i l l } Definition . We consider the differential operators acting on tensor densities , namely , Notation periodusual The formula Diff open : parenthesis M closing parenthesis hyphen module of differential operators on M with the action open parenthesis 4 period 2 closing parenthesis \ beginis denoted{ a l i g Dn ∗} sub lambda comma mu period For every k comma the space of differential operators of order less or (A) ∗−1 ∗ equalA:F k is a Diff{\ openlambda parenthesis}\ Mrightarrow closing parenthesisfλ, µF hyphen{\= fmu ◦}A ◦.f \,tag ∗{$ ( 4 . 1(4. )2) $} \endsubmodule{ a l i g n ∗} of D sub lambda comma mu comma denoted D sub lambda comma mu to the power of period to the power where f ∗ is the natural Diff (M)− action on F . of k λ Notation . The Diff (M)− module of differential operators on M with the action ( 4 \noindentIn this paperTheDiff we will only $( deal with M the ) special− $ case action lambda on = mu $D and use ( the M notation ) D , sub $ lambda depending on two parameters . 2 ) is denoted D . For every k, the space of differential operators of order ≤ k is a Diff $ \forlambda D sub lambda$ and comma $ \mu lambdaλ,µ, and $ f is lambda defined for f lambda by the comma lambda period usual formula(M)− : The modules D sub lambda comma mu havek already been considered in classical works .. open parenthesis see .. open submodule of D , denoted D . square bracket 1 6 closing squareλ,µ bracket closingλ,µ parenthesis .. and In this paper we will only deal with the special case λ = µ and use the notation D for \ beginsystematically{ a l i g n ∗} studied in a series of recent papers open parenthesis see .. open square bracket 4 commaλ .. 9 comma .. D and fλ for fλ, λ. 1f 0 comma\lambda .. 3 commaλ,λ , 5 closing\mu ˆ{ square(A) bracket ..} and= references f ˆ{ ∗ − 1 }\ circ A \ circ f ˆ{ ∗ } The modules D have already been considered in classical works ( see [ 1 6 ] ) , \thereintag ∗{$ closing ( parenthesis 4 . 2 period ) $λ,µ} and \end4 period{ a l i g n 2∗} period .. Projectively equivariant symbol map period From now on comma we suppose that the manifold systematically studied in a series of recent papers ( see [ 4 , 9 , 1 0 , 3 , 5 ] and M is endowed with a projective structure period It was shown in open square bracket 1 0 closing square bracket that references therein ) . there\noindent exists awhere open parenthesis $ f ˆ{ unique ∗ }$ is the natural Diff $( M ) − $ action on $F {\lambda } 4 . 2 . Projectively equivariant symbol map . From now on , we suppose that the . $up to normalization closing parenthesis projective ly equivariant symbol map comma that is comma a linear bijection manifold M is endowed with a projective structure . It was shown in [ 1 0 ] that there exists sigma sub lambda a ( unique up to normalization ) projective ly equivariant symbol map , that is , a linear Notationidentifying . the The space Diff D open $ ( parenthesis M ) M closing− $ parenthesis module of with differential the space of symmetric operators contravariant on $M$ tensor fields with the action ( 4 . 2 ) bijection σ identifying the space D(M) with the space of symmetric contravariant tensor onis M denoted period $ D {\λ lambda , \mu } . $ For every $ k , $ the space of differential operators of order fields on M. Let us give here the explicit formula of σ in the case of second order differential $ \Letleq us givek$ here isaDiff the explicit formula $( of M sigma ) sub− lambda$ in theλ case of second order differential ope hyphen ope - rators period In coordinates of the projective structure comma sigma sub lambda associates to a differential operator rators . In coordinates of the projective structure , σ associates to a differential operator \noindentEquation:submodule open parenthesis of 4 $ period D {\ 3 closinglambda parenthesis, \ ..mu A =}λ ij a, sub $ 2 denoted partialdiff divided $Dˆ{ byk partialdiff} {\lambda x to the power, \mu of i partialdiffˆ{ . }}$ divided by partialdiff x to the power of j plus i a sub 1 partialdiff divided by partialdiff x to the ∂ ∂ ∂ power of i plus a sub 0 comma A = ij + i + a , (4.3) Inwhere this a paper to the we power will of i only sub 1 ldeal to the with power thea of2 ∂x period speciali ∂xj perioda1 case∂x periodi $0 i\ sublambda l in C to= the\ powermu $ of and infinity use open the notation parenthesis$ D {\lambda M closing}$ parenthesis with l = 0 comma 1 comma 2 comma the tensor field : where ai1 `...i` ∈ C∞(M) with ` = 0, 1, 2, the tensor field : f oEquation: r $ D open{\lambda parenthesis 4, period\lambda 4 closing}$ parenthesis and $ .. fsigma\lambda sub lambda$ open f o r parenthesis $ f \ Alambda closing parenthesis, \lambda =. ij $ a-macron sub 2 partialdiff sub i oslash partialdiff sub j plus i macron-a sub 1 partialdiff sub i plus a-macron sub 0 comma σλ(A) = ija¯2 ∂i ⊗ ∂j + ia¯1 ∂i +a ¯0, (4.4) \ hspaceand is∗{\ givenf i by l l }The modules $ D {\lambda , \mu }$ have already been considered in classical works \quad ( see \quad [ 1 6 ] ) \quad and ij a-macron suband 2 is = given ij a sub by 2 Equation: open parenthesis 4 period 5 closing parenthesis .. i a-macron sub 1 = i a sub 1\noindent minus 2 opensystematically parenthesis n plus studied 1 closing parenthesis in a series lambda of plus recent 1 divided papers by n plus( see 3 partialdiff\quad [ ij 4 a sub , \ 2quad divided9 , \quad 1 0 , \quad 3 , 5 ] \quad and references byt h partialdiff e r e i n ) x . to the power of j a-macron sub 0 = a sub 0 minus lambda partialdiff i a sub 1 divided by partialdiff x to ij = ij the power of i plus lambda open parenthesis n plus 1 closing parenthesisa¯ lambda2 plusa2 1 divided by n plus 2 partialdiff to 4 . 2 . \quad Projectively equivariant symbol map(n + . 1) Fromλ + 1 ∂ij now on , we suppose that the manifold the power of 2 ij a sub 2 divided by partialdiff x to the power of i partialdiff x toa the2 power of j ia¯1 = ia1 − 2 (4.5) $M$The main is property endowed of the with symbol a projective map sigma sub structure lambda is that .n It it+ commutes was3 shown∂xj with in open [ parenthesis1 0 ] that lo cally there defined exists a ( unique up to normalization ) projective ly equivariant symbol map2 , that is , a linear bijection closing parenthesis ∂ia (n + 1)λ + 1 ∂ ija a¯ = a − λ 1 + λ 2 $ \SLsigma open parenthesis{\lambda n plus}$ 1 comma R0 closing0 parenthesis∂xi hyphenn + 2 action∂x periodi∂xj In other words comma the formula openidentifying parenthesis 4 the period space 5 closing $ D parenthesis ( M .. does ) $ not withchange the under space linearhyphen of symmetric contravariant tensor fields on $ Mfractional . $ coordinate changes open parenthesis 2 period 2 closing parenthesis period Let us give here the explicit formula of $ \sigma {\lambda }$ in the case of second order differential ope −

\noindent rators . In coordinates of the projective structure $ , \sigma {\lambda }$ associates to a differential operator

\ begin { a l i g n ∗} A = i j { a { 2 }}\ f r a c {\ partial }{\ partial x ˆ{ i }}\ f r a c {\ partial }{\ partial x ˆ{ j }} + i { a { 1 }}\ f r a c {\ partial }{\ partial x ˆ{ i }} + a { 0 } , \ tag ∗{$ ( 4 . 3 ) $} \end{ a l i g n ∗}

\noindent where $ a ˆ{ i { 1 }}{\ e l l }ˆ{ . . . i {\ e l l }}\ in C ˆ{\ infty } ( M ) $ with $ \ e l l = 0 , 1 , 2 ,$ thetensorfield:

\ begin { a l i g n ∗} \sigma {\lambda } ( A ) = i j {\bar{a} { 2 }}\ partial { i }\otimes \ partial { j } + i {\bar{a} { 1 }}\ partial { i } + \bar{a} { 0 , }\ tag ∗{$ ( 4 . 4 ) $} \end{ a l i g n ∗}

\noindent and is given by

\ begin { a l i g n ∗} i j {\bar{a} { 2 }} = i j { a { 2 }}\\ i {\bar{a} { 1 }} = i { a { 1 }} − 2 \ f r a c { ( n + 1 ) \lambda + 1 }{ n + 3 }\ f r a c {\ partial i j { a { 2 }}}{\ partial x ˆ{ j }}\ tag ∗{$ ( 4 . 5 ) $}\\\bar{a} { 0 } = a { 0 } − \lambda \ f r a c {\ partial i { a { 1 }}}{\ partial x ˆ{ i }} + \lambda \ f r a c { ( n + 1 ) \lambda + 1 }{ n + 2 }\ f r a c {\ partial ˆ{ 2 } i j { a { 2 }}}{\ partial x ˆ{ i }\ partial x ˆ{ j }} \end{ a l i g n ∗}

The main property of the symbol map $ \sigma {\lambda }$ is that it commutes with ( lo cally defined ) SL $ ( n + 1 , R ) − $ action . In other words , the formula ( 4 . 5 ) \quad does not change under linear − fractional coordinate changes ( 2 . 2 ) . The main property of the symbol map σλ is that it commutes with ( lo cally defined ) SL (n + 1, R)− action . In other words , the formula ( 4 . 5 ) does not change under linear - fractional coordinate changes ( 2 . 2 ) . 22 .. S period BOUARROUDJ AND V period YU period OVSIENKO \noindent4 period 322 period\quad .. DiffS . open BOUARROUDJ parenthesis AND M closingV . YU parenthesis . OVSIENKO hyphen module .. of s econd order differential operators period In this section we will \ hspacecompute∗{\ thef i Diffl l }4 open . 3 parenthesis . \quad MD closing i f f $ parenthesis ( M hyphen ) − action$ module f lambda\quad given byof open s econd parenthesis order 4 period differential operators . In this section we will 2 closing parenthesis with lambda = mu on the space D sub lambda to the power of 2 open parenthesis of second \noindentorder differentialcompute22 operatorsS . BOUARROUDJ the Diff open parenthesis AND $ ( V . YU M 4 . periodOVSIENKO ) 3− closing$ a cparenthesis t i o n $ actingf \lambda on lambda$ hyphen given densities by ( closing 4 . 2 ) with parenthesis$ \lambda period= 4 .\mu 3 . $Diff on ( theM)− spacemodule $Dˆ of s econd{ 2 } order{\ differentiallambda } operators( $ o . fIn second this section orderLet us differential givewe here will the explicit operators formula ( of 4 Diff . open 3 ) parenthesis acting on M closing$ \lambda parenthesis− $ hyphen densities action in ) terms . of the 2 projectively compute the Diff (M)− action fλ given by ( 4 . 2 ) with λ = µ on the space Dλ ( of second \ hspaceinvariant∗{\ symbolforder i l l } Let sigmadifferential us to give the operators power here of ( the 4 lambda . 3 explicit ) acting period on Namelyλ formula− densities comma of ) . Diff we are $ looking ( M for the ) operator− $ actionf-macron in terms of the projectively lambda = sigma sub lambdaLet us circgive f here lambda the explicit circ open formula parenthesis of Diff sigma (M)− subaction lambda in terms closing of the parenthesis projectively to the power of \noindent invariant symbolλ $ \sigma ˆ{\lambda } . $ Namely¯ , we are looking−1 for the operator minus 1 open parenthesisinvariant symbol such σ . Namely , we are looking for the operator fλ = σλ ◦ fλ ◦ (σλ) ( such $ \thatbar{ thef }\ diagramthatlambda the below diagram is= commutative below\sigma is commutative closing{\lambda parenthesis ) : }\ : circ f \lambda \ circ ( \sigma {\lambda } ) ˆEquation:{ − 1 open} ( parenthesis $ such 4 period 6 closing parenthesis .. D lambda sigma arrowvertex-arrowvertex-arrowbt to the powerthat of the 2 lambda diagram arrowbt-arrowvertex below is commutative to the power ) of : f lambda lambda from arrowvertex-arrowvertex-arrowbt sigma D2λ ↓fλ λ↓σλ (4.6) sub lambda to D to the power of 2 λσ ↓ D2 \ beginwhere{ a S l to i g then ∗} power of 2 oplus S to the power of 1 oplus S to the power of 0 is the to the power of S sub space to the D {\lambda {\2 sigma1 0}\downarrowS 2 ⊕S 1 ⊕ }ˆ{2S{\0 ¯ lambda }}\downarrowS ˆ2{ f \lambda⊕S1⊕S0 }\lambda ˆ{\downarrow power of 2 to thewhere powerS ⊕S of oplus⊕S Sis sub the ofspace to theof powersecond of 1fλarrowvertex to the power of− oplusarrowbt second− o torder thecontravariant power of S to the power of\sigma 0 macron-f{\ lambdalambdatensor arrowvertex-arrowbt-o fields}} ({ 4D . 4 ˆ ){ on2M.}}\The subtag following rder∗{$ to ( the st atement 4 power . of , Swhose 6 sub contravariant proof ) $} is straightforward to the power , shows of 2 to the power of\end oplus{ a Sl i tog n the∗}how power the ofcocycles 1 oplus ( 2S . to 3 the ) and power ( 3 . of 6 0) are tensor related fields to open the module parenthesis of second 4 period - order 4 closing differential parenthesis on M period operators . \noindentThe followingwhere st atement $Proposition S ˆ comma{ 2 }\ whose 4 .oplus 1 proof. If isdimS straightforward ˆM{ ≥1 2}\, the aoplus comma ction of showsSDiff ˆ{ (howM0 )}on the$ th cocycles i es space $ open the of the parenthesisˆ{ S }ˆ{ 22 } { space }ˆ{\oplus S }ˆ{ 1 } { o f }ˆ{\2 oplus } second ˆ{ S ˆ{ 0 }}\bar{ f }\lambda { arrowvertex−arrowbt−o } { rder }ˆ{ S }ˆ{ 2 } { contravariant }ˆ{\oplus period 3 closingspace parenthesisDλ S ˆand{ 1 open}\ parenthesisoplusof s econd 3S - operiod ˆ rder{ 0 differential 6}} closing$ tensor parenthesis operators fields , are defined related (4 by . ( to 4)on4 .the 2 , module 4 . $M6 ) ofis assecond . fo $ l lows hyphen : order differential operatorsThe following period st atement , whose proof is straightforward , shows how the cocycles ( 2 . 3 ) andProposition ( 3 . 6 4 period ) are 1 period related If dim to M the greater module equal 2 of comma second the a− ctionorder of Diff differential open parenthesis operatorsM closing parenthesis . ¯ a¯ ij ∗ ij on th e space of the space D sub lambda to the power of 2 (fλ 2) = (f a¯2) \ hspace ∗{\ f i l l } Proposition 4 . 1 . If dim $M \geq 2 ,$ theaction of Diff $( of s econd hyphen o rder differential¯ a¯ i operators∗ commai defined by openn + parenthesis 1 i −1 4∗ periodkl 2 comma 4 period 6 closing (fλ 1) = (f a¯1) + (2λ − 1) ` (f )(f a¯2) (4.7) Mparenthesis ) $ is on as th fo l e lows space : of the space $Dˆ{ 2 } {\nlambda+ 3 kl }$ parenleftbig f-macron lambda to the power¯ a¯ of macron-a∗ 2 parenrightbig2λ(λ − 1) to the−1 power∗ ofkl ij = open parenthesis f to the \noindent o f s econd − o rderfλ differential0 = f a¯0 operators− ,S definedkl(f )(f bya¯2) ( 4 . 2 , 4 . 6 ) is as fo l lows : power of * a-macron sub 2 closing parenthesis to the power of ij Equation:n + 2 open parenthesis 4 period 7 closing parenthesis

.. parenleftbig f-macron∗ lambda to the power of macron-a 1 parenrightbig to the power of i = open parenthesis f to the power\ begin of{ *a a-macron l i g nwhere∗} subf 1is closing th e natural parenthesis action to ofthe powerf on the of i symmetric plus open parenthesiscontravariant 2 lambda t ensor minus fields .1 closing parenthesis n( plus 1\bar divided{ f }\ by nRemark pluslambda 3 l sub.ˆ{\ In k l the tobar the one{a power}} - dimensional of2 i open ) case parenthesisˆ{ ,i the j } formula f= to the ( ( 4 power . 7 f ) ˆof holds{ minus ∗ true } 1 \closing ,bar recall{aparenthesis that} { 2 } open) ˆ{ i j }\\ ( \bar{ f }\lambda`(f) ≡ ˆ{\bar{a}} 1 ) ˆ{ i } = ( f ˆ{ ∗ } \bar{a} { 1 } ) ˆ{ i } parenthesis f to the power of−1 * macron-a∗ kl sub 2− closing1 ∗ parenthesis to the power of k l f-macron lambda to the power of +macron-a ( 0 2 = f0 to\ andlambda theS powerkl(f of−)(f * a-macrona¯21) = ) S sub(\f f r 0) af minus c {a¯2 nwith 2 lambda + the operator 1 open}{ n parenthesis of multiplication + 3 lambda}\ e by minus l l theˆ{ 1 classical closingi } { parenthesisk l } divided( fby ˆ{ n − plusSchwarzian1 2 S} sub) k derivative l ( open f parenthesis in ˆ{ the ∗ right } f \ to handbar the{ sidea power} ({ cf of2 . minus[} 3 ] ) .1 ˆ This closing{ k shows parenthesis l }\ thattag the∗{ open cocycle$ ( parenthesis ( 4 3 . . f to 7 the power) $}\\\ of *bar macron-a{6f )}\ is sub, indeedlambda 2 closing , itsˆ natural parenthesis{\bar generalization{a to}} the power0 . = of k lf ˆ{ ∗ } \bar{a} { 0 } − \ f r a c { 2 \lambda ( where\lambda f to the power− 1 ofNote * is ) th also}{ e naturaln that +the action formula 2 } of f (S on 1 .the{ 5k ) symmetric is a lparticular} contravariant( case f ˆ of{ ( − 4t ensor. 71 ) .} fields) period ( f ˆ{ ∗ } \barRemark{a} period{ 2 } In4 the) . 4 ˆ one{ .k hyphen Module l } dimensional of differential case operators comma the as formula a deformation open parenthesis . The space 4 period of differential 7 closing parenthesis \end{ a l i g n ∗}op - holds true comma recall that2 l open parenthesis f closing parenthesis equiv 0 and S suberators k l openDλ parenthesisas a module f to over the the power Lie algebra of minus of 1 vector closing fields parenthesis Vect (M open) was parenthesis first studied fto in the power of * a-macron\noindent subwhere 2[ closing 4 ] , it parenthesis$ was f ˆ shown{ ∗ to } that$ the is powerthis th module of e k natural l = can S open be naturally action parenthesis considered of f to $ the f$ aspower a on deformation of the minus symmetric 1 closing of the parenthesis contravariant t ensor fields . f to the power ofmodule * macron-a of tensor sub fields 2 with on theM. operatorProposition of multiplication 4 . 1 extends by this the result classical to the level of the diffeo - \ hspace ∗{\ fmorphism i l l }Remark group . Diff In (theM). The one formula− dimensional ( 4 . 7 ) shows case that , the the Diff formula (M)− module ( 4 of . second7 ) holds true , recall that Schwarzian derivative in the right hand side open2 parenthesis cf period open square bracket 3 closing square bracket closing$ \ e l l parenthesis(order f period differential ) This\equiv shows operators that$ ontheM cocycleDλ is a opennontrivial parenthesis deformation 3 periodof 6 closingthe module parenthesis of tensor fields T 2 generated by the cocycles ( 2 . 3 ) and ( 3 . 6 ) . is comma indeed comma its natural generalization period 1 \noindentNote also that0 and theIn formula $the S one{ open -k dimensional parenthesis l } ( case 1 period, f the ˆ{ Diff − 5 closing (S 1)−} parenthesismodules) ( of differential is f a particular ˆ{ ∗ operators } case \bar of{ and opena} the{ parenthesis2 } ) 4 ˆ{ k periodl } = 7 closing Srelated parenthesis ( higher f ˆ{ period − order1 analogues} ) of f the ˆ{ Schwarzian ∗ } \bar derivative{a} { was2 } studied$ with in [ 3the ] . operator of multiplication by the classical Schwarzian4 period 4 period derivative ..Acknowledgments Module inof differential the right . operatorsIt hand is a pleasure as side a deformation to( acknowledge cf . [ period 3 ] numerous The ) . space This fruitful of shows differential discussions that op thehyphen cocycle ( 3 . 6 ) iserators , indeed D subwithlambda , Christian its naturalto the Duval power generalizationand of his 2 as constant a module interest . over in the this Lie work algebra ; of vector we fieldsare also Vect grateful open toparenthesis M closing parenthesisPierre was Lecomte first studied for fruitful in discussions . \ centerlineopen square{ bracketNote also 4 closing that square the bracket formula comma ( 1 it was . 5 shown ) is that a particular this module can case be naturally of ( 4 considered . 7 ) . as} a deformation of the \ hspacemodule∗{\ of tensorf i l l }4 fields . 4 on . M\ periodquad Module Proposition of 4 differential period 1 extends thisoperators result to theas levela deformation of the diffeo hyphen . The space of differential op − morphism group Diff open parenthesis M closing parenthesis period The formula open parenthesis 4 period 7 closing parenthesis\noindent showse r a t that o r s the $ Diff D ˆopen{ 2 parenthesis} {\lambda M closing}$ parenthesis as a module hyphen over module the of Lie second algebra of vector fields Vect $ (order M differential ) $ wasoperators first on M studied D sub lambda in to the power of 2 is a nontrivial deformation of the module of tensor fields T to the power of 2 generated by the cocycles open parenthesis 2 period 3 closing parenthesis and open parenthesis 3\noindent period 6 closing[ 4 parenthesis ] , it was period shown that this module can be naturally considered as a deformation of the moduleIn the one of hyphen tensor dimensional fields on case comma$M the . $ Diff Proposition open parenthesis 4 S . to 1 the extends power of this 1 closing result parenthesis to the hyphen level of the diffeo − modulesmorphism of differential group Diff operators $ and( the M ) . $ The formula ( 4 . 7 ) shows that the Diff $ ( M)related− higher$ order module analogues of second of the Schwarzian derivative was studied in open square bracket 3 closing square bracket period \noindentAcknowledgmentsorder period differential It is a pleasure operators to acknowledge on $M numerous D ˆ fruitful{ 2 } discussions{\lambda }$ is a nontrivial deformation of the module of tensor with Christian Duval and his constant interest in this work semicolon .... we are also grateful to \noindentPierre Lecomtef i e l d for s fruitful $ T ˆ discussions{ 2 }$ period generated by the cocycles ( 2 . 3 ) and ( 3 . 6 ) . \ hspace ∗{\ f i l l } In the one − dimensional case , the Diff $ ( S ˆ{ 1 } ) − $ modules of differential operators and the

\noindent related higher order analogues of the Schwarzian derivative was studied in [ 3 ] .

\ hspace ∗{\ f i l l }Acknowledgments . It is a pleasure to acknowledge numerous fruitful discussions

\noindent with Christian Duval and his constant interest in this work ; \ h f i l l we are also grateful to

\noindent Pierre Lecomte for fruitful discussions . SCHWARZIAN DERIVATIVE .. 23 \ hspaceReferences∗{\ f i l l }SCHWARZIAN DERIVATIVE \quad 23 open square bracket 1 closing square bracket .. L period V period Ahlfors comma .. Cross hyphen ratios and Schwarzian\ centerline { References in R to the} power of n comma in : comma Birkh a-dieresis user comma Boston comma 1 989 period [ 1open ] \ squarequad L bracket . V 2. closing Ahlfors square , \ bracketquad Cross .. R period− ratios Bott comma and SchwarzianSCHWARZIAN On the characteristic DERIVATIVE derivatives classes23 in of groups $ R ˆ of{ n } diffeomorphisms, $ in : Complex comma EnseignAnalysis period , Math period 23References Birkhopen parenthesis $ \ddot[ 1{ 1a ] 977} $ L closing . V user . Ahlfors parenthesis , Boston, commaCross , 1 - 209 989 ratios endash . and Schwarzian 2 20 period derivatives in Rn, in : Complex open squareAnalysis bracket , 3 Birkh closinga¨ squareuser , Boston bracket , 1 .. 989 S period . Bouarroudj and V period Ovsienko comma Three cocycles on[ 2 Diff ] open\quad parenthesisR[ . 2 ]Bott S R to . ,Bott the On power the, On of the characteristic 1 characteristic closing parenthesis classes classes generalizingof groups of of diffeomorphisms thegroups Schwarzian of diffeomorphisms, Enseign . Math , Enseign . Math . 23 (derivative 1 977 ) comma. , 23 209 (Internat 1−− 9772 ) , 20 209 period –. 2 20 Math . period Res period Notices 1 998 comma No period 1 comma 25 endash 39 period [ 3 ] S . Bouarroudj and V . Ovsienko , Three cocycles on Diff (S1) generalizing the [ 3open ] \ squarequad Schwarzian bracketS . Bouarroudj 4 closing derivative square and, Internat bracket V .Ovsienko . .. Math C period . Res Duval , . Notices Three and 1 Vcocycles 998 period , No . Ovsienko 1 ,on 25 – Diff 39 comma . $ Space ( S of ˆ second{ 1 } order) $ lineargeneralizing differential operators the[ 4 ] Schwarzian C as . Duval a moduleand V . Ovsienko , Space of second order linear differential operators as a derivativeover the Liemodule algebra , Internat of over vector the . Liefields Math algebra comma . of Res vector Adv . period fields Notices in, Adv Math 1 . in 998 period Math , 1 .No32 1 32 open . ( 1 1 9 parenthesis , 97 25 ) ,−− 3 1 639 1 – 9 333 97 . .closing parenthesis comma 3 1 6 endash[ 5 333 ] period C . Duval and V . Ovsienko , Conformally equivariant quantization , Preprint CPT [ 4open ] \ squarequad , bracketC 1 998 . Duval . 5 [ closing 6 ] and A square . V A . .Kirillov bracket Ovsienko .. C, Infinite period , Space dimensional Duval of and second V Lie period groups order Ovsienko : their linear orbits comma , invariants Conformallydifferential and equivariant operators as a module quantizationover the comma Lierepresenta algebra Preprint - tions CPT of . vector The comma geometry 1 fields 998 of period moments , Adv, . Lect in . Math Notes in . Math 1 32 . , ( 9 701 Springer9 97 ) - Verlag, 3 1 1 6 −− 333 . open square9 bracket 82 , 1 1 6 – 1closing 23 . [ square 7 ] S bracket . Kobayashi .. A periodand C A. Horst period, KirillovTopics in comma complex Infinite differential dimensional geometry Lie groups : their[ 5 orbits ] \quad comma,C in : . invariants Complex Duval Dif and - representa ferentialV . Ovsienko Geometry hyphen , , Birkh Conformallya¨ user Verlag equivariant , 1 9 83 , 4 – 66 quantization . , Preprint CPT , 1 998 . [tions 6 ] period\quad TheA[ 8 geometry . ] A . S . KirillovKobayashi of moments ,and comma Infinite T . Nagano Lect period dimensional, NotesOn projective in Math Lie periodconnections groups comma :, their J 9 . 70 ofSpringer Math orbits . and hyphen , invariants Verlag and representa − 1tions 9 82 comma . The 1Mech 1 geometry endash . 1 3 23( of period 1 964 moments ) , 2 1 5– , 2 Lect 35 . . Notes in Math . , 9 70 Springer − Verlag 1 9 82 , 1 1 −− 1 23 . [open 7 ] square\quad bracketS[ 9 . ] Kobayashi 7 P closing . B . A square . Lecomte and bracket C .,P. Horst ..Mathonet S period , Topics Kobayashiand E in. Tousset complex and C, periodComparison differential Horst of comma some modules geometry Topics in complex , in : Complex Dif − differentialferential geometry Geometryof the comma Lie algebra , in Birkh : Complex of vector $ \ Dif fieldsddot hyphen{,a Indag} $ . userMath . Verlag , N . S . , , 7 1 ( 1 9 996 83 ) , , 461 4 −− – 47166 . . [ 1 0 ] ferential GeometryP.B.A. commaLecomte Birkhand a-dieresis V . Ovsienko user Verlag, Projectively comma 1 invariant9 83 comma symbol 4 endash calculus 66, period Lett . Math . [ 8open ] \ squarequad Phys bracketS . . Kobayashi, to 8 appear closing . square and[ 1 1 ] T bracket . R Nagano . Molzon .. S period , and\quad Kobayashi K . POn . Mortensen projective and T period, The connections Nagano Schwarzian comma derivative , .. J On . projective of Math . and Mech . \quad 1 3 connections( 1 964 comma )for , 2 maps J 1 period 5 between−− of2 Math manifolds 35 . period with and complex Mechperiod projective .. 1 connections 3 , Trans . of the AMS 348 ( 1 996 open parenthesis) , 3015 1 964 – 3036 closing . [parenthesis 1 2 ] B comma . Osgood 2 1and 5 endash D . Stowe 2 35 period, The Schwarzian derivative and [ 9open ] \ squarequad conformal bracketP . B 9 . mapping closing A . Lecomte square of Rie - bracket mannian , P .. . P Mathonet manifolds period B period, andDuke E AMath period . Tousset . J .Lecomte 67 ( 1 , 992 comma Comparison ) , 57 P – 9period 9 . of [ Mathonet 1 some modules and of the ELie period algebra Tousset3 ] comma of V vector . Ovsienko Comparison fields, Lagrange of some , Indag modules Schwarzian . Math of thederivative . , N and . symplectic S . , 7 Sturm ( 1 theory 996 ) . ,Ann 461 . Fac−− 471 . [Lie 1 0 algebra ] \quad of. vector SciP . Toulouse . fields B . comma A Math . Lecomte . Indag 6 ( 1 period993 and ) , noMath V . 1. period, Ovsienko 73 – 96 comma . [ 14 , N ] Projectively period V . Retakh S periodand comma invariant V . Shander 7 open symbol parenthesis calculus 1 , Lett . Math . 996Phys closing . , parenthesis to, appearThe comma Schwarz . 461 derivative endash for 471 noncommutative period differential algebras . Unconventional Lie [open 1 1 square ] \quadalgebras bracketR . 1, 0 MolzonAdv closing . Soviet square and Math K bracket . . P 1 7 . .. ( MortensenP 1 period 9 93 ) B, 1 period 39 , – The 1 A 54 period Schwarzian . [ 1 Lecomte 5 ] S derivativeand . Tabachnikov V period Ovsienko for, maps comma between manifolds Projectivelywith complex invariantProjective projective symbol connections calculus connections , group comma Vey Lett cocycle , period Trans , and Math deformation . ofperiod the quantiza AMS 348 - tion ( 1 . 996Internat ) , . 3015Math .−− 3036 . 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