POISSON GEOMETRY \ centerlineBANACH CENTER{POISSON PUBLICATIONS GEOMETRY } comma VOLUME 5 1 INSTITUTE OF MATHEMATICS \noindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 5 1 WARSZAWA 2 0 0 \ centerlineSCHWARZIAN{INSTITUTE DERIVATIVE 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 connection{ 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 differential operator $ 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 derivatives{ References in R to the} power of n comma in : Complex Analysis 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 . [Phys 1 2 period ] \quadRes comma .B Notices . to Osgood appear 1 996 period, andNo . 14D , . 705 Stowe – 722 . , [\quad 1 6 ]The E . J Schwarzian . Wilczynski derivative, Projective differential and conformal mapping of Rie − mannianopen square manifoldsgeometry bracket of1 ,1 curves closingDuke and Math square ruled .surfaces bracket J . 67, .. Leipzig R ( period 1 -992 Teubner Molzon ) , - 57 1 and 906−− K . 9 period 9 . P period Mortensen comma The Schwarzian[ 1 3 ] derivative\quad V for . maps Ovsienko between , manifolds Lagrange Schwarzian derivative and symplectic Sturm theory . Ann . Fac . Sciwith . complex Toulouse projective Math connections . 6 ( 1 993 comma ) Trans, no period . 1 , of 73 the−− AMS96 348 . open parenthesis 1 996 closing parenthesis comma[ 14 3015 ] \quad endashV 3036 . Retakh period and V . Shander , \quad The Schwarz derivative for noncommutative differential a lopen g e b r square a s . bracket\quad 1Unconventional 2 closing square bracket Lie algebras .. B period , Osgood Adv . and Soviet D period Math Stowe . 1 comma 7 ( 1.. The9 93 Schwarzian ) , 1 39 −− 1 54 . derivative[ 1 5 ] and\quad conformalS . mapping Tabachnikov of Rie hyphen , Projective connections , group Vey cocycle , and deformation quantiza − tionmannian . Internat manifolds comma . Math Duke . Res Math . period Notices J period 1 99667 open , No parenthesis . 14 , 1 992705 closing−− 722 parenthesis . comma 57 endash 9[ 9 period 1 6 ] \quad E . J . Wilczynski , Projective differential geometry of curves and ruled surfaces , Leipzig − Teubneropen square− 1 bracket 906 . 1 3 closing square bracket .. V period Ovsienko comma Lagrange Schwarzian derivative and symplectic Sturm theory period Ann period Fac period Sci period Toulouse Math period 6 open parenthesis 1 993 closing parenthesis comma no period 1 comma 73 endash 96 period open square bracket 14 closing square bracket .. V period Retakh and V period Shander comma .. The Schwarz derivative for noncommutative differential algebras period .. Unconventional Lie algebras comma Adv period Soviet Math period 1 7 open parenthesis 1 9 93 closing parenthesis comma 1 39 endash 1 54 period open square bracket 1 5 closing square bracket .. 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