SOME GEOMETRIC ASPECTS OF VARIATIONAL PROBLEMS IN FIBRED MANIFOLDS

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1 Electoc tascptos Mathematcal Isttute Slesa Uvesty Opava Czech Republc August Ths tet s a electoc tascpto of the ogal eseach pape D Kupa Some Geometc Aspects of Vaatoal Poblems Fbed Mafolds Fola Fac Sc Nat Uv Pu Buess Physca 4 Bo (Czechoslovaa) 973 Typogaphcal ad spellg eos have bee coected SOME GEOMETRIC ASPECTS OF VARIATIONAL PROBLEMS IN FIBRED MANIFOLDS DEMETER KRUPKA Depatmet of Theoetcal Physcs J E Puyě Uvesty Bo Czechoslovaa Uvesta J E Puyě v Bě

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3 Cotets Itoducto 5 A few toductoy emas Fudametal stuctues ad deftos The Eule mappg Fst vaato fomula Ivaat vaatoal poblems Some fomulas Notato Fudametal stuctues 8 Fbeed mafolds Fbeed chats Jets The taget budle of J Y The total taget budle The budle of vetcal vectos o J Y Mophsms of locally cove mafolds Addtoal emas The budle of covaat atsymmetc tesos o J Y 3 Jet pologatos of vecto felds 5 Pojectable vecto felds Jet pologatos of local automophsms Jet pologatos of vecto felds Ifte jet pologatos of local automophsms ad vecto felds Total ad vetcal vecto felds o J Y Eample Jet pologatos of the Le bacet 4 Hozotal ad pseudovetcal dffeetal foms o jet pologatos of fbeed mafolds Hozotal ad pseudovetcal foms Ma popetes of hozotal ad pseudovetcal foms Eample: Hozotal ad pseudovetcal foms o JY A eteso of h to ( + ) -foms Dffeetal foms o J Y Eample Two fomulas fo the fte pull-bac of foms 5 The Eule mappg 9 Eplct fomula fo d whee s a -fom hozotal wth espect to Eplct fomula fo hd Lepaga foms Lagaga foms ad the Eule foms A ote o a classcal poposto The Eule mappg Lagage fuctos of degee leadg to zeo Eule fom Eamples 6 Caocal vaatoal poblems 38 Vaatoal poblems o compact base mafolds A ote o the theoy of vaatoal poblems Vaatoal poblems o o-compact base mafolds Fst vaato fomula: Local cosdeatos Local fst vaato fomula: Some specal vaatos Fst vaato fomula Lepaga foms assocated wth a gve hozotal Lagaga fom Ctcal pots of caocal vaatoal poblems

4 7 Ivaat vaatoal poblems 53 Classes of symmety tasfomatos Ivaat tasfomatos Geealzed vaat tasfomatos Symmety tasfomatos Eample Geeal covaace Dffeetal cosevato laws 8 Summay 63 Acowledgmets 64 Refeeces 64

5 FOLIA FACULTATIS SCIENTIARUM NATURALIUM UNIVERSITATIS PURKYNIANAE BRUNENSIS TOMUS XIV PHYSICA 5 OPUS 973 Itoducto A few toductoy emas Ths wo cotas some otes o the geometc stuctue of the calculus of vaatos fbeed mafolds Let ( Y X) be a fbeed mafold ( JY X) ts -jet pologato (Secto ) = dm X Deote by Γ( ) the set of all coss sectos of the fbeed mafold ( Y X) If thee s gve a -fom o JY we ca tegate the -fom j o X obtaed by meas of the -jet pologato j of ay coss secto Γ( ) ad get to each compact doma c X the umbe ( ) = j c c Ths wo taes otce of the behavo of the eal fucto () Γ( ) c ( ) R We ae teested especally the most smple ad fequetly appled case = The wo deals wth the followg questos: ) what ae the easoable udelyg geometc stuctues fo the calculus of vaatos of fuctoals (); what ae the objects appeag the stadad classcal calculus of vaatos fom the geometcal pot of vew; ) how ca oe obta codtos fo a coss secto Γ( ) to be a etemal of the fuctoal () a completely vaat mae; how to obta vaat fst vaato fomula; 3) what s the geometcal stuctue of the Eule equatos ow fom the vaatoal calculus; what -foms lead to "detcally satsfed" Eule equatos; 4) what ae the so-called caocal vaatoal poblems fbeed mafolds ad how ca be a gve poblem tasfomed to the caocal fom; 5) what ae the symmety tasfomatos of a vaatoal poblem how ca oe fomulate vaous poblems coceg the behavo of () ude some mappgs of the fbeed mafold ( Y X) ; 6) what ae the so-called geeally vaat vaatoal poblems ad what ae ecessay ad suffcet codtos fo to defe such a poblem I addto some eplct calculatos ae pefomed ad the coespodg fomulas ae gve Fo some othe questos we do ot touch elated to the fuctoal () we efe to [] [3] 5

6 Fudametal stuctues ad deftos Ou appoach to the calculus of vaatos s based o the theoy of jets As the ma souce we use fo ths [7]; see also [] [3] [] [3] [3] [35] [38] [39] I may of these wos the theoy of jets s appled to the calculus of vaatos (eg [] [] [3] [35] [38]) We also use the budle of fte jets (see [39]) As a effect the fst vaato fomula (6) may be wtte a compact fom also fo o-vetcal vaatos (compae wth the fomula (63)) As a seous dffculty the use of the fte jets a moe systematc way we feel that geeal ou cocept of the jet pologato of pojectable vecto felds the fte jet pologato has some uusual popetes: ts "hozotal" ad "vetcal" pats do ot geeate local oe-paamete goups of tasfomatos (at least the sese we wo wth; compae wth the eposto Secto 3) I Secto 3 we toduce the oto of a pojectable vecto feld o Y ad polog such a vecto feld to the jet spaces Ths s motvated by Tautma's cosdeatos coceg symmetes of vaatoal poblems [38] who has used them a coodate fom (see also [35] ad []) Ths tme t s clea that such vecto felds ae vey useful fo the calculus Secto 4 cotas deftos of hozotal ad pseudovetcal foms o the jet spaces JY The fst defto s obvously ow (see eg []) whle the secod oe appeag [] s motvated by what s sometmes called the Lepage's equvalece elato (see [4] [7] [35]) The elato assgs -foms defg the same vaatoal poblem ad ca be epessed fbeed mafolds by meas of a lea mappg h (4) fom the space of geeal foms to the space of hozotal foms The mappg h ca be eteded fom -foms to ( +) -foms (see (45)); ths way we come to the oto of the so-called Lepaga -foms These ae closely elated to the Eule fom (59) ad at the same tme to the Eule equatos fo etemals As fo the oto of vaatos we ote that the oly type of vaatos used ths wo s that oe defed by vecto felds Moe pecsely we shall wo oly wth such "defomatos" of coss sectos that ca be defed as local oe-paamete goups of tasfomatos of the udelyg mafolds The Eule mappg Roughly speag by the Eule mappg we mea the mappg assgg to each -fom (o to each Lagage fucto) the set of Eule epessos (left-had sdes of the Eule equatos fo etemals cosdeed as fuctos o a jet space) o as we say the Eule fom (see (53)) We gve a complete chaactezato of the Eule mappg descbg ts eel (Secto 5) Ths s doe tems of eteo foms ad the eteo dffeetal The esult pesets a geealzato of a well ow classcal theoem [8] coceg the so-called dvegece epessos to the case of moe "depedet vaables" o whch s the same to the case of multple tegals () A eplct fomula fo Lagage fuctos leadg to zeo Eule epessos s deved (see (56)) Fst vaato fomula Leag o the theoy of Lepaga -foms (Secto 5) the basc fomula of the calculus of vaatos of fuctoals () the fst vaato fomula s deved a completely vaat way Ths s the fst vaato fomula of the so-called caocal vaatoal poblems If the gve -fom s ot 6

7 Lepaga oe ca apply the pocedue of Satyc [35] geealzg É Cata's [6] ad Lepage's appoach to the vaatoal poblems ad thus obta a equvalet vaatoal poblem the caocal fom (see also [4]) The pocedue of Satyc s eplaed a somewhat dffeet way Secto 5 (47) Aothe "caozato" pocedue s descbed by (67) We ote that ou cocept of caocal vaatoal poblems s essetally due to Hema [3] [4] Ivaat vaatoal poblems Cosdeatos coceg vaat vaatoal poblems ae also of geometcal atue Fo stace the mpotat fomula (75) o (76) used fo a classfcato of symmety tasfomatos s deved a completely tsc mae wthout efeeces to a coodate system (compae wth [38]) The classfcato of symmetes of the caocal vaatoal poblems s studed due to Tautma The poblem of ctcal pots (etemals) wth pescbed symmety tasfomatos s stated ad ts soluto s obtaed athe a smple fom (see (7) (74)) It s chaactezed as the soluto of a system of Eule equatos depedg o geeatos of the pescbed oe-paamete goups of tasfomatos As a teestg specal case that may appea the vaatoal calculus teso budles we dscuss the so-called geeally covaat vaatoal theoes ad obta codtos fo the -fom defg such theoy (7) (See especally [38] [35] [] ad also [] [3] [5] [8] [9] [] [5] [8] [9] [3] [34] [36] [37]) Some fomulas I ths paagaph thee ae collected some fomulas fequetly eeded calculatos We ofte use them wthout eplctly metog t Let f: X Y be a mophsm of mafolds X Y let be a dffeetal p-fom o Y; the the pull-bac f of by f s a p-fom o X defed as f = ( o f) ( Tf) p (see eg [6]) Let Z be aothe mafold ad gz : Xa mophsm The ( f o g) = g f f d= df If s aothe fom o Y the f ( ) = f f I these fomulas Tf deotes the taget map to f d ad deote the eteo dffeetal ad the wedge poduct of foms espectvely Let be a vecto feld o Y ad t ts oe-paamete local goup of tasfomatos of Y The Le devatve ( ) of by s defed as d ( ) t dt = { } Tae a pot y Y ad taget vectos K p to Y at the pot; the the cotacto of by s a ( p ) -fom o Y defed as 7

8 ( ) K = ( ) K p p Hee deotes the atual pag of foms ad vecto felds Let be two abtay foms o Y ad let be of degee The the toduced opeatos obey the followg ules (omttg the stadad bleaty ad leaty codtos): ( ) = ( ) d+ d( ) ( )( ) = ( ) + ( ) ( ) d= d ( ) ( )( ) = ( ) + ( ) ( ) If s aothe vecto feld o Y the ( ) ( ) = ( ) ( ) (Fo deftos ad popetes of these opeatos see eg [7] [] [3] [6] [36]) Notato All thoughout the wo the followg stadad otato s used: R - -dmesoal eal Eucldea space R = R - the eal le ( X) - a atlas o a mafold X t - the local oe-paamete goup geeated by a vecto feld X - a -dmesoal eal oetable paacompact mafold ( TX X X) - the taget budle of X ( Y X) - a fbeed mafold - a volume elemet fom o X Γ( ) - the set of all coss sectos of ( Y X) ( J X) - the -jet pologato of ( Y X) All mafolds ad dffeetable mappgs (mophsms) ae supposed to be of class C The composto of mappgs f ad g s deoted by o; we also wte go f = gf The -jet pologato of a coss secto Γ( ) s defed by ( j )( ) = j Fally we ote that we use the usual summato coveto thoughout the wo but we show summato detal f eeded especally moe complcated fomulas If the symbol of summato s omtted we suppose that the summato s obvous Fudametal stuctues Fbeed mafolds By a fbod mafold we mea each tple ( Y X) whee Y ad X ae dffeetable mafolds ad :Y X s a sujectve submeso A mo- 8

9 phsm : X Y satsfyg o = d X wll be called a coss secto of the fbeed mafold ( Y X) The set of all coss sectos of ( Y X) wll be deoted by Γ( ) Thoughout the wo we suppose that we ae gve a fbeed mafold ( Y X) wth fte-dmesoal Y ad X; we deote = dm X + m= dm Y whee ad m ae some postve teges Fbeed chats By defto of to each y Y oe ca fd a chat ( V ) wth cete y such that s of the fom () ( y) ( ( ( y)) ( y)) = fo some chat o X ( ( V ) ) wth cete ( y ) Chats of the descbed popetes wll be called fbeed chats [5] We usually wte ( y ) m fo the coodate fuctos defed by () ad = ( ) Jets I ths pape we feely use the symbols / D fo patal devatves ad D fo devatves the sese of [] [9] Let X Γ ( ) ad let be a abtay atual umbe The coss sectos ae sad to be -equvalet at the pot f ( ) = ( ) ad f thee ests a fbeed chat ( V ) wth cete ( ) = ( ) such that D ( ( )) = D ( ( )) fo all = K ; hee = ( ) The class of -equvalece at cotag a coss secto s deoted j ad called the -jet of at the pot The set of all j whee X ad Γ( ) s deoted by JY o just J f thee s o dage of cofuso We put JY= Yad defe mappgs s s ad by the fomulas () s ( j) = j ( j) = If Γ( ) we defe j ( ) = j s ad call the mappg j the -jet pologato of s m Deote by L ( R R ) the vecto space of -lea symmetc mappgs fom R to R m Let ( V ) be a fbeed chat o Y = ( ) ad cosde a pa ( Z ) whee Z = ( V) ad fo j Z (3) ( j ) = ( ( ) ( ( )) D ( ( )) K D ( ( ))) m m Evdetly maps Z to R R L( R R ) K L ( R R ) It s ow that m s 9

10 thee ests a uque mafold stuctue o JY such that fo ay choce of ( V ) ( Z ) s a chat We shall always cosde JY as a dffeetable mafold wth ths stuctue s It ca be show the that the tples ( JY s JY) ( JY X) ae fbeed mafolds ad each j defed by Γ( ) a coss secto of ( JY X) We call ( Z ) the caocal chat o JY (assocated wth ( V )) The fbeed mafold ( JY X) s called the -jet pologato of ( Y X) We also spea about JY as the -jet pologato of Y; o cofuso ca possbly ase fom ths If we wte ( y ) fo the coodates o Y defed by a fbeed chat ( V ) the fo the sae of smplcty of otato coodates o JY defed by ( Z ) wll be deoted by (4) = ( y z z K z K ) whee K s (Compae wth [] [7] [] [3]) s Ifte jets Now we wsh to dscuss the case whe = The the set J Y s defed the same mae as the pevous paagaph the set JY J Y wll be gve a fte dmesoal mafold stuctue modeled o a locally cove Fechet space Sce thee appea oly a few fte-dmesoal mafolds ths pape t seems to be most coveet to toduce them wthout efeeces to the geeal deftos (see [39] [3]) m Let us cosde the fte dmesoal vecto space L s ( R R ) of -lea symmetc mappgs fom R to R m Defe m m s F = R R L ( R R ) = ad cosde F the stuctue of a locally cove space wth the topology of pojectve lmt of spaces R R m m L( R R ) L m s ( R R ) K by the atual pojectos Wth the descbed stuctue F s a Fechet space [33] (fo detals see [5] [33]) Let ( Y X) be ou fbeed mafold Fo a fbeed chat ( V ) o Y let us put Z = ( V) whee J Y Y deotes the atual pojecto Put fo j Z : (5) ( j ) = ( ( ) ( ( )) D ( ( )) D ( ( )) K) The mappg taes values the Fechet space F If ( Y ) s a atlas o Y fomed by fbeed chats we ca defe ( Z ) fo each ( V ) A( Y) the same mae as above Cosde J Y wth the topology of the pojectve lmt by the mappgs The the followg codtos hold: ) U Z = J Y ) :Z F s homeomophsm of Z oto the ope set ( Z ) F 3) If Z Z the the mappg

11 s of the fom : ( Z Z ) ( Z Z ) ( ( ) D( ) D ( ) K) ( ( ) ( ( )) D ( ( )) o( d D ( )) od ( ( )) K) ad s theefoe dffeetable the sese of the dffeetal calculus locally cove spaces ([39] III 3) I othe wods the set ( J Y) of all pas ( Z ) s a locally cove atlas of class C o J Y ad ca be used to defe the stuctue of a locally cove dffeetable mafold of class C J Y modeled o the Fechet space F We always cosde J Y as ths dffeetable mafold ad call each pa ( Z ) the caocal chat o J Y (assocated wth ( V )) Cosde the mappgs ; t s easly see that they ae submesos the sese of the theoy of locally cove mafolds [39] Ths s why we shall spea about the tples ( J Y J Y) ( J Y X) = K as about fbeed mafolds The fbeed mafold ( J Y X) wll be called the fte jet pologato of ( Y X) Smlaly as the fte dmesoal case we shall also call J Y the fte jet pologato off Y ad wte J Y = J (Deftos ad poofs ca be cosulted wth [] [3] [6] [3]) The taget budle of J Y Cosde the locally cove mafold J Y ad the atlas ( J Y) o t defed befoe Let j J Y be a abtay pot Let us cosde the set of the tples ( Z w) whee j Z ( Z ) A( J Y) ad w F We have a equvalece elato ~ the set defed as follows: (6) ( Z w )~( Z w ) D ( ( j )) w = w I ageemet wth the usual temology [6] the equvalece class wth espect to the equvalece elato s called the taget vecto to the mafold J Y at the pot j I ths Secto we shall wte { Z w} j fo the taget vectos cotag the tple ( Z w) The set of all taget vectos at a pot j s called the taget space to the mafold J Y at j ad s deoted T J Y j I the followg we shall wo wth the vecto space Lb ( F F) of cotuous lea mappgs fom F to F wth the topology of ufom covegece o all bouded subsets of F; t ca be checed that Lb ( F F) s a Fechet space [33] (see also [5]) Put TJ Y = U T J Y ({ Z w} ) = j j j j ad fo a abtay fbeed chat ( V ) A( Y) wth the coespodg caocal chat ( ) o J Y (7) Z j j T { Z w} = D ( ( j )) w T ({ Z w} ) = ( ( j ) T { Z w} ) j j j

12 Cosde the set TJ Y wth the uquely detemed topology defed by the assumpto that fo each ( Z ) A( J Y) ( Z ) s a ope set ad T s a homeomophsm The the tple ( TJ Y J Y) has the followg popetes: ) If ( J Y) = {( Z )} s a atlas o J Y the ( TJ Y) = {( ( Z) T)} s a atlas o TJ Y ( the sese of [39]) ) Fo each j J Y the mappg T j defed by chat ( Z ) wth cete j s a bjecto ad defes T J Y the stuctue of a locally cove space j somophc to F 3) Deote p: Z F F the atual pojecto To each thee s a somophsm : ( Z) Z F such that o ( Z ) = p o ad that the estcto j of to ( j ) s a somophsm of locally cove spaces Futhemoe fo Z Z the mappg j b Z Z j ( o ) L ( F F) s a mophsm (e s dffeetable) The poof of all these assetos s staghtfowad ad s based o [39] [33] [5] Wth the defed dffeetable stuctue the tple ( TJ Y J Y) s sad to be the taget budle of J Y The total taget budle Let j J Y be a pot ad let ( Z ) be a caocal chat wth cete j assocated wth a fbeed chat ( V ) o Y Let us wte = ( ) as befoe ad cosde a vecto w = ( v w w K ) F satsfyg w = D ( ( )) v w = D ( ( )) v K o whch s the same (8) w= DJ ( ( )) v It follows fom the cha ule fo devatves that the followg asseto holds: the class of equvalece (wth espect to the elato (6)) cotag ( Z w) satsfyg w= DJ ( ( )) v s equal to the equvalece class cotag ( Z w ) f ad oly f w= D ( ( j )) w e f ad oly f v = D ( ( )) v Ths meas that the codto (8) s vaat; each taget vecto { Z w} j satsfyg (8) s called the total taget vecto o J Y We shall wte tt J Y fo the set of all total taget vectos at a pot j ad toduce the j otato ttj Y = U tt J Y j j Let t deotes the estcto of to the set ttj Y I ths paagaph we shall befly descbe ceta popetes of the tple ( ttj Y t J Y) Cosde a atlas ( J Y) o J Y cosstg of caocal chats Let ( ttj Y) be the set of all pas ( ( Z ) tt ) whee tt s a mappg fom ( Z ) to ( Z ) R defed as

13 j (9) tt ({ Z DJ ( ( )) v} ) = ( ( j D ( ( )) v) The: ) U ( Z ) = ttj Y ) Fo each the mappg tt s bjectve 3) The mappg tt o ( tt ) s of the fom () ( ( j ) D ( ( )) v) ( ( ( j )) D ( ( )) D ( ( )) v) ad s theefoe a mophsm To put t dffeetly the tple ( ttj Y t J Y) has the popetes of the vecto budle (compae wth [6]) We call t the o J Y (See [39] [6] [6] ad compae wth appoach [3]) The budle of vetcal vectos o J Y Wth smla agumets as the pevous paagaph we call each taget vecto { Z w} TJ Y satsfyg w= ( w w K ) a vetcal vecto o J Y The set of all vetcal vectos at a pot j s deoted vt J Y ad we put j vtj Y = U vt J Y j j The tple ( vtj Y v J Y) wth v deotg the estcto of to vtj Y has aalogous popetes as say ( TJ Y J Y) ( vtj Y s of couse a mafold modeled o m m the locally cove space F R Ls ( R R )) We call the tple ( vtj Y J Y) the budle of vetcal vectos o J Y v Mophsms of locally cove mafolds A mophsm f: P Q of locally cove mafolds s defed fomally the same way as the fte-dmesoal case If f s a mophsm oe ca defe the taget mophsm Tf : TP TQ Let p P be a pot ( U ) a chat wth cete p ( V ) a chat wth cete f( p) The () Tf ({ U u} ) = { V Df ( ( p)) u} p Let us ow defe what we mea by a mophsm of (fte dmesoal) fbeed mafolds Let ( Y X) ad ( Y X) be two fbeed mafolds A pa ( f f ) s called a mophsm of these fbeed mafolds f f: Y Y ad f : X X ae mophsms of dffeetable mafolds ad o f = f o (See [6] [39]) j f( p) 3

14 Addtoal emas It ca be show that both ( ttj Y t J Y) ad ( vtj Y v J Y) ca be cosdeed as subbudles of the taget budle ( TJ Y J Y) ( the sese of deftos aalogous to the case of mafolds modeled o Baach spaces [6]) The space T J Y s the topologcal dect poduct of j ts subspaces tt J Y ad vt J Y ; we wte o () 4 j j = v j j j T J Y tt J Y T J Y TJ Y = ttj Y v TJ Y Let us cosde the pull-bac ( TX X J Y) [6] It s mmedately clea that the mappg TX( j { U u} ) Tj { U u} (3) = { Z D j ( ( p)) u} ttj Y (togethe wth the detty of J Y ( TX X J Y) ( ttj Y t J Y) j ) s a somophsm of vecto budles The budle of covaat atsymmetc tesos o J Y We ae ow gog to cosde aothe type of budles epeseted by the budle of multlea foms o J Y Sce we use oly some dffeetal foms of a specal type o J Y patculaly those whch ca be obtaed as the pull-bac by the mappgs we ae ot eed to gve qute geeal deftos We defe what could be called the budle of p-lea cotuous atsymmetc foms o J Y (also sepaately cotuous foms could be teated) Let F be ou locally cove space It ca be poved that the vecto space L p a ( F R ) of all cotuous atsymmetc p-lea foms o F wth the topology of ufom covegece o all bouded subsets of F s a Fechet space [5] [33] Let j J Y ad cosde the set of all tples ( Z u) whee ( Z ) s a chat o J Y wth cete j p ad u La ( F R ) I ths set thee s a equvalece elato ~ defed as follows: (4) ( Z u) ~ ( Z u) uo( D ( ( j Y)) KD ( ( j Y))) = u The equvalece class cotag ( Z u) s deoted by { Z } (5) u j { Z u} { Z w} { Z w } K { Z w } j j j p j = uw w K w p Futhe we set whee the ght-had sde s equal to uw ( w K w p ) Thus the equvalece class { Z u} j defes a p-lea cotuous atsymmetc fom o T J Y The set of j p all such classes s deoted LT J Y ad we wte a j p p LTJ a Y= U LT a J Y ({ Z u} ) = j j j j

15 If ( Z ) s a chat o J Y wth cete j defe p = j (6) ({ Z u} ) ( ( j ) u ( D ( ( j )) K D ( ( j )))); the mappg p taes values ( Z ) La ( F R ) If ( J Y) = {( Z )} s a atlas o J p Y deote ( LTJ a Y) the set of all pas ( ( Z) p ) The followg codtos tae place: ) U p ( Z ) = LaTJ Y ) p s a bjecto fo all p p 3) The mappgs o ( ) ae mophsms p p I othe wods ( LTJ a Y) s a locally cove atlas o LTJ a Y; wth the stuctue defed by ths atlas ad wth the obvous popety 4) the estcto of p to ( j ) j Z s a lea somophsm of the p Fechet spaces LT a J Y L p F j a( R ) p we call the tple ( LTJ a Y J Y) the budle of covaat atsymmetc tesos o J Y p 3 Jet pologatos of vecto felds Pojectable vecto felds Cosde a fbeed mafold ( Y X) Let be a vecto feld o Y ad a vecto feld o X The pa ( ) s sad to be -elated f fo ay yy (3) T ( y) = ( ( y)) The equalty s also wtte as To = o Deote t (es t ) the local oe-paamete goup of tasfomatos geeated by (es ) The pa ( ) s -elated f ad oly f t t (3) o = o Let ( V ) = ( ) be a fbeed chat o Y () The the pa ( ) s -elated f ad oly f ad ae epessed by meas of the fbeed chat as (33) = = + y It s clea that f to a gve vecto feld o Y thee ests a vecto feld o X such that the pa ( ) s -elated the s uquely detemed by ths codto Each vecto feld o Y wth the popety that such the does est s called pojectable Jet pologatos of local automophsms Let ( ) be a local automophsm of the fbeed mafold ( Y X) Ths meas that s defed o a ope set V Y s defed o ( V) X ad ae somophsms ad accodg to the def- 5

16 to of mophsms of fbeed mafolds o o = o V Let be a postve tege ( ) defes a local automophsm ( j ) of the fbeed mafold ( J X) by the fomula (34) j ( j ) = j ( ) Let ( V ) be a fbeed chat o Y wth cete ( ) ad ( V ) a fbeed chat wth cete ( ( )) The we ca wte by meas of the coespodg caocal chats o J (3) ( j ) = ( ( ( )) ( ( ( ))) (35) ( ) D o o )( ( ( ))) K D o o )( ( ( ))) It follows that ) the ght-had sde of (34) eally depeds oly o j (by the cha ule [9]) ) j depeds dffeetably o j 3) j s a somophsm o ( V) Futhemoe the elatos (36) o j = o s (37) o j = j o s s hold o ( V) The pa ( j ) s thus a local automophsm of the fbeed mafold ( J X) ; we call t the -jet pologato of the local automophsm ( ) of ( Y X) I the pevous paagaph thee wee toduced the so called pojectable vecto felds o Y They ae chaactezed by the popety that the local oe-paamete goups ae just local automophsms of ( Y X) We ca polog the local automophsms by the descbed pocedue ad thus obta some local oe-paamete goups of local automophsms of ( J X) I tu we wll be led to ceta vecto felds defed o the space J Jet pologatos of vecto felds Let be a pojectable vecto feld o Y Its - jet pologato s a vecto feld o J defed by d (38) j j dt j ( ) = { t t t ( ) } Let ( V ) be a fbeed chat o Y ( y ) the coodates defed by ths chat ad cosde the caocal chat o J assocated wth ( V ) Assume that s epeseted by (33) By defto (39) ( j ) = ( ( )) t ( ) t ( ) t t t t t t y ( j ) = y ( ( ( ))) 6

17 z ( j ) = D( y o )( ( ( ))) L t ( ) t t t t t K t t t t t t ( ) z ( j ) = D D KD ( y o )( ( ( ))) whee D deotes the -th patal devatve Evdetly the detty holds so that q s t t t DDD KD( y o )( ( ( ))) l s t t t = D( D D KD ( y o ) o )( K ) q l t t D ( o )( ( ( ))) { } d dt DD D y K ( s ot t )( ( t ( ))) = { d dt D D D D y ( K ( o ) o )( s K ) } s d zl dt D + Ks { ( s l o t )( ( t ( )))} l t t t l Hee l stads fo the Koece symbol But (3) { } + = d dt D ( l o t )( ( t ( ))) D s s l ad we ca wte d dt DD D y K ( o t t )( ( t ( ))) s (3) { } = { } z s s t t t D d dt DD K D ( y o ) o ) ( K ) D l Ks s l If we ow wte (3) j = K+ y z K z K K fo the coodate epesso of j we see that (3) epesets a ecuet fomula fo the compoets K of the vecto feld Fo the sae of smplcty of otato toduce the abbevato df f f (33) d y z f = + + K+ z K z K K ( f - abtay fucto o ( V) ) Accodg to [3] we shall call the epesso 7

18 (33) the fomal devatve of f by Some useful popetes of the fomal devatve ae deved [3] Retug to ou coodate epesso (3) we see that d K s l (34) s z K = l K d Applyg the fomula we mmedately have (35) j d = z d l d d l l l = zl zl zlj d d j j j (Compae wth [] [38]) Ifte jet pologato of local automophsms ad vecto felds Let ( ) be a local automophsm of the fbeed mafold ( Y X) It has bee show that fo each postve tege oe ca costuct a local automophsm ( j ) of the fbeed mafold ( J X) such that fo s (37) s s s o j = j o Cosde the ope sets V ad ( V) = U o whch ad ae defed espectvely Each mappg j ca be composed wth the pojecto Thus fo each we get a mophsm j : ( V) j ( ( V)) ; fo s s o s s s s j o = j o o = o( j o ) satsfyg the cod- Ths shows that thee s a uque mophsm j : ( V) J to (36) o j = j o fo each (see [39] III 3) The same agumet may be appled to The cosdeatos show that j s a somophsm commutg wth Aalogously as the fte case the pa ( j ) wll be called the fte jet pologato of the local automophsm ( ) Let be a pojectable vecto feld o Y wth defed by the codto To = o Cosde the cuve t j t t ( ) t J The cuve s a mophsm so that thee ests ts taget vecto d (37) j j dt j ( ) t t t ( ) 8 = { } Ths gves se to a vecto feld j j ( j) o J whch s called the fte jet pologato of the pojectable vecto feld It s clea that (38) T o j = j o

19 Total ad vetcal vecto felds o J Y Total vecto felds o J ae coss sectos of the fbeed mafold ( ttj Y t J Y) Vetcal vecto felds o J ae coss sectos of the fbeed mafold ( vtj Y v J Y) Let be a pojectable vecto feld o Y ad defe by the elato To = o We have at ay pot j J a detty j ( j ) = T j ( ) + ( j ( j ) T j ( )) Let us deote t( j )( j) = Tj ( ) (39) vj ( )( j) = j( j) Tj ( ) Clealy these vectos ae well-defed e do ot deped o the epesetatves used fo the deftos We also use the otato (3) ( j ) = t( j )( j ) It ca be poved that the mappgs j ( j) j = v( j )( j) ae mophsms Thus s a total vecto feld o J ad vj ( ) s a vetcal vecto feld o J Oe must be caeful howeve whe wog wth these vectosfelds I geeal (whe ) the oto of the local oe-paamete goup caot be joed wth them at least ot the usual sese We shall oly eed the fomula (3) j = + v( j ) fo the vaat decomposto of j to the total ad vetcal pats Eample It s ow how the local oe-paamete goup of a vecto feld o a dffeetable mafold ca be used to costuct a vecto feld o each teso budle ove ths mafold ([36] Chap II 8) Sce the vecto felds obtaed by ths pocedue ae pojectable (wth espect to the atual pojecto o the dffeetable mafold) they ca be pologed to the coespodg jet spaces Such vecto felds ae of geat teest the calculus of vaatos fbeed mafolds We wsh to wte up eplctly the coodate epessos fo the -jet pologato It s ot so dffcult to pefom the calculatos full geealty Let deote the type of tesos whch wll be cosdeed; detal may be egaded as the set { K ; + + K + s} of dces whch some of them say the fst ae egaded as cotavaat The coespodg teso budle of a mafold X s deoted ( TX X) To smplfy ou otato we wte { K ; j j K js} fo the base vectos defed by a chat o X wth cete ; { K ; j j K js} s smply some teso poduct of the base vectos / d j wth the ode pescbed by Notce that the symbols ae used fo the cotavaat dces Let be a vecto feld o X By meas of a chat ( U ) = ( K ) ts local oe-paamete goup t s epessed as ( o t )( ) = f( t K ) It s ow that the duced oe-paamete goup actg o each space of tesos o 9

20 X (the acto beg wtte multplcatvely) satsfes the ules ) t ( uv) = ( t u) ( t v) ) f t = 3) t f j d j = d l l tag place fo ay tesos u ad v (of couse we must have md that the tesos / d j ae defed at the pot whle the same symbols / d j o the ght-had sde ae used fo tesos at the pot t ( )) It follows that t { K ; j j K js} f f f f j f j f js = K K K K l l ls { ; l l l s } If y{ K ; l l K l s } stads fo the coodate fuctos o TX defed by the base vectos { K ; l l K ls} the duced mappg deoted ow by t has the followg coodate epesso: o t = f { K l l K l o ; s } t y f f f f = y{ K ; j j K j K s} j l f j l f K js ls Dffeetatg wth espect to t at t = we get d dt { ot } = d { y{ K dt l l Kl o ; s } t } = ( pq K jljl Kjl + s s pqk jl 3 3 jl K + K+ K K ) { K ; K } p jl jl j l jp lq y s s s s j j js q If we wte { K ; j j K js } { K ; l l K l s } pq fo the bacet the last epesso ad fo the vecto feld geeated by the local tasfomatos of TX the calculato leads to the epesso = + y { K ; j j K js} { K ; j j K js} { K ; l l K l s } pq y { K ; l l K l s } If we futhe wte A B K fo the sets { K ; l l K ls} of " -admssble - p q jl ss

21 dces" ad Bq Ap = { ; } { ; } K j j K js K l l Kl s pq the fomula becomes Bq p (3) = + Ap yb y q A whee the costats Bq Ap ae completely detemed by the teso chaacte of T X As metoed befoe the vecto feld plays a mpotat ole the vaatoal calculus o fbeed mafolds It s used fo the defto of so called geeally vaat (o covaat) vaatoal poblems (see [38]) We shall etu to ths vecto feld Secto 7 Notce that depedetly of the type of the teso budle s always costucted the same way fom (patculaly by meas of the devatves of ) Ths meas that the pocedue of the -jet pologato of such vecto felds wll be the same fo all types of Fom the geeal ule (35) we obta j Bq p = + Ap y B y q Bq p y Bq p Ap B Ap z B z l + la + z q q A eplct epessos fo abtay -jet pologatos may ow easly be deved As a eample oe ca tae the so called vaatoal vecto feld o the taget space TX ; t s of the fom = + y j j y s used the vaatoal calculus of oe depedet vaable [36] Jet pologatos of the Le bacet The followg poposto holds: Let ( Y X) be a fbeed mafold ad let be two vecto felds o Y If both ad ae pojectable the so s [ ] ad fo ay j [ ] = [ j j ] The mappg j defed o pojectable vecto felds o Y s a R-lea somophsm (Ths ca be poved by a staghtfowad calculato local coodates) A ; 4 Hozotal ad pseudovetcal dffeetal foms o jet pologatos of fbeed mafolds Hozotal ad pseudovetcal foms Let ( J X) be the -jet pologato of

22 a fbeed mafold ( Y X) Cosde a eteo dffeetal p-fom o J + + Choose j J ad defe h( ) by the elato (4) + h( )( j ) K p = ( j ) T j T T j T K T j T p + + whee K p ae abtay vectos fom T J It s clea that h( ) s a p- j fom o J + + wth the popety that h( )( j ) K p s vashg wheeve oe of the vectos K p s vetcal (e T + = ) Such foms ae usually called hozotal Put + (4) p( ) = h( ); the p-fom p( ) o J + fulflls + (43) j p ( ) = fo ay coss secto Γ( ) Accodgly we defe: A dffeetal p-fom o J s sad to be pseudovetcal f fo ay Γ( ) (44) j = (compae wth []) The epesso of h( ) ad p( ) caocal coodates o J + may be dectly deved fom the defto Notce that h( ) s uquely detemed by the codto (4) Thus f we gve a coodate fom satsfyg (4) we gve at the same tme the coodate epesso fo h( ) Wte fo some caocal coodates o J ( s ug ove a set of admssble dces) ad (45) = p f K d d K d p p! fo a p-fom ths coodates The f we deote by q q= K = the coespodg coodates o J + we have (46) h( ) = f d d d p p p K K K p p! Fo the fom p( ) we obta (47) p( ) = f ( d d d Kp K p p! K d d Kd pp p ) A useful eample of h( ) ad p( ) wll be dscussed oe of the et paagaphs Ma popetes of hozotal ad pseudovetcal foms The followg poposto descbes some useful popetes of hozotal ad pseudovetcal p-foms

23 If s a p-fom o J the thee est uquely detemed p-foms h( ) ad p( ) o J + such that ) + = h( ) + p( ) ) h( ) s hozotal ad p( ) s pseudovetcal The mappg p( ) s lea (ove the g of fuctos) ad ts eel s fomed by all hozotal p-foms o J Fo ay p-fom ad q-fom o J (48) p( ) = p( ) p( ) + h( ) p( ) + p( ) h( ) If p > =dm X the h( ) = Both ad h( ) ae defed o J f ad oly f s hozotal wth espect to the pojecto Fally let ( ) be a automophsm of the fbeed mafold ( Y X) ; the + (49) pj ( ) = j p ( ) (All these assetos mmedately follow fom the deftos) Eample: Hozotal ad pseudovetcal -foms o JY We shall llustate the above poposto by gvg eplct fomulas fo -foms o J (= dm X) whch ae hozotal wth espect to the pojecto (oe should emembe that ths s a ecessay ad suffcet codto fo both ad p( ) to be defed o J ) Let be a -fom o J hozotal wth espect to Sce ay coodate epesso of does ot cota dz j (( y z zj ) beg the coodates o J defed by a caocal chat) t wll be useful to toduce the otato Wtg (4) = z = y = z = y = z l l = fdd Kd + = s< s< K< s K q q K q f! s q q s K s q d d Kd d d Kd d ds +K d s q s + s q whee q q K q ad the summato s obvous we see at oce that the epesso o the ght-had sde s the most geeal -fom o J hozotal wth espect to Clealy the fuctos f q s q s s q K ca be supposed atsymmetc the lowe couples of the dces q By defto s (4) h f f q s s ( ) = + q q sq sq sq d d d K K K whee the summato s the same as (4) The decomposto of to hozotal ad pseudovetcal foms s thus gve eplctly as s (4) = f + f q s s q q K K d d d K sq sq sq 3

24 s + f q ( s s q q K d Kd d d K K! K d d Kd ) sq sq sq q q A eteso of h to ( +) -foms We shall eted the mappg h( ) defed o -foms o the -jet pologato of a fbeed mafold to ( +) -foms defed o hghe jet pologato of the same fbeed mafold Ths s based o the followg poposto: Let be a ( +) -fom o J Thee ests oe ad oly oe ( +) -fom o J + satsfyg 4 + h (( ) ) = ( ) fo all vetcal vecto felds (wth espect to + ) o J + To pove t let us wo wth the coodates q toduced befoe (see (46)) Let us eame the codto ( ) = tag place fo a ( +) -fom o J + ad fo all vetcal vecto felds (wth espect to + ) Suppose = fk d d d Kd ; ( + )! the (compae wth a fomula [7]) ( ) = f d d d K K! ad the codto leads to the equaltes f K = tag place fo all K ( K de the coodates o X) Sce f K ae atsymmetc ths meas that f K = wheeve oe of the dces K s dffeet fom K But f K whee K vashes detcally whch shows that = I othe wods f the ( +) -fom fom the poposto ests t s uque As fo the estece t suffces to fd oe such fom o each coodate eghbohood Let us ow wo wth some caocal coodates o J Put ad defe wth = f K d d d Kd ( + )! = (43) = fk K d ddk d The equalty h (( ) + = ( ) follows by a dect calculato (see [7] p 49

25 fomulas of Secto ad (46)): ( ) + = f d d d K K! h (( ) + ) = f d d d K K K! = f K d d K d = ( ) K Ths completes the poof We ote that the fom ca be vaatly defed by meas of the elato + ( j ) K = ( j ) T j ot T j ot = K TjoT T TjoT K Tj T whch K ae abtay vectos fom the taget space T J Put j + (44) h ( ) = ; the elato (43) taes the fom + (45) h (( ) ) = ( ) ( h) Dffeetal foms o J Y The oly foms we shall wo wth o the fte jet pologato of a fbeed mafold wll be the pull-bacs by the pojectos The pull-bacs wll be defed a full aalogy to the case of mafolds modeled o Baach spaces [6] Let be a p-fom o J ad let K p be taget vectos at the pot j J ; defe ( )( j ) : (46) (46) p ( )( j ) K = ( j ) T T K T ( )( j ) K p p = ( j ) T T K T p o symbolcally p = ( o ) ( T ) The mappg j ( )( j) thus asg s a p-fom o J e a coss secto p of the fbeed mafold ( LTJ Y J Y) defed Secto a 5

26 Eample I ths paagaph we wsh to llustate the descbed deftos ad methods ad tae otce of oe of the elatos to the calculus of vaatos We pose the followg poblem: Suppose we have a hozotal -fom o J (wth espect to ) Fd a - fom Θ o J wth sutable satsfyg the followg codtos: ) I some caocal coodates (4) o J Θ s of the fom (47) Θ= + f + fjj whee j f = f j = ddkd ( dy zd) d+ d+ Kd j = ddkd ( dz j zjd ) d+ d+ Kd patcula h( Θ ) = ) hdθ ( ) s hozotal wth espect to + It wll be poved the et Sectos that the poblem s fact motvated by the calculus of vaatos We just ote that because of the assumpto ) the (+)-fom hdθ ( ) well coespods to the Eule equatos of the calculus of vaatos detemg the etemals Notce that f the aswe s postve e f a fom satsfyg ) ad ) does est we have poved as a patal esult that the mappg h( ) maps oto the space of all hozotal -foms o J Let us ow cosde a -fom Θ of the eeded fom (47) By (45) we shall eame the epesso h (( ) + dθ ) stead of ( ) ( h dθ ) fo a abtay vetcal vecto feld o J + I the caocal coodates j ( ) = ( ) d d Kd ˆ Kd ( ) = ( ) d d Kd ˆ Kd j ; 6 ( ) d = ddkd ( ) = d d Kd j j Obvously these fomulas ˆd meas that d s mssg ad K deote the compoets of the vecto feld gve by (48) = + + K y z z Let us wte ( = dm X) = d d K d K K

27 Now dθ= d ddkd + df + fd + df jj + f jd j But the foms ad j ae pseudovetcal so that accodg to the ule h( ) = h( ) h( ) followg fom (48) df df j h (( ) + dθ) = + f y d z d + z f f + z < j j j j j d d K d Remembe that d / d deotes the fomal devatve toduced Secto 3 (34) Now codto ) dectly leads to the equaltes defg f ad f j Sce f j f j = we get f f f j = j zj = z d d = z d z d z j j j the summato beg obvous We see that Θ s defed o J 3 It mght be of teest to obta a coodate epesso fo hdθ ( ); we get d d d hd ( Θ ) = (49) y d z + d d j z j j dy d d d The fucto (4) E K d d d = y d z + d d z j j j defed locally o J 4 s usually temed the Eule epesso assocated wth the Lagage fucto (The Eule epessos ae othg but left-had sdes of the Eule equatos of the calculus of vaatos The Eule epessos wll be fequetly used et Sectos) If s defed o J the the coespodg Eule epessos ae E d = y d z 7

28 I ths case the -fom Θ has bee obtaed [35] fo a specal case Ou appoach was motvated pat by [4] Let us summaze the esults Thee s oe ad oly oe -fom Θ o J 3 satsfyg codtos ) ad ) If some caocal coodates ( y z z z ) the fom s epessed as (4) = d d K d the (4) j j d d Θ= ddkd + z d z j d j z j + + z j z j j Two fomulas fo the fte pull-bac of foms We shall eed the followg fomulas: ) Fo ay p-fom o J (43) + h ( ) = h ( ) ) Fo ay (+)-fom o J ad ay pojectable vecto feld o Y (44) vj (( )) ( h ) = h ( ( j ) ) The fst oe s a dect cosequece of deftos To pove the secod fomula we wte (3) j = + v( j ) ad h (( ) )( j ) K = ( )( j ) T j ( ) T j T K T j T = ( j )( ) ( ) T T K T = fo ay vectos K at a pot j J I these fomulas the cotacto by a vecto feld o J ad the opeato h ae appled to -foms o J a evdet way Now by the vaat defto of ( h ) ( )( h j ) v ( j )( j ) K Ths fshes the poof = = ( )( j ) v( j )( j ) Tj ot K Tj ot hv (( ( j )) )( j ) K = h (( j ) )( j ) K 8

29 5 The Eule mappg Eplct fomula fo d whee s a -fom hozotal wth espect to At the begg of ths Secto we gve two coodate fomulas coceg -foms defed o J Let us cosde a fbeed mafold ( Y X) ad deote by J ts -jets pologato Let be a -fom o J whee = dm X I the caocal coodates (4) o J = gdd Kd (5) + = s < s < K< s K ss Ks g d d d K! s dy d Kd dy d Kd s + s s + We tae to accout ths fomula that s hozotal wth espect to ad we ss Ks assume that the fuctos g K ae atsymmetc wth espect to the subscpts (compae wth the eample gve Secto 4) The oe ca deve by a dect but tedous calculato the followg fomula fo the dffeetal d: If s epessed by (5) the (5) s g g d = y + s dy ddkd + ssks ssks g K g K! y + = s s s < < K< K < s ss Ks ss Ks g K K g + + K+ s s < < > s s s+ s s+ dy ddk d dy d Kd dy d Kd K K g + dy dy dy Kdy! y K g + z + dz d d Kd = s < s < K< s K ss Ks K g! z s s+ K s s+ K dz d d K d dy d d dy d d As fo summato otce that we ofte do ot eplctly desgate the summato ove the dces ad Eplct fomula fo ( hd ) Comg out fom the pevous paagaph we pove 9

30 the followg mpotat poposto: Let be a -fom o J ad suppose that s hozotal wth espect to Let be a volume elemet o X Wte the caocal coodates (4) o J (53) =ddkd ss Ks g g K (54) A = + zs zs zs z K z = s < s < < K s K (55) h( ) = G The G d G da (56) hd ( ) = dy + dy + Adz y d z d Fo the poof let us stat by the fomula (43); we have hdy ( ) = dy hdy ( ddkdy Kdy Kd ) h s s = ( d d d ( )! ) K K K K K K K + Ks = Ks K K d K K K j I the last fomula K lm K s the completely atsymmetc symbol (see eg [7] p 36) Clealy all tems wth K vash so that hdy ( d d K dy K dy K d) K K K K K K s s = K dy K K K KK K Ks K s K K KK K dy s s K dy K ( s s K s s s K s K K s s s K s K s s s s ss s s = K K K K ) dy ss ss ss s s s s ss Ks dy = K ( K K K K ) whee we passed to the summato ove K stead of s s K s Wtg = z we obta the desed fomula Aalogous epessos fo d gve by (5) ca be obtaed the same mae Oe gets togethe hdy ( d d K dy K dy K d) 3

31 ss Ks = z z z z z z K ( K K K z z Kz z ) dy hdy ( dy dy Kdy ) K = K ( z z Kz zz Kz K z z Kz z ) dy hdz ( ) = dz hdz ( d d Kdy Kdy Kd ) ss Ks Thus ( hd ) becomes = z z z dz z z z K ( K ( K K + + z z Kz z ) dy ) (57) s g g hd ( ) = y dy g +! y = s< s< K< s K s ss Ks K ss Ks K ( ssks g K = s< s< K< s z z Kz z z Kz K z z Kz z ) dy + K+ > s z g z K ss Ks g + dz + z ( + )! ss Ks K < s ( + z z Kz s< < s g Kz K z z Kz z ) dy = s< s< K< s K g! z ss Ks K ss Ks K ss Ks ( z z Kz + K+ z z Kz z ) dy ) It s see fom ths epesso that the tems cotag dz ae just equal to A dz Futhe G y g = + y = s< s< K< s K g ss Ks K y z z Kz ad t emas to deteme the emag tems Wte fo ths G = A + B z s s s 3

32 whee B ss Ks = g K = s< s< K< s K z ( z z Kz ) s s s It suffces to show that 3 G db hd ( ) G = dy + B dz y d z e that the emag tems ( hd ) ae equal to db = + + d dy B B B z z y z dy By a athe dffcult computato we ave at the epessos B s g = + s s Ks K ( + )! s = s< s< K< s K < s ss Ks ss Ks g g g K K ss Ks + + K+ K s< < s > s ( z z K z z K K z z z z Kz z ) ss Ks B g K ss Ks z = y K! = s < s < < s y K K ( z z Kz + K+ z z Kz z ss Ks B g K ss Ks z = z! = s s s z < < K< K ( z z Kz + K+ z z Kz z ) Ou asseto ow follows by compaso of these epessos wth (57) We ote that the decomposto of the fom ( hd ) to two tems (see (56)) s vaat as ca be checed by a dect calculato Notce that the epesso the fst bacet (56) e the coeffcet at dy s just the Eule epesso defed by (4) Ths s the ma easo why those foms ( hd ) that ae hozotal wth espect to ae of specal teest fom the pot of vew of the vaatoal calculus Lepaga foms Usg the esults of the pevous paagaph we ae led to the followg defto: Let be a -fom o J ; s sad to be a Lepaga fom f the (+)-fom ( hd ) s hozotal wth espect to the pojecto + )

33 p p I ths wo X ( J ) Y ( J ) wll deote the spaces of hozotal p-foms wth espect to ad hozotal p-foms wth espect to espectvely The Lepaga foms o J ae chaactezed by the followg: Fo a -fom Ω Y ( J ) the et fou codtos ae equvalet: ) s Lepaga ) I ay coodates defed by a caocal chat o J (58) A = 3) ( hd ) depeds oly o h( ) 4) Thee s oe ad oly oe pseudovetcal -fom EΩ ( J ) such that (59) hd ( ) = E The poof s based o the elato (56) Codtos ) ad ) ae obvously equvalet Codto 3) meas fact that the mappg hd ( ) s costat o the sets of foms wth a gve hozotal pat It could be efomulated moe eactly by sayg that s a Lepaga f ad oly f fo ay Y ( J ) such that h( ) = we have ( hd ( )) = But t becomes clea that the 3) s equvalet wth ) Let us eame 4) If s Lepaga we tae G d (5) E = F y d G dy z d z ( ) whee s a volume elemet o X ad the fucto F > s defed by the elato = F Oe may pove that ths epesso s vaat wth espect to coodate tasfomatos J If we admt that thee s aothe -fom E satsfyg 4) we ave at the equalty ( E E) = ad the by the assumpto of the pseudovetcalty of both E ad E E = E Thus 4) follows fom ) Covesely f ( hd ) s of the desed fom we apply (56) ad see at oce that A = Ths completes the poof (Fo the classcal appoach to the poblem of a vaat devato of the Eule equatos see [7] [4] Ou eposto s fact equvalet wth the appoach setched [4] A mode teatmet has bee gve [35] Notce that the appoach gve hee s based upo the estece of the mappgs h ad h eplacg the equvalece elatos of Lepage [7] [4]) Lagaga foms ad the Eule fom Accodg to ou pevous defto [] by a Lagaga fom of degee o a fbeed mafold ( Y X) we mea each -fom o J (emembe that = dm X) Real fuctos o J ae called Lagage fuctos of degee o ( Y X) Clealy to each Lagaga fom of degee thee coespods a Lagage fucto L of degee + defed by the elato (5) h ( ) = L + Y 33

34 I some specal cases (compae wth the poposto of Secto 4 dealg wth the popetes of hozotal ad pseudovetcal foms) eg the case whe s hozotal wth espect to the Lagage fucto L ca be cosdeed as defed o J Let us ow cosde the mappg hd ( ) estcted to the set of Lepaga foms o J As we ow fom the above theoem the mappg s costat o the classes of Lepaga foms wth the same hozotal pats O the othe had the mappg h s sujectve (e maps oto Ω X ( J ) see (4)) Ths meas that the fomula (5) hd ( ) = Eh ( ( )) gves se to the mappg (53) Ω ( J ) E( ) Ω ( J ) 34 X Y whch wll be called the Eule mappg Each E( ) wth Ω X ( J ) wll be called the Eule fom defed by the Lagaga fom Evdetly the Eule fom mght be cosdeed as defed by the coespodg Lagage fucto L whch s accodg to the hozotalty of Lepaga foms wth espect to also of degee (see (5)) The fucto E( ) s lea (ove R) I ths Secto we ae gog to gve a chaactezato of ts eel o whch s the same to fd codtos ude whch the Eule equatos of the vaatoal poblem defed by a Lagaga fom of degee ae detcally fulflled A ote o a classcal poposto Let UR be a ope set ad cosde the fbeed mafold ( U R p U) ad ts -jet pologato ( U R R p U) wth the atual pojectos o U deoted by the same symbol p Let L be a dffeetable fucto o U R R e a Lagage fucto of degee o ( U R p U) The coespodg Eule equato the ca be wtte as L d EL ( ) = y d L y = It s usually cosdeed as a patal dffeetal equato fo a fucto :U R (see say [8] []) It s ow that the equato s detcally fulflled (e fo all ) f ad oly f L s a dvegece epesso [8] Ths meas that the atual coodates ( K y y y K y ) o U R R f f L y y df = + = d fo some fuctos f f K f of the vaables K y I the lteatue thee appeas a asseto that a aalogous theoem s also tue fo moe fuctoal agumets K m (e fuctos wth values R) The et eample shows howeve that thee ae Lagage fuctos whch ae ot of the dvegece type but do lead to the vashg Eule epessos

35 Let UR be a ope doma ( U R p U) the fbeed mafold defed by 4 the atual pojecto J = U R R ts -jet pologato L a dffeetable fucto o J We shall wte ( y y z z z z ) fo the atual coodates o J The Lagage fucto gves se to the followg equatos (the Eule equatos) L y d d L = = z whee summato ove = taes place Let us cosde the case of L defed by meas of a abtay fucto f of vaables ( y y ) by the fomula f L = f + y z z f f + y z The dect substtuto shows that d L d L E = y d z d z = d L d L E = y d z d z = z whle L s ot of the "dvegece type" (t depeds blealy o the z ) It s see fom ths eample that the case of moe fuctoal agumets essetally dffes fom that oe of oe fuctoal vaable It s ot ow to the autho whethe the poblem of detcal vashg of the Eule epessos o alteatvely the poblem of a chaactezato of the eel of the Eule mappg has bee dscussed ts full geealty (e egadless of the umbe of the fuctoal agumets ) I the et two paagaphs we ae gog to eted the metoed classcal esult of Couat ad Hlbet [8] so much used theoetcal physcs to the geeal case The Eule mappg Let us tu to the Eule mappg (53) As a cosequece of the poved theoem we have the followg poposto: Cosde the Eule mappg Ω X Y ( J ) E( ) Ω ( J ) A ecessay ad suffcet codto fo E( ) = s that thee ests a -fom o Y such that ) h( ) = ) d = To each wth E( ) = thee ests oe ad oly oe satsfyg ) ad ) Suppose E( ) = ad wo wth a caocal chat o J Wte ( y z zj ) j fo the coespodg coodates Put = d d K d 35

36 The fucto s supposed to satsfy the system of patal dffeetal equatos d y d equvalet wth = z + = z = z z z z y z y z l l It follows that must be of the fom ss Ks K = s< s< K< s K = f + f z z Kz s s s ss Ks K ss Ks K whee f ad f do ot deped o z j ad f K The secod codto the ca be wtte as ae atsymmetc f y ss Ks ssks K f K f + f = s< s< K< s K ss Ks K s < < s > s sssks ss Ks s 3 f f 3K K y + z z K f y < s ss Ks ss Ks K f f K K y y z s s K K 3K K f f f K K 3K K z Kz f K y y y Kz = s Sce the coeffcets should vash sepaately oe gets some codtos fo f ad ss s f K K : f y f (54) f = y ss Ks ss Ks sssks ss Ks s K K 3 f f f K 3K K y y y ssks ss Ks f K f K s < s< < s > s f f ss Ks y f = K K f K y y y K = whee Suppose that we have some fuctos f f the codtos (54) ad costuct a local -fom defed as 36 ss Ks K satsfyg all

37 (55) = fd dkd + f! = s < s < K< s K d d Kdy Kdy Kd ss Ks K (compae wth (5)) A mmedate compaso wth fomula (5) shows that d = Moeove s defed o Y ad h( ) = by (4) Suppose that thee s aothe fom such that h( ) = ad that s defed o Y The fo all Γ( ) j h( ) = = j ( ) = ( ) whch mples that = Ths poves uqueess It follows that s depedet of the choce of patcula coodates To pove the covese tae (55) wth f ss Ks f K ot depedg o z apply the fomula (5) ad the use (54) Ths completely poves the theoem Lagage fuctos of degee leadg to zeo Eule fom As we oted befoe the questo o the fom of the Lagage fuctos leadg to zeo Eule epessos has bee studed some smple cases classcal lteatue [8] ad s fequetly dscussed coecto wth applcato of the calculus of vaatos theoetcal physcs (see eg [37] [38]) It s maly of pactcal teest to have a chaactezato of such fuctos sce vaatoal poblems ae usually defed by meas of them ad ot by meas of eteo dffeetal foms A coodate descpto of such fuctos ca be deved fom ou theoem o the Eule mappg by meas of the Pocaé lemma (see eg [6]) Cosde a fom o Y such that d = By the Pocaé lemma ca be wtte locally as d fo a sutable ( ) -fom o Y The oly thg we eed s just to deteme the fucto L fom the codto hd ( ) = L The t s clea that L wll gve zeo Eule epessos Covesely the pocedue maes t possble to obta all Lagage fuctos of degee wth the popety Let us pass to the fomulato of the poposto Let be a hozotal Lagaga fom of degee o a fbeed mafold ( Y X) Wte ( y z ) fo coodates o J coespodg to a caocal chat Let = Ld d Kd be the coodate epesso fo The E( ) = f ad oly f thee est fuctos f s s K s ; K + p = p s s K s K p m of coodates y such that (56) fs s s f L = + s< s< K< s K p K+ + K p p ssksk p z z Kz K ; K p s s K s; K p y z 37

38 I ths epesso summato ove K p taes place Fo the poof t suffces to compute the coodate epesso fo hd ( ) wth a geeal ( ) -fom o Y wtte as (See []) = fs d d d s K s ; K s s K s s < s < K< s K dy dy Kdy + p = p p Eample Sce fomula (56) s athe complcated t seems to be moe effectve to poceed fom the begg each dvdual case If eg dm X = the -foms o Y ae epessed as = fd + gdy Afte some calculato whch defes L as f g f g hd ( ) = + z j + zz j j d d y y f g f g L = + z j + y y z z j j If dm X = we stat wth a -fom e wth a fucto F o Y ad get df hdf d d L df ( ) = = d Hee deotes a coodate o X ad d / d s the fomal devatve wth espect to the coodate 6 Caocal vaatoal poblems Vaatoal poblems o compact base mafolds Accodg to Hema [3] [4] we say that thee s gve a vaatoal poblem f we have the followg objects: ) a compact oeted mafold X wth bouday X ; deote = dm X ad suppose that X has the duced oetato; ) a mafold Y wth dmy = + m m ; 3) a dffeetal -fom o Y; 4) a dffeetal deal I of dffeetal foms o Y (e a deal wth espect to the eteo algeba stuctue) 38