Functor and natural operators on symplectic manifolds

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1 Fuctor ad atural operators o symplectc mafolds CONSTANTIN PĂTRĂŞCOIU Faculty of Egeerg ad aagemet of Techologcal Systems, Drobeta Turu Sever Uversty of Craova Drobeta Turu Sever, Str. Călugăre, No. ROANIA.patrascou@yahoo.com Abstract. Symplectc mafolds arse aturally abstract formulatos of classcal echacs because the phase spaces Hamltoa mechacs s the cotaget budle of cofgurato mafolds equpped wth a symplectc structure. The atural fuctors ad operators descrbed ths paper ca be helpful both for a ufed descrpto of specfc propretes of symplectc mafolds ad for fdg ls to varous felds of geometrc objects wth applcatos Hamltoa mechacs. Key-words: Fuctors, Natural operators, Symplectc mafolds, Comple structure, Hamltoa.. Itroducto. The geometrcs objects le as vectors, covectors, tesors, metrcs, e.t.c.) o a smooth mafold, are the elemets of the total spaces of a vector budles wth base. The felds of such geometrcs objects are the secto correspodg vector budles. By eample, the vectors o a smooth mafold are the elemets of total space of vector budles T,π, ); the covectors o are the elemets of total space of vector budles T*, p, ). A feld of vectors o the mafold s a secto of the vector budles T,π, ); a feld of covectors o the mafold s a secto of the vector budles T*, p, ). A dfferetal form or a eteror form of degree or a -form s a secto of the vector budle Λ T *, p,) I fact, the vector budle T,π, ) s the value of the fuctor, T: a VB, where a s the category of smooth mafoldsthe morphsms of ths category s smooth maps betwee mafolds) ad VB s the category of vector budles, the morphsms of ths category are morphsms of vector budles []. So, the taget fuctor T, assocates to each mafold, the taget budle T,π, ) ad to each smooth map f: N, a vector budle morphsm Tf: T TN, whch covers f. I the case of the cotaget fuctor T* we ca ot use the whole category a, we use oly mafolds of the same dmeso The cotaget fuctor T* : a m) VB assocates to each m-dmesoal mafold, the cotaget budle T*,π*, ) ad to each local dffeomorphsm f : N, a vector budle morphsm T*f : T* T*N, whch covers f, where T * f ) = T f ) )*: T * T N f ) * So, the cotaget fuctor, T* : a m) VB ad more geeral Λ T * : am) VB, are the budle fuctors from am) to VB, where am) subcategory of a) s the category of m- dmesoal smooth mafolds, the morphsms of ths category are local dffeomorphsms betwee mafolds. Recall that a operator s a rule trasformg the sectos of a fber budles E, p, ) to sectos of aother fber budle E', p ', '). Regardg to the budle fuctors Λ T *, the eteror dervatve d, trasforms sectos of + Λ T * to sectos of Λ T * for every mafold ad d commutes wth local dffeomorphsms. So, d s a atural operator from the fuctor Λ T * to fuctor Λ + T * ad wrtte: d : Λ T * Λ + T *, N ad d : + Λ T * Λ T * for ay am).natural operator from taget to cotaget fuctors o symplectc mafolds. Recall that the couple,ω ) s a almost symplectc mafold f s a smooth mafold ad ω s a almost symplectc form.e. a odegeerate -form o the mafold. If a almost symplectc form ω Ω ) s closed, ω s called symplectc form ad the couple,ω ) s called symplectc mafold. ISBN:

2 If,ω ) s a symplectc mafold. The, each taget space T, ω ) s symplectc vector space ad the mafold s ecessarly of eve dmeso. If s the dmeso of mafold, the product ω = ω ω... ω -factors) ever vashes, thus s oretable ad ay symplectc dffeomorphsm preserve the volume. By Darbou s Theorem such a -dmesoal mafold loos locally le R C wth the stadard symplectc form ω = d dy, 0 where,,...,, y, y,..., y ) are coordates. R C. So symplectc mafolds, cotrast to Remaa mafolds, have o local varats. A symplectomorphsm betwee -dmesoal symplectc mafolds, ω) ad, ω ) s a dffeomorphsm f : satsfyg the codto: f * ω = ω. We deote Smp), the category of -dmesoal symplectc mafolds, the morphsms ths category are the symplectomorphsms. The Smp) s a subcategory of a). We wll cosder the restrcto of taget ad cotaget fuctors to category Symp). Let T,π, ) be the taget budle ad T*,π*, ) the cotaget budle of symplectc mafold. The mafold s edower wth a symplectc structure.e. a odegeerate closed -form ω Ω ). The, each taget space T, ω ), s symplectc vector space. For each we ca defe the map Φ : T T*, X Φ X ) = ω =ω X,. ) Sce ω s odegeerate ths map s a somorphsm betwee the taget space T ad cotaget space T*. The, the map Φ : T T*, Φ /T =Φ, s a somorphsm of taget fber budles T ad cotaget fber budle T*. Let X ) be the set of vector felds of the sectos of taget budle ) ad Ω ) the set of -forms of the sectos of cotaget budle T*,π*, )). There s a oe-to-oe correspodece betwee vector felds ad -forms of mafold, gve by the map Φ : X ) Ω ), Φ X) = Φ o X = X ω. X So, Φ : T T*, s a operator from T to T*. oreover, Φ s a regular operator because every smoothly parameterzed famly of vector felds s trasformed to a smoothly famly of covector felds. Let F ad G be two budle fuctors over mafolds, a smooth mafold, F ad G the fber budle correspodg to ; ΓF) ad ΓG), the set of smooth secto of ths fber budle. Recall that a atural operator A : F G s a system of regular operators A : ΓF) ΓG) satsfyg followg codtos: ) For every secto s ΓF) ad every somorphsm f : N category of mafold t holds A N Ff o s o f - ) = Gf oa s of ) A U s U ) = A s) U for every secto s ΓF) ad every ope submafold U of. Let be T ad T*, the restrcto of taget ad cotaget fuctors to category Symp). Proposto. Φ : T T* s a atural operator betwee the two budle fuctors T ad T*. Proof. Φ s a system of regular operators Φ : X ) Ω ), Ob Symp) Let be, N Ob Symp), ω ad ω N the correspodg sympletc forms. The codto ) from prevously defto s Φ N Tf o X o f - ) = T*f oφ X of for every vector feld X X ) ad for every symplectomorphsm f : N Let be, f) = y N, Z X N) [Φ N Tf o X o f ] Z) y = ω N yt f X, Z y ) [T*f oφ X of ]Z) y = T*f ω )Z y ) X = ω X, Tf Z y ) = f*ω N y) X, T y f Z y ) = ω N yt f X, T f T y f )Z y ) = ω N yt f X, Z y ) I the prevous calculus we have used the equalty ω =f* ω N due the fact that f s a symplectomorphsm. The codto ) s satsfed because the geometrc objects mpled defto of Φ do ot deped o the chages of coordates. A vector feld X X ) s called symplectc f s closed. ω X ISBN:

3 If X X ) s a vector feld ad L X the Le dervatve alog X, the vector feld X X ) s symplectc f ad oly f L X ω =0. Ideed, we ow that L X = X o d + do X. Because ω s closed we have dω =0. But, X s symplectc f ad oly f ω s closed.e. f ad oly f X d X ω) = 0 L X ω = X o d) ω+ do X )ω = X d ω)+d X ω) = d X ω) = 0 f ad oly f X s symplectc. We deote the space of symplectc vector felds by X,ω ) Proposto. Let be X X ) ad L X the Le dervatve The vector feld X s symplectc vector feld X X,ω )) f ad oly f the Le dervatve L X commute wth atural operator Φ : T T*.e. f ad oly f the dagram X ) Φ Ω ) L X L X X ) Φ Ω ) s a commutatve oe. Proof. Φ o L X )Y) = Φ L X Y)) = Φ [X,Y] ) ω = [ X, Y ] = L X o Y - o L Y X )ω = L X Y ω) - L Y X ω). If the vector feld X s symplectc L X ω = 0 the Φ o L X )Y) =L X Y ω) = L X Φ Y)) = L X o Φ )Y). So the equalty holds. Coversely, f the dagram commute. L X ω = 0 ad X s symplectc vector feld. 3. Codferetal operator ad De Rham laplaca. Le for Remaa geometry we defe the De Rham laplaca Hodge laplaca or Beltram operator). Recall that J s a almost comple structure o a mafold f J s a secto of EdT) such that J = -Id,.e. J s a smooth feld of comple structures o the taget spaces,.e. ad J : T T lear ad J = Id. J The almost comple structure J o the mafold s tamed by the symplectc form ω f ωx,jx)>0, X T) -{0}; f moreover ω s J-varat, J s sad to be calbrated. We ow that ay symplectc mafold have a lot a almost comple structure, the space of almost comple structures o a gve symplectc mafold,ω) whch are tamed resp. calbrated) by ω s oempty ad cotractble partcular these spaces are coected). Let J be a almost comple structure o the mafold, tamed by the symplectc form ω. We defe the map gx,y) = ωx,jy) - ωjx,y), X,Y T). Because blearty of ω ad learty of J follow that g s a blear map. However g has the followg propertes: gx,x) = ωx,jx) - ωjx,x) = ωx,jx)>0, X T)-{0}; gjx,jy) = ωjx,j Y) - ωj X,JY) = ωjx,-y) - ω-x,jy) = -ωjx,y) + ωx,jy) = gx,y), X,Y T); gy,x) = gjy,jx) =ωjy,j X) - ωj Y,JX) = ωjy,-x) - ω-y,jx) =ωx,jy) - ωjx,y) =gx,y), X,Y T). The, g s a J-varat Remaa metrc. Let be a symplectc -dmesoal mafold ad ω Ω ) the symplectc form. Let ω be the volume caoc form o, J a almost comple structure o the mafold tamed by the symplectc form ω ad the J-varat Remaa metrc gx,y) = ωx,jy)- ωjx,y). Let F) be the set of real fuctos defed o. We ca defe the map F:Ω ) Ω - ) F) 0. α,β) Ω ) Ω - ) Fα,β) = s Ω o ) such that α β =sω. The real fucto s s well defed because the space of -forms s -dmesoal. So, Fα β)) s a real umber such that α β)) = Fα β))ω ), for ay two -forms α, β ad for ay pot. Proposto. For ay oegatve teger, there s a somorphsm Ψ : Ω ) Ω - ). Proof. The map f : Ω ) Ω - ))* ISBN:

4 α Ω ) f α ) =Fα,. ) Ω - ))* s a somorphsm. There s a somorphsm muscal somorphsm) f :Ω - ) Ω - ))* determed by the Remaa metrc g. The, Ψ =f of : Ω ) Ω - ) s the somorphsm. Let U,u) be a local coordate chart o, u : U u) =,,,, y,y,,y ) R such that the symplectc form ω U = d ) ω = ) dy d dy. We deote e dy. The,! d dy d = d, e = dy ; =,,,. For ay postve tegers s < s <... < s, t < t <... < t such that s t,,j) {,,,} {,,,-} j we have Ψ e s e s s e ) t =±! e t e t e, ± s the sg of the permutato s, s,..., s, t, t,..., t ). If α Ω ), the, Ψ α) s the uque - form such that X, X,, X - T. Ψ α)x, X,, X - ) ω = α θ θ θ -, where θ ss, s =,,,- s the - forms correspodg to X, =,,,-, va duced by muscal somorphsm. Evdetly, f β = fω we have Ψ β)=f ; Ψ ) = ω, Ψ ω )=, Ψ ω ) = ω -. For ay α Ω ), Ψ Ψ α ))=-) -) α Ψ oψ = -) -) Id. The, Ψ s a vertble ad Ψ - =-) -) Ψ. Wth the help of Ψ, we ca defe the operator: δ: Λ T * Λ T *, δ :Ω ) Ω - ) δ α =-) Ψ - dψ α))) =Ψ dψ α))), α Ω ), ad the fberwse scalar product <<, >>:Ω ) Ω ) Ω o ) such that f α,β Ω ), α Ψ β)=β Ψ α)=<<α,β >>ω. If s compact, α,β ) = def. = Ψ <<α ) >> blear form o Ω ) ad α = β ), β ω s odegeerate symmetrc Ψ α, Ψ β) = Ψ α Ψ Ψ =-) -) Ψ α) β =-) -)+-) β Ψ = Ψ β ) α =α,β ). ) β )) α ) Thus,.,.) s a scalar product o Ω ) varat to Ψ. The operator δ: Λ T * Λ T * s a adjot operator for the eteror dfferetal operator d,.e. α, d β ) = δ α, β ) for ay forms α Ω ) ; β Ω - ). Because ω s closed form, Ψ ω)=ω -, dω - = -)dω ω - ad we have δ ω = 0 The De Rham laplaca Hodge laplaca) wll be: Θ =d+δ) =dδ+δd. The De Rham laplaca s self adjo for.,.).e. α, Θβ) = Θα,β). Summarzg, f Λ T * s the atural fuctor o the category of symplectc mafold to the category of vector budles, the: d s atural operator from the fuctor Λ T * to the fuctor Λ + T * ad wrte d : Λ T * Λ + T * ; δ s atural operator from the fuctor Λ + T * to the fuctor Λ T * ad wrte δ : Λ + T * Λ T * ; Θ s atural operator from the fuctor Λ T * to the fuctor Λ T * ad wrte Θ : Λ T * Λ T *. We call the form α Ω ) harmoc f Θ α =0.e. f ad oly f ths form s closed d α =0) ad coclosed δ α =0). From precedet relatos hold that the symplectc form ω s harmoc. If the mafold s compact, the Hodge theorem hold: For ay form α Ω ), < there s three uque forms: β Ω - ), γ Ω ), σ Ω + ) such that α =d β + γ +δ σ, wth γ - harmoc form. The for ay form α Ω ), < there s the forms β Ω - ) ad σ Ω + ) such that δ α = δ d β ad d α = d δ σ. We remar that the -form α=d β + γ +δ σ s ISBN:

5 closed f ad oly f the form δ σ s closed ad the - form α s eact f ad oly f there s a - form σ such that α = d δ σ. The vector feld X o the mafold s a symplectc vector feld f ad oly f X ω = d f + γ +δ σ, wth f Ω o ), γ - harmoc - form, σ Ω ) ad the form δ σ s closed. The vector feld X o the mafold s a Hamltoa vector feld f X ω s eact.e. there s a smooth fucto H : R such that X ω = dh. The fucto H s called Hamltoa fucto of X. So, The vector feld X o the mafold s a Hamltoa vector feld f ad oly f there s a -form σ such that: X ω =d δ σ. The, δ σ : R, s a Hamltoa fucto of X. If q, q,..., q, p, p,..., p ) are local coordates of mafold ad symplectc form ths local coordates s d q d p ad H = δ σ : R s the Hamltoa fucto of X, the δ σ δ σ X = H p q q p The curve qt), pt)) s a tegral curve for X H f δ σ δ σ q& t) =, p& t) = Hamlto p q equatos) Remar. The atural fuctors ad operators ca be used to fd ew propertes of geometrc objects based o some already ow. By eample, usg a sem Remaa metrc g o mafold,.e. a smooth symmetrc tesor feld of type 0, ) whch assgs to each pot a odegeerate er product g o the taget space of sgature, r), we ca defe the eergy of the vector feld X T): f : R, f ) = g X, X ). So, the vector feld X s: tme le, f f < 0 ; ospacele or causal, f f 0 ; ull or lghtle, f f = 0 ; space le, f f > 0. The set of tme-le vector felds ad the set of space-le vector felds are the cove coe. The set of ospacele or causal vector felds s a T poted cove coe. So, taget budle of a symplectc mafold we have atural felds of coes. By meas of atural operator Φ : T T* betwee two budle fuctors T ad T*, we ca eted ths classfcato of vector felds to - forms ad allow to defe felds of coes o a correspodg mafold, to assocate felds of coes o the -taget budle, to defe atural felds of coes o the -cotaget budle, to defe atural, postve ad mootoe operators betwee the -taget ad the -cotaget budle ad. 4. Cocluso. Symplectc mafolds are specal cases of a Posso mafold, arse aturally abstract formulatos of classcal echacs. So, the study of symplectc mafolds s motvatg because the phase spaces Hamltoa mechacs s cotaget budle of cofgurato mafolds the set of all possble cofguratos of a dyamcal system), equpped wth a symplectc structure. The atural fuctors ad operators descrbed ths paper ca be helpful both for a ufed descrpto of specfc propretes of symplectc mafolds ad for fdg ls to varous felds of geometrc objects wth applcatos Hamltoa mechacs. Refereces:. Arfe B.G., Weber J. H., athematcal ethods for Physcsts, 6th edto Harcourt: Sa Dego, 005).. Kolar I., chor P. W., Slova J., Natural operatos dfferetal geometry, Electroc edto. Orgally publshed by Sprger-Verlag, Berl Hedelberg 993, ISBN cduff D., Salamo D., Itroducto to Symplectc Topology, secod ed., Oford athematcal oographs, Claredo Press, Oford Uversty Press, New Yor, Pătrăşcou, C.; Ttrez, A.. Felds of coes o a symplectc mafold, A. Uv. Tm»soara, Sera matematca, vol. XLI, f, 003, Pătrăşcou, C.; Feld of coes o,r)-coveloctes vector budle o a mafold. Bala J. Geom. Appl. 3, No., ). SC Papuc, D.I. ; Feld of coes ad postve operators o vector budle, A. Uv.Tmsoara, Sera matematca, vol. XXX, f, 99, P apuc, D.I. The geometry of a vector budle edowed wth a coe feld, Proceedgs of the 3rd Iteratoal Worshop o Dfferetal Geometry ad ts applcatos, Sbu, vol. V, 997, Puta, Hamltoa echacal Systems ad Geometrc Quatzato, Kluwer 993 ISBN: