DECOMPOSITION OF SYMPLECTIC STRUCTURES

Transcription

1 Transnatinal Jurnal f Mathematical Analysis and Applicatins Vl., Issue, 03, Pages 7-44 ISSN Published Online n Nvember 30, Jyti Academic Press DECOMPOSITION OF SYMPLECTIC STRUCTURES NOBUTAKA BOUMUKI, TAKUYA SHIMOKAWA, KYOJI SUGIMOTO and TOMONORI NODA Department f Mathematics Tky University f Science -3 Kagurazaka Shinjukuku, Tky Japan bumuki@rs.tus.ac.jp j0068@ed.kagu.tus.ac.jp j0073@ed.kagu.tus.ac.jp General Educatin and Research Center Meiji Pharmaceutical University -5- Nshi Kiyse-shi, Tky Japan nda@my-pharm.ac.jp Abstract We cnsider the decmpsitin f symplectic structures by Pissn structures r presymplectic structures and give an example which admits n nn-trivial cmpact decmpsitins. 00 Mathematics Subject Classificatin: Primary 53D05; Secndary 7B0. Keywrds and phrases: symplectic, Pissn, presymplectic, cmpact exceptinal Lie grup. Received Octber 0, 03

2 8 NOBUTAKA BOUMUKI et al.. Intrductin Symplectic structures n differentiable maniflds are interesting in bth physics and mathematics (e.g., Abraham and Marsden [], Guillemin and Sternberg [3]). In physics, it gives the framewrk f classical mechanics and is studied at the present time (e.g., Cantrijn et al. [9], Nrris [8], Paufler and Remer [9], Sardanashvily [0]). On the ther hand, in mathematics, symplectic structures appear in nt nly differential gemetry but als tplgy (e.g., Atiyah [], Grmv []), cmplex gemetry (e.g., Atiyah and Btt [3], Dnaldsn [0], Mabuchi [5]), and statistics (e.g., Barndrff-Nielsen and Jupp [4], Friedrich [], Nakamura [7]). A symplectic structure is a structure n a manifld pssessing the integrability and nn-degenerateness. As generalizatins f this structure there are Pissn and presymplectic structures. These are cnsidered as degenerate symplectic structures and have resemblance and different pints. Fr example, these tw structures define fliatins f the manifld but the directins f leaves are different. In this paper, we discuss hw t cnstruct a symplectic structure n a manifld by degenerate structures, i.e., Pissn structures and presymplectic structures. This is cnsidered as a decmpsitin f a symplectic structure and is described by states f fliatins defined by these structures. In general, let F and F be fliatins f a manifld M. Dente by D ( x) and D ( x) tangent spaces f leaves f F and F passing thrugh x M, respectively. We define F = F if D ( x) = D ( x) fr any x M and F F = M if D ( x) D ( x) = Tx M fr any x M. Therem.. (i) Let i, i =,, be regular Pissn structures n M and let F be the fliatin f M defined by. Assume that rank( ) i + rank( ) = dim M. Then i, i =,, define a symplectic structure n M if and nly if F F = M. (ii) Let ωi, i =,, be presymplectic structures with cnstant rank n M and let F be fliatins defined by ω. Assume that ω i i i

3 DECOMPOSITION OF SYMPLECTIC STRUCTURES 9 rank( ω ) + rank( ω ) = dim M. Then ωi, i =,, define a symplectic structure n M if and nly if F F = M. ω (iii) Let and ω be a regular Pissn structure and a presymplectic structure with cnstant rank n M and let F and F ω be fliatins f M defined by and ω, respectively. Assume that rank ( ) + rank( ω) = dim M. Then and ω define a symplectic structure n M if and nly if F = F ω and there exists an integrable distributin D = { D( x) } x M n M such that the restrictin f ω t D is nndegenerate at each pint in x M. ω Cnversely, if there exists a fliatin F f a symplectic manifld ( M, Ω) such that the restrictin f Ω t every leaf f F is nndegenerate, then there is a regular Pissn structure n M. In additin, if there exists a fliatin F f M such that F F = M, then there exist regular Pissn structures, and presymplectic structures ω, ω with cnstant rank n M such that is cmpatible with ω and is cmpatible with ω. Thugh we may cnsider inexpensively the case f (iv) F F = M; (v) F = F ; (vi) F = F, ω ω ω by means f cnditins f fliatins, we lack infrmatin t define a symplectic structure n M. The case (iv) was studied f Vaisman [] and applied t the gemetric quantizatin (in this case, we have a hrsymplectic structure as in De Barrs [5]. Cf. dual P-p structures in Błaszak and Marciniak [6]). In ur wrk, a result in Vaisman [] is crucial. Fr (v) and (vi), we nly nte that by the equivariant Darbux therem the lcal thery is cmpletely determined (e.g., Guillemin and Sternberg [3]).

4 30 NOBUTAKA BOUMUKI et al. By Therem., there exists a decmpsitin f a symplectic structure Ω n M by Pissn structures r presymplectic structures, if and nly if there exist fliatins F and F f M such that F F = M and the restrictin f Ω t n fewer than leaves f ne f fliatins (and then all leaves f bth fliatins) is nndegenerate. The pair ( F, F ) f these fliatins is called a decmpsitin f Ω. If all leaves f F and F are cmpact, then the decmpsitin ( F, F ) is called cmpact. By taking a direct prduct f cmpact symplectic maniflds, we btain an example f cmpact decmpsitins. We easily have an example f nn cmpact decmpsitins n 4-dimensinal trus 4 T by -dimensinal planes with irratinal slant. It is well knw that every cmpact hmgeneus symplectic manifld with Kirillv-Kstant- Suriau frm is expressed as a cadjint rbit f suitable cadjint actin. In view f Therem., we have the fllwing: Therem.. On cmpact Kählerian hmgeneus space ( G H, Ω), there exist n nn-trivial cmpact decmpsitins f Ω by invariant structures. Nte that H in the therem abve is either U ( ) r T and the secnd Betti number f G T is tw.. Pissn and Presymplectic Structures Let M be an n-dimensinal differentiable manifld. We dnte by p Λ ( TM ), the space f all p-vectrs, i.e., anti-symmetric cntravariant 0 tensr fields f type ( p, 0), where Λ ( TM ) = C ( M ). The Lie derivative, X ( TM q LX Λ ), n M acts n Λ ( TM ) by d q ( LXQ )( x) = { exp ( tx ) Q( exp tx ( x) )}, x M, Q Λ ( TM ). dt t= 0 (.)

5 DECOMPOSITION OF SYMPLECTIC STRUCTURES 3 Fr X,, X p Λ ( TM ), we define [ X, Q] : L Q, i = X i p p, i p i Q i= i+ [ X X Q ] : = ( ) X Xˆ X [ X, ], (.) then we have the peratr p [, ] : ( M q ) ( M p+ q Λ Λ ) Λ ( M ), (.3) satisfying (.), which is called the Schuten-Nijenhuis bracket. (Fr detail, see, e.g., Vaisman [3]). A -vectr Λ ( TM ) is called a Pissn structure, if [, ] = 0. If we lcally express as n = ij i, j= i x j x, then [, ] = 0 is equivalent t hi x h jk + hj x h ki + hk x h ij = 0. We next give an alternative definitin f Pissn structures. Fr f, g C ( M ), we define the bilinear frm {, }: C ( M ) C ( M ) C ( M ) n C ( M ) by f g { f, g} : = ( df, dg) = ij ( x), x x n i, j= and call the Pissn bracket. This bracket satisfies { f, { g, h } + { g, { h, f } + { h, { f, g } = 0, (.4) fr any f, g, h C ( M ) if and nly if is a Pissn structure n M. i j

6 3 NOBUTAKA BOUMUKI et al. Fr f C ( M ), the vectr field X f X( M ) defined by X f g = { g, f}, is called the Hamiltnian vectr field f f. Define a map : T M TM, by ( df ) =. X f If a Pissn structure is defined frm a symplectic structure n M, then this map is nn-degenerate (linear ismrphic at each pint in M), while this is degenerate in general. The additive structures t define the inverse f is given in the next sectin (Lemma 3.). We define the rank f at x M as the rank f the linear map ( x) and dente by rank( ( x) ). Nte that rank ( ( x) ) is always even. A Pissn structure whse rank is cnstant n M is especially called regular. Fr regular Pissn maniflds, we have the fllwing: Lemma.. Fr every pint x in a regular Pissn manifld ( M, {, }), there exists a lcal chart ( U, x,, x, y,, y, z,, z ) f x such that where r : = rank. i j i j i j { x, y } = δ, { x, z } = { y, z } = 0, ij i j i j i j { x, x } = { y, y } = { z, z } = 0, i In this lemma, { z = cnst. } defines a glbal fliatin F f M. Indeed, fr x M, if we set D( x) : = { X T M; f C ( M ) s.t. X ( x) X }, x f = then rank ( ( x) ) = dim D( x) and D( x) defines the fliatin F. x M Here the integrability f D( x ) fllws frm [, ] = 0 r the x M r r n r

7 DECOMPOSITION OF SYMPLECTIC STRUCTURES 33 Jacbi identity (.4). Mrever, induces a symplectic structure n each leaf f F. Nte that n general Pissn manifld, the distributin defined abve is integrable and define a fliatin called the symplectic fliatin. Lemma.. Let M be a n-dimensinal differentiable manifld, Ω be a nndegenerate -frm n M, and F be a fliatin f M. Assume that all leaves f F are f dimensin k and the restrictin f Ω t every leaf defines a symplectic structure n the leaf. Then, there exists a Pissn structure n M induced frm the symplectic structures n the leaves f F. Fr the prf, see Vaisman [3]. Nte that this lemma hlds fr mre general situatin and the Pissn structure cnstructed in the lemma is called a Dirac structure. We next review the definitin and prperties f presymplectic structures. The pair ( M, ω) f a differentiable manifld M and -frm ω n M is a presymplectic manifld, if ω is clsed -frm n M. Because ω is a -frm, at each x M, we have the linear map ω : Tx M Tx M X i( X ) ω, where i dentes the interir prduct. In the case f symplectic maniflds, this map is ismrphic. By the clsedness f ω, we have a fliatin n M. Indeed, let V ( x) : = { X Tx M; i( X ) ω = 0}, x M. We define the rank f ω at x M by the cdimensin f V ( x) and dente by rank( ω ( x) ). Nte that rank ( ω( x) ) is always even. Frm nw n, we assume that the rank f x M ω is cnstant n M. It fllws that V := V ( x) defines a distributin n M by the Frbenius therem. Dente the maximal cnnected integral submanifld f V thrugh x M by N ( x). The fliatin defined by N ( x), x M, is called the null fliatin (in Vaisman [], it is called the vertical fliatin). We dente by M N the leaf space f the fliatin. In general, M N admits n manifld structures.

8 34 NOBUTAKA BOUMUKI et al. Let ( S, ω S ) be a symplectic manifld and N be a manifld, and set M : = S N. Then the pull-back pr ωs f ω n S with respect t the prjectin nt the first factr pr : M S defines a presymplectic structure n M. Cnversely, every presymplectic manifld with cnstant rank lcally has this frm. Indeed, the fllwing Darbux type therem hlds fr presymplectic maniflds. Lemma.3. Fr every pint x in a presymplectic manifld ( M, ω), r r n r there exists a lcal chart ( U; x,, x, y,, y, z,, z ) such that ω = r i= r r i i a ( x,, x, y,, y ) dx dy. i Here r is the half f the rank f ω. 3. Prf f Therem. We first cllect prperties abut Pissn and presymplectic structures befre giving the prf f Therem.. Pissn and presymplectic structures may be cnsidered as degenerate symplectic structures. We first see differences between these structures. If ω is a presymplectic structure with cnstant rank n a manifld M, then there is a map ω : TM T M. On the ther hand, if we have a Pissn structure n M, then we have : T M TM. Althugh in bth cases, we have maps between TM and T M, directins f maps are different. Furthermre, Pissn and presymplectic structures define fliatins f M. Hwever, the directins f leaves are different (fr Pissn structures, the resulting fliatins are symplectic fliatins, whereas fr presymplectic structures these are null fliatins).

9 DECOMPOSITION OF SYMPLECTIC STRUCTURES 35 Fr a regular Pissn structure and a presymplectic structure ω with cnstant rank (ranks are bth r), if ω Im ( ) = id and ω Im( ω ) = id, (3.) hld, then is cmpatible with ω r ω is cmpatible with. The existence f cmpatible structures is given by Lemma 3. (Vaisman []). (i) Let ( M, ω) be a presymplectic manifld and V be the distributin n TM defined by ω. If there exists an integrable distributin D n M satisfying D V = TM, then there is a ω -cmpatible Pissn structure n M. (ii) Let ( M, ) be a Pissn manifld and set V : = ( T M ). If there exists an integrable distributin D n M satisfying D V = TM, then there is a -cmpatible presymplectic structure ω n M. Prf. (i) By TM = D V, we have T M = D V, where D = { α T M; α( v) = 0 fr all v V}. Fr any α T M, we set α = α P + α p D V, and define : T M TM and -vectr by P P P ( α) : = ( ω ) ( α ), ( α, β) : = ω(( ω ) ( α ), ( ω ) ( β )). Since D is integrable, satisfies (.4) and then define a Pissn structure n M (cf. Lichnerwicz [4]). Obviusly is ω -cmpatible. We als btain (ii) the same as (i). Fr a presymplectic manifld M, a Pissn bracket is defined n nt hle C ( M ) but the subalgebra C ( M N ) : = { f C ( M ); f is cnstant n any leaf f the null fliatin}. The reasn why a Pissn bracket cannt be defined is that ω is degenerate and lack f infrmatin abut

10 36 NOBUTAKA BOUMUKI et al. null directins. A sufficient cnditin t define a Pissn bracket n C ( M ) is (i) f Lemma 3., and (ii) f Lemma 3. is a cnditin that a Pissn structure determines a clsed -frm. By Lemma 3., we may prve Therem. as fllws. (i) and (ii) f Therem. is easy t check by the definitin f symplectic structures and (iii) is reduced t (i) r (ii) by Lemma 3.. The latter half f Therem. fllws frm Lemma.. Remark 3.. (i) Let F and F be fliatins n a symplectic manifld ( M, Ω) such that F F = M. Assume that Ω ( x ) is nndegenerate fr all x M. In this situatin, Ω ( D ( x) D ( x)) = { 0}, x,, M des nt hld in general. Fr example, let D M = T = R Z, Ω = 3 4 dx dx + dx dx, where 3 4, x, x, x x is the crdinates n T induced frm the 4 cannical crdinates f R. If we define = span{, }, D = span{ + 3, + 4 }, D where i : = ( i =,, 3, 4), then Ω D and x i Ω D are nndegenerate; Ω( X, Y ) = fr X : = L and Y : = + 3 L. (ii) Fr a decmpsitin f symplectic structure Ω n M by presymplectic structures ω and ω, i.e., Ω = ω + ω( F ω F = M ), ω there des nt necessarily exist symplectic maniflds ( M, ω ), ( M ω ) such that M = M M.,

11 DECOMPOSITION OF SYMPLECTIC STRUCTURES Prf f Therem. Let D = {( D ) p } p G H and D = {( D ) p } p G H be nn-trivial invariant integrable distributins n a cmpact Kählerian hmgeneus space G H n which G acts effectively. T prve Therem., we may shw the fllwing: Let L i ( ) dente the leaf f D i thrugh the rigin G H ( i, ). If bth L ( ) and L ( ) are cmpact, then = T ( G H ) = ( D ) ( D ) ( direct sum), cannt hld. In rder t shw this, we first see the fllwing: Prpsitin 4.. Let D = { D p } p G H be a nn-trivial, invariant, integrable distributin n a cmpact Kählerian hmgeneus space G H n which G acts effectively. Suppse that the leaf L ( ) f D thrugh is cmpact. Then, there exists a cmpact, cnnected subgrup K f maximal rank f G, which satisfies tw cnditins: (i) H K ; G (ii) D = d ( T ( K H )) fr any pint p = τ g ( ) G H, p τ g where τ g G, dentes a transfrmatin f G H defined by g, τ ( ah ) : gah fr ah G H. g = Prf. Let g : = Lie( G ) and h : = Lie( H ). By Therem in Matsushima [6, p. 56], H is the centralizer f a trus f G and then rank G = rank H. (4.) It is clear that G H is a reductive hmgeneus space and s there exists a vectr subspace m f g, which satisfies tw cnditins: ( a) g = h m and ( b) Ad( h) m m fr all h H.

12 38 NOBUTAKA BOUMUKI et al. Hereafter, we identify T ( G H ) with m and assume D is a subspace f m. Nte that { 0 } D m because D is nn-trivial. Since D = { D } p p G H is invariant and integrable, we are able t deduce that Ad ( h) D D fr all h H, [ D, D ] D. Define a subspace k f g by h (4.) k : = h D. Then (4.) implies that k is a subalgebra f g. Let K be the cnnected Lie subgrup f G with Lie( K ) = k. By virtue f h k g, we see that H0 K G. Here H 0 dentes the identity cmpnent f H. Therefre, the first cnditin (i) hlds; H K G, (4.3) because H is cnnected (see Matsushima [6, Therem, p. 56]). Since H is clsed in G and since the identity mapping f K int G is cntinuus, H is clsed in K. Frm nw n, we are ging t verify that L ( ) cincides with K H. In terms f k = h D, we cnclude that T ( K H ) =. Accrdingly, the secnd cnditin (ii) hlds; D D p = dτ ( T ( K H )) fr any pint p = τ ( ) G, g g H because D = { D p } p G H is invariant. Hence bth L ( ) and K H are cnnected integral maniflds f D thrugh G H. It fllws that K H L( ), because L ( ) is the leaf f D thrugh. Nte that K H is an pen submanifld f L ( ). Let us shw that the cnverse inclusin is als true. Since L ( ) is cmpact, it is a cmplete Riemannian manifld with respect t the metric induced frm G H. Thus, any pint x L( ) can be jinted t by a brken gedesic f L ( ). Since K H is a cmplete

13 DECOMPOSITION OF SYMPLECTIC STRUCTURES 39 ttally gedesic submanifld f L ( ), we cnfirm that K H cntains this brken gedesic. Hence x K H. This shws L( ) K H. We have verified that L ( ) = K H. Since bth L ( ) and H are cmpact (see Matsushima [6, Therem, p. 56] again), K is als cmpact. It fllws frm (4.) and (4.3) that rank G = rank K, namely, K is a subgrup f maximal rank f G. Cnsequently, we have prved Prpsitin 4.. It is knwn that a cmpact Kählerian hmgeneus space either G H is G U( ) r G T (4.4) (cf. Brdemann et al. [7, p. 643]), where T is a maximal trus f G. It is als knwn that a cnnected clsed subgrup K f maximal rank f G is ismrphic t ne f the fllwing: SU ( 3), SU( ) SU( ), U( ), T (4.5) (cf. Brel and de Siebenthal [8, p. 9]). Taking (4.4) and (4.5) int cnsideratin, we will demnstrate that ( G H ) = ( D ) ( D ) T cannt hld in tw cases G H = G ( ) and G H = G. U T Remark 4.. Tw actins f G n G U( ) and n G T are almst effective but nt effective. Hwever, we hereafter assume that tw actins f G n G U( ) and n G T are effective. Because, if necessary, ne may cnsider tw effective actins f G Z n ( G Z ) ( U( ) Z ) and n ( G Z ) ( T Z ) instead f them, and cnsider SU ( 3) Z, ( SU( ) SU( ) ) Z, U( ) Z, T Z instead f (4.5), where Z dentes the center f G.

14 40 NOBUTAKA BOUMUKI et al. Case G H = G U( ). Let D = {( D ) p } p G U ( ) and D = {( D ) p } p G U( ) be tw, nn-trivial, invariant, integrable distributins n G U( ) such that bth L ( ) and L ( ) are cmpact, where L i ( ) is the leaf f D i thrugh the rigin G U( ). A cnnected clsed subgrup K f maximal rank f G satisfying U( ) K is either SU ( 3) r SU ( ) SU( ) by (4.5). Therefre, Prpsitin 4. assures that ( D ) = T ( SU( 3) U( ) ) r T (( SU( ) SU( ) ) U( ) ). i Since dim SU ( 3) U( ) = 4, dim( SU( ) SU( ) ) U( ) = and dimg U( ) = 0, we cnclude that T ( G U( )) = ( D ) ( D ) cannt hld. T Case G H = G. Let D = {( D ) } and p p G T D = {( D ) } be tw, nn-trivial, invariant, integrable p p G T distributins n T G such that bth L ( ) and L ( ) are cmpact, where L i ( ) is the leaf f D i thrugh the rigin G T. A cnnected clsed subgrup K f maximal rank f G satisfying T K is either f the SU ( 3), SU( ) SU( ) r U ( ) by (4.5). By Prpsitin 4., we cmprehend that ( D ) = T ( SU( 3) T ), T (( SU( ) SU( )) T ) r T ( U( ) T ). i It is natural that dim SU ( 3) T = 6, dim( SU( ) SU( )) T = 4, dim U ( ) T =, and dim G T. If ( G T ) = ( D ) ( D ) = = T T hlds, then ( D ) = ( D ) T ( SU( 3) ). Hwever, such a case cannt T ccur by Lemma 4.3 belw. Therefre, ( G T ) = ( D ) ( D ) cannt hld and Therem. hlds.

15 DECOMPOSITION OF SYMPLECTIC STRUCTURES 4 Lemma 4.3. Let D = {( D ) } be an invariant distributin n T p p G T G such that T ( G T ) = ( D ) D. Then D cannt be integrable. Here D is the invariant integrable distributin { } { dτ ( T ( SU( 3) T ))} ( ). = g τg G T Dp p G T Prf. Let t : = Lie( T ). Then t is a maximal abelian subalgebra f g, and g is decmpsed int the direct sum g 6 = t Vθ j j=, where V θ is an ad t -invariant subspace f g and dim V = fr j θ j each j {,, 6}; mrever, there exist a basis { X, j X, j } 6 f 6 j= V and rts θ j : t R, which satisfy θ j 0 θ j ( Z ) λ ad ( ) (,, ), ( ) 0 Z Y = X j X j θ j Z µ, j= fr any Z t and Y = λx, j + µ X, j V (cf. Tda and Mimura [, θ j + Chapter 5]). Let { α, α } dente the set f simple rts in : = { θ j } j =. We assume that the Dynkin diagram f { α, α } is as fllws: 6 Nte that in this case + is described as + = { α, α + α, α + α, 3α + α, 3α + α, α}. In the setting, we are ging t prve that the distributin D cannt be 6 integrable. Let us identify T ( G T ) with V. Since D and D j= θ j

16 4 NOBUTAKA BOUMUKI et al. ) + are invariant, ( D and D are given by cmbinatins f V, θ j. Direct cmputatin enables us t see that the cmbinatin f rts in generating su ( 3) = Lie( SU( 3) ) is ne f the fllwing: (){ α, α + α, }, α ( ) { α, α + α, α + }, α ( ) { α, α + α, 3α + }, 3 α ( ) { α + α, α + α, 3α + }, 4 α ( ) { 3α + α, 3α + α, }. 5 α Accrdingly, ne f the fllwing five cases nly ccurs () = su () 3 t = V V V, D α α +α α ( ) = su ( 3) t = V V V, D α α +α α +α ( 3) = su ( 3) t = V V V, D α α +α 3α +α ( 4) = su ( 3) t = V V V, D α +α α +α 3α + α ( 5) = su ( 3) t = V V V. D 3α +α 3α + α Since D = { D } is integrable, it must satisfy [ D, D ] t D. p p G T Therefre, the last case (5) nly ccurs. In this case, ( D is as fllws: α ( ) = V V V, D α α +α α + α because T ( G T ) = ( D ) D and ( D ) is given by a cmbinatin f V θ. Thus, it is impssible fr ( D ) j t satisfy [( D ) ( D, ) ] t ( D ) in this case. Cnsequently, D = {( D ) } cannt be integrable. p p G T ) θ j +

17 DECOMPOSITION OF SYMPLECTIC STRUCTURES 43 References [] R. Abraham and J. E. Marsden, Fundatins f Mechanics, nd Editin, Reading, Massachusetts, 978. [] M. F. Atiyah, New Invariants f 3-and 4-Dimensinal Maniflds, The Mathematical Heritage f Hermann Weyl (Durham, NC, 987), 85-99, Prc. Symps. Pure Math., 48, Amer. Math. Sc., Prvidence, RI, 988. [3] M. F. Atiyah and R. Btt, The Yang-Mills equatins ver Riemann surfaces, Phils. Trans. Ry. Sc. Lndn Ser. A 308(505) (983), [4] O. E. Barndrff-Nielsen and P. E. Jupp, Ykes and symplectic structures, J. Statist. Plann. Inference 63() (997), [5] C. M. de Barrs, Sur la gémetrie différentielle des frmes différentielles extérieures quadratiques, In: Atti Cnvegn Intern. Gemetria Differenziale, Blgna (967), 7-4, Blgna: Ed. Zanichelli. [6] M. Błaszak and K. Marciniak, Dirac reductin f dual Pissn-presymplectic pairs, J. Phys. A 37(9) (004), [7] M. Brdemann, M. Frger and H. Römer, Hmgeneus Kähler maniflds: Paving the way twards new supersymmetric sigma mdels, Cmm. Math. Phys. 0 (986), [8] A. Brel and J. de Siebenthal, Les sus-grupes fermés de rang maximum des grupes de Lie cls, Cmment. Math. Helv. 3 (949), 00-. [9] F. Cantrijn, A. Ibrt and M. de León, On the gemetry f multisymplectic maniflds, J. Austral. Math. Sc. (Series A) 66 (999), [0] S. K. Dnaldsn, Symmetric Spaces, Kähler Gemetry and Hamiltnian Dynamics, Nrthern Califrnia Symplectic Gemetry Seminar, 3-33, Amer. Math. Sc. Transl. Ser., 96, Amer. Math. Sc., Prvidence, RI, 999. [] T. Friedrich, Die Fisher-infrmatin und symplectische Strukturen, Math. Nachr. 53 (99), [] M. Grmv, Pseud hlmrphic curves in symplectic maniflds, Invent. Math. 8 (985), [3] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge, 984. [4] A. Lichnerwicz, Les variétés de Pissn et leurs algèbres de Lie assciées, J. Diff. Gem. (977), [5] T. Mabuchi, Sme symplectic gemetry n cmpact Kähler maniflds (I), Osaka J. Math. 4 (987), 7-5. [6] Y. Matsushima, Sur les espaces hmgènes Kählériens d un grupe de lie réductif, Nagya Math. J. (957),

18 44 NOBUTAKA BOUMUKI et al. [7] Y. Nakamura, Cmpletely integrable gradient systems n the maniflds f Gaussian and multinmial distributins, Japan J. Indust. Appl. Math. 0 (993), [8] L. K. Nrris, Generalized Symplectic Gemetry n the Frame Bundle f a Manifld, In Prc. Symp. Pure Math., R.E. Green and S. T. Yau Eds., 54 (993), [9] C. Paufler and H. Remer, Gemetry f Hamiltnian n-vectrs in Multisymplectic Field Thery, arxiv: math-ph/ [0] G. Sardanashvily, Generalized Hamiltnian Frmalism fr Field Thery, Cnstraint Systems, Wrld Scientific, Singapre, 995. [] H. Tda and M. Mimura, Tplgy f Lie Grups, I and II, American Mathematical Sciety, Prvidence-Rhde Island, 99. [] I. Vaisman, Gemetric quantizatin n presymplectic maniflds, Mnatsh. Math. 96(4) (983), [3] I. Vaisman, Lectures n the Gemetry f Pissn Maniflds, Prgress in Mathematics, 8, Birkhäuser Verlag, Basel, 994. g