Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 034, 23 pages Geometric Realizations of Bi-Hamilto nian Complete ly In tegrab le Systems ⋆ Gloria MAR ´ I BEFF A Dep artment of M athematics, University of Wisc onsin, Madison, WI 53705, USA E-mail: marib eff@math.wisc.e du URL: http://www.mat h.wisc.e du/ ~ maribeff / Received Nov em ber 14, 2007, in f inal form March 13, 2008 ; Published online March 27, 2008 Original article is av ailable a t ht tp://w ww.em is.de/journals/SIGMA/2008/034/ Abstract. In this pap er we present an ov erview o f the connection b e t ween co mpletely in- tegrable systems and the background geometry of the f low. This r elation is b etter seen when using a g roup-bas ed concept of moving fra me intro duced by F els and Olver in [ A cta Appl. Math. 51 (199 8), 161– 213; 55 (1999 ), 127–2 08]. The pap er dis cusses the clo s e connection betw een dif ferent types of geometries and the type o f equations they re alize. In particular, we descr ib e the dir ect relation be t ween symmetric space s and equations of KdV-type, a nd the po ssible geometric o rigins of this c o nnection. Key wor ds: inv ariant evolutions of curves; Hermitian symmetr ic spa ces; Poisson br ack ets; dif ferential inv ariants; pro jective dif ferential inv aria nts; equations of KdV type; c o mpletely int egrable PDEs; moving fra mes; g e o metric realizations 2000 Mathematics Subje ct Classific ation: 37K25; 53 A55 1 In tro duction Example 1. One of the simp lest examples of a geometric realization of a completely in tegrable system is that of the nonlinear Sc hr¨ odinger equation (NLS) realized by th e self-induction V ortex Filamen t f low (VF). The VF f low is a f lo w in the Euclidean space SO(3) ⋉ R 3 / SO(3) (see [3]). In [21] Hasimoto show ed th at, if u ( x, t ) is a f lo w solution of the VF equation u t = κB , where κ is the Euclidean curv ature of the curve u ( · , x ) ∈ R 3 , x is the arc-length and B is the binormal, then the ev olution of the curv ature and torsion of u is equiv alent to the NLS equation via th e Hasimoto transformation Φ = κe i R τ . The Hasimoto trans formation ( κ, τ ) → ( ν , η ), with Φ = ν + iη , is, in fact, ind u ced by a change from classical Euclidean mo ving frame to the natur al moving fr ame ( ν and η are the natur al cu rvatur es , see [28]). Th us, VF is a Euc lide an ge ometric r e aliza tion of NLS if we use natur al m o ving fr ames. Equiv alent ly , NLS is the invariantization of VF. The relation to the Eu clidean geometry of the f low go es further; consider the ev olution u t = hT + h ′ κ N + g B , (1) where { T , N , B } is the classical E u clidean mo ving f r ame and h and g are arb itrary smo oth functions of the curv at ure, torsion and their deriv ativ es. Equation (1) is the general form of ⋆ This pap er is a contribution to the Pro ceedings of the Seventh International Conference “Sy mmetry in Nonlinear Mathematical Physics” (June 24–3 0, 200 7, Kyiv, Ukraine). The full collection is a v ailable at http://w ww.emis.de/j ournals/SIGMA/symmetry2007.h tml 2 G. Mar ´ ı Bef fa an arc-length pr eserving evolutio n of space curve s, inv arian t under the action of the E u clidean group (i.e., E ( n ) tak es solutions to solutions). Its inv arian tizatio n can b e wr itten as  κ τ  t = P  g h  , (2) where P d ef ines a Poi sson br ac k et generated by the se c ond Hamiltonian structur e for NLS via the Hasimoto transform ation, i.e. the Hasimoto transformation is a Poisson map (see [28, 38, 39]). (F or more in formation on in f inite d im en sional Poisson brac k ets see [40], and for more information on P see [38].) Cle arly , P can b e generated using a classical Euclidean moving frame and the in v ariant s κ and τ . T he NLS equation is a bi-Hamiltonian system , i.e., Hamiltonian with resp ect to t w o compatible Hamiltonian structures. One of the stru ctures is inv ertible and a recurs ion op erator can b e constru cted to generate int egrals of the system (see [30] or [40]). The f irst Hamiltonian str ucture for NLS is inv ertible and can also b e pro v ed to b e generated by the geometry of the f lo w, although it is of a d if ferent c haracter as we will see b elo w. Example 2. A second example is th at of the Kortew eg–de V ries (KdV) equation. If a 1-pa- rameter family of functions u ( t, · ) ∈ R evo lv es follo wing the Schwarzian KdV e quation u t = u ′ S ( u ) = u ′′′ − 3 2 ( u ′′ ) 2 u ′ , where S ( u ) = u ′′′ u ′ − 3 2  u ′′ u ′  2 is the Schwa rzian derivative of u , then k = S ( u ) itself ev olv es follo wing the Kd V equation k t = k ′′′ + 3 k k ′ , one of the b est kno wn completely integrable nonlin ear PDEs. The KdV equ ation is Hamiltonian with resp ect to t w o compatible Hamiltonian structures, namely D = d dx and D 3 + 2 k D + k ′ (called resp ectiv ely f irst and second KdV Hamiltonian stru ctures). As b efore, if w e consider the general curv e ev olution giv en by u t = u ′ h, (3) where h is any smo ot h fun ction dep end ing on S ( u ) and its deriv at iv es with r esp ect to the parameter x , then k = S ( u ) ev olv es follo wing the evo lution k t = ( D 3 + 2 k D + k ′ ) h. (4) This time S ( u ) is th e generating differ ential i nv ariant asso ciated to the action of PSL(2) on R P 1 . That is, any pro jectiv e dif f er ential inv arian t of curve s u ( x ) is a f u nction of S ( u ) and its deriv a- tiv es. Equation (3) is th e most general form for ev olutions of reparametrizations of R P 1 (or parametrized “curve s”) inv arian t u nder the action of PSL(2). Evo lution (4) can b e view ed as the inv arian tizat ion of (3). Equiv alen tly , the family of evol utions in (3) p r o vides R P 1 geomet- ric realizatio ns for the Hamiltonian ev olutions def ined by (4). Th us, on e can obtain geometric realizatio ns in R P 1 not only for KdV, but also for any system w hic h is Hamiltonian w ith resp ect to the second KdV Hamiltonian str u cture. F or example, the S a w ada–Kote rra equation k t = ( D 3 + 2 k D + k ′ )  2 k ′′ + 1 2 k 2  = 2 k (5) + 5 k k ′′′ + 5 k ′ k ′′ + 5 2 k ′ k 2 is bi-Hamiltonian with resp ect to the same Hamiltonian structur es as KdV is. Its Hamiltonian functional (4) is h ( k ) = R ( 1 6 k 3 − ( k ′ ) 2 ) dx . Therefore, the Sa w ada–Kote rra equation has u t = u ′  2 S ( u ) ′′ + 1 2 S ( u ) 2  Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 3 as R P 1 realizatio n. (Inciden tally , Saw ada–Koterra has a second realizat ion as an equi-af f ine f lo w, see [41].) The manifold R P 1 is an example of a p ar ab olic homo gene ous sp ac e , i.e., a manifold of the form G/P with G semisimple and P a p ar ab olic sub gr oup . F rom (3) and (4) we can see that the second Hamiltonian structure for Kd V can b e generated with the sole knowle dge of u ′ (a classical pro jectiv e mo ving fr ame along u ) and S ( u ), its p r o jectiv e dif ferential in v ariant. Th e f irst Kd V s tructure is also similarly generated, although, again, it is of a dif feren t n atur e. These t w o simple examples illustrate the close relationship b et ween the classical geometry of cu rv es and b i-Hamiltonian completely in tegrable P DEs. In the last y ears man y examples of geometric realizatio ns for most known completely integrable systems ha v e b een app earing in the literature. Some are linke d to the geometric inv arian ts of the f lo w (see for example [1, 2, 11, 13, 17, 25, 26, 28 , 29, 31, 36, 38, 39, 46, 47, 48, 49, 51]). Th is list is, by n o means, exhaustiv e as this pap er is n ot mean t to b e an exhaustiv e r eview of th e su b ject. P erhaps the s implest wa y to understand the close r elationship b et ween d if ferential in v ariants and integrable sys tems is through the AKNS rep resen tation on one hand and group -based m oving frames on the other. If G is a Lie group, a G -AKNS repr esen tation of a nonlinear PDE k t = F ( k , k x , k xx , . . . ) (5) is a linear system of equations ϕ x = A ( t, x, λ ) ϕ, ϕ t = B ( x, t, λ ) ϕ, ϕ ( t, x, λ ) ∈ G, A ( x, t, λ ) , B ( x, t, λ ) ∈ g suc h that the compatibilit y condition for the existence of a solution, A t = B x + [ B , A ] , is ind ep endent of λ and equiv alent to the nonlinear PDE (5). Suc h a representa tion is a basis f or generating solutions and int egrating the sy s tem. Indeed, most in tegrable systems hav e an AKNS represent ation. Geometrically , this is th ou ght of as h a ving a 2-parameter f lat connection d ef ined b y − d dx + A and − d dt + B along the f lo w (see [22]). Th e brid ge to dif f eren tial inv arian ts and dif f er ential geometry a-la-Cartan app ears when one realizes that this 2-parameter connection is a redu ction of the Maur er–Cartan c onne ction of G along the f lo w ϕ and the AKNS system could b e interpreted as the Serret–F r en et equations and the t -ev olution of a group-based (righ t) mo ving fr ame ( ϕ ) along a f lo w u : R 2 → G/H in a certain h omogeneous space. This is explained in the next section. In th is pap er we describ e ho w the bac kground geometry of affine and some symmetric mani- folds generates Hamiltonian structur es and geometric realizations f or some completely int egrable systems. Our af f ine manifolds will b e homogeneous m anifolds of the form G ⋉ R n /G with G semisimple. Th ey in clud e Euclidean, Mink o wski, af f ine, equi-af f ine and symplectic geometry among others. W e will also discu s s related geometries, lik e the cen tro-af f ine or geometry of star- shap ed curves, for which the action of the group is linear instead of af f ine. On the other h and, our symmetric manifolds are lo cally equiv alen t to a homogeneous manifold of the form G/H where g , the Lie algebra asso ciated to G , has a gradation of the form g − 1 ⊕ g 0 ⊕ g 1 and wh ere g 0 ⊕ g 1 = h is th e Lie algebra of H . These includes pro jectiv e geometry ( G = PSL( n + 1)), the Grassmannian ( G = S L( p + q )), the conformally f lat M¨ obius sphere ( G = O ( n + 1 , 1)), the La- grangian Grassman n ian ( G = Sp(2 n )), the manifold of r educed p ure spinors ( G = O ( n, n )) and more. W e will see how the Cartan geometry of cu r v es in these manifolds induces a Hamiltonian structure on th e space of d if ferential inv arian ts. In the last part of the pap er we lo ok closely at the case of s ymmetric spaces. W e def in e differ ential invariants of pr oje ctive typ e as those genera- ted b y the action of th e group on second ord er fr ames. W e then describ e ho w, in most cases, the 4 G. Mar ´ ı Bef fa reduced Hamiltonian structure can b e fur th er reduced or restricted to the sp ace of curves w ith v anishin g non-pr o jectiv e dif f eren tial in v ariant s. On this manifold the Hamiltonian structure has a (geometrically d ef ined) compatible P oisson compan ion. They def ine a bi-Hamiltonian p encil for some integrable equations of Kd V-t yp e, an d they provide geometric realiza tions for them. W e f inally state a conjecture by M. East w oo d on wh at the pr esence of these f lows m igh t say ab out the geometry of curves in symmetric s paces. As group based m o ving frames are relativ ely new, our next section will d escrib e them in detail. W e will also describ e their role in AKNS representat ions. 2 Mo ving frames The classical concept of moving frame was dev elop ed by ´ Elie Cartan [8, 9 ]. A classical moving frame along a curve in a manifold M is a cur ve in the frame bu ndle of th e manifold o v er the curv e, inv arian t under the action of the transformation group un der consideration. This metho d is a ve ry p o werful to ol, bu t its explicit application relied on in tuitiv e c hoices that were not clear on a general setting. Some id eas in Cartan’s w ork and later work of Grif f iths [19], Green [20] and others laid the f ou n dation for the concept of a group-based mo ving f rame, that is, an equiv arian t map b et ween the jet space of cu rv es in the manifold and the grou p of transformations. Recen t w ork by F els and Olv er [14, 15] f inally ga v e the precise d ef inition of the group -based moving frame and extended its application b ey ond its original geome tric picture to an astonishingly large group of applications. In this section we w ill describ e F els and O lv er’s moving frame and its relation to the classical m oving frame. W e will also introd uce some def initions that are useful to the study of P oisson brack ets and bi-Hamiltonian nonlinear PDEs. F rom n o w on we will assu me M = G/H with G acting on M via left multiplicatio n on repr esen tativ es of a class. W e w ill also assume that curves in M are p ar ametrize d and, th erefore, the group G do es not act on the parameter. Definition 1. Let J k ( R , M ) th e space of k -jets of cur ves, that is, th e set of equ iv alence classes of curve s in M up to k th order of con tact. If w e denote by u ( x ) a curve in M and by u r the r deriv ative of u with resp ec t to the p arameter x , u r = d r u dx r , the jet space has lo cal co ordinates that can b e repr esen ted by u ( k ) = ( x, u, u 1 , u 2 , . . . , u k ). The group G acts naturally on parametrized curv es, therefore it acts natur ally on the j et space via the formula g · u ( k ) = ( x, g · u, ( g · u ) 1 , ( g · u ) 2 , . . . ) , where by ( g · u ) k w e mean th e formula obtained when one dif f eren tiates g · u and then writes the result in terms of g , u , u 1 , etc. Th is is usually called the pr olonge d action of G on J k ( R , M ). Definition 2. A fun ction I : J k ( R , M ) → R is called a k th order differ ential invariant if it is inv arian t with resp ect to the prolonged action of G . Definition 3. A map ρ : J k ( R , M ) → G is called a left (resp. righ t) moving fr ame if it is equiv arian t with resp ect to the prolonged action of G on J k ( R , M ) and the left (resp. righ t) action of G on itself. Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 5 If a group acts (lo cally) effe ctively on subsets , then f or k large enough th e prolonged action is lo cally free on regular jets. This guaran tees the existence of a movi ng fr ame on a neigh b orho o d of a regular jet (for example, on a neigh b orho o d of a generic curve, see [14, 15]). The group-based mo ving frame already app ears in a familiar metho d for calculating the curv atur e of a curv e u ( s ) in the Euclidean plane. In this metho d one uses a translation to tak e u ( s ) to the origin, and a rotation to mak e one of the axes tangent to the curve. T he curv atur e can classically b e found as the coef f icien t of the second order term in the expansion of the cur v e around u ( s ). Th e crucial observ atio n made by F els and O lv er is that the element of the gr oup carrying out the translation and r otation dep end s on u and its deriv ative s and so it def ines a m ap from the jet space to the group . Th is map is a right moving fr am e , and it carries all the geometric inf ormation of the curve. In fact, F els and O lver develo p ed a similar normalization pro cess to f ind righ t mo ving frames (see [14, 15] and our next theorem). Theorem 1 ([14, 15]) . L et · denote the pr o longe d action of the gr o up on u ( k ) and assume we have normalization e quations of the form g · u ( k ) = c k , wher e c k ar e c onstants (they ar e c al le d norma lization c onstants ). A ssu me we have enough normaliza tion e quations so as to determine g as a function of u, u 1 , . . . . Then g = ρ is a r ight invariant moving fr a me . The d irect r elation b etw een classical mo ving fr ames and group-based moving fr ames is stated in the follo wing theorem. Theorem 2 ([32]) . L et Φ g : G/H → G/H b e define d b y multiplic ation by g . That is Φ g ([ x ]) = [ g x ] . L et ρ b e a gr oup-b ase d left moving fr ame with ρ · o = u wher e o = [ H ] ∈ G/H . Iden- tify d Φ ρ ( o ) with an element of GL ( n ) , wher e n i s the dimension of M . Then, the matrix d Φ ρ ( o ) c ont ains in its c olumns a classic al moving fr ame. This th eorem illustrates h o w classical moving fr ames are describ ed only by the action of the group-based mo ving frame on f ir st ord er frames, while the action on higher order frames is left out. Accordingly , those inv arian ts determined by the action on higher order frames will b e not b e found with the u se of a classical m o ving frame. Next is the equiv ale n t to the classical S erret–F renet equations. Th is concept if fund amen tal in our P oisson geometry stud y . Definition 4. Consider K dx to b e the horizon tal comp onen t of the pullback of the left (resp . righ t)-in v arian t Maurer–Cartan form of the group G via a group-based left (resp. righ t) moving frame ρ . That is K = ρ − 1 ρ x ∈ g (resp . K = ρ x ρ − 1 ) ( K is th e co ef f icien t matrix of the f irst order dif feren tial equation satisf ied by ρ ). W e call K the left (r esp. right) Serr et–F r enet e quations for the mo ving frame ρ . Notice that, if ρ is a left mo ving fr ame, then ρ − 1 is a right mo ving f rame and their Serret– F renet equations are the negativ e of eac h other. A complete set of generating dif feren tial in- v ariants can alw a ys b e foun d among the co ef f icien ts of group-based Serret–F renet equatio ns, a crucial dif ference w ith the classical picture. The follo wing theorem is a d irect consequence of the results in [14, 15]. A more general resu lt can b e found in [23]. Theorem 3. L e t ρ b e a (left or right) moving fr am e along a curve u . Then, the c o efficients of the (left or right) Serr et–F r enet e q uations for ρ c ontain a b as is for the sp ac e of differ ential invariants of the curve. That is, any other differ ential invariant for the curve i s a function of the entries of K and their derivatives with r esp e ct to x . 6 G. Mar ´ ı Bef fa Example 3. Assume G = PSL(2) so that M = R P 1 . T he action of G on R P 1 is giv en b y fractional transformations. Assume ρ =  a b c d  ∈ G is a (right ) m o ving fr ame satisfying the normalization equations ρ · u = au + b cu + d = 0 , ρ · u 1 = au 1 cu + d − ( au + b ) cu 1 ( cu + d ) 2 = 1 , ρ · u 2 = au 2 cu + d − 2 acu 2 1 ( cu + d ) 2 + au + b ( cu + d ) 3  cu 2 ( cu + d ) + 2 c 2 u 2 1 ( cu + d ) 3  = 2 λ. Then it is straigh tforw ard to chec k that ρ is completely d etermined to b e ρ =  1 0 1 2 u 2 u 1 − λ 1  u − 1 / 2 1 0 0 u 1 / 2 1 !  1 − u 0 1  . A movi ng f r ame satisfying this n ormalization will hav e the follo wing r ight Serret–F renet equation ρ x =  − λ − 1 1 2 S ( u ) + λ 2 λ  ρ. (6) This equation is gauge equiv al en t to the λ = 0 equation via the constan t g auge g =  1 0 λ 1  . (7) This gauge g will take the second normalization constan t to zero. F ur thermore, if u is a solution of (3), it is kno wn (see [31]) that the t -evo lution indu ced on ρ is giv en by ρ t = − 1 2 h x − λh − h 1 2 h xx + λh x + λ 2 h + 1 2 S ( u ) h 1 2 h x + λh ! ρ. (8) W e can now see the link b et w een the AKNS represen tatio n of KdV and the evo lution of a r igh t mo ving frame. Assume a completely integrable system (5) has a geometric r ealization whic h is inv arian t u nder the action of the geometric group G u t = f ( λ, u, u 1 , u 2 , . . . ) . (9) Then, under regularit y assu mptions of the f lo w, the in v ariant ization of (9) is the in tegrable system (5). A righ t mo ving frame along u will b e a solution of its S erret–F renet equation ρ x = K ( t, x, λ ) ϕ and the time evo lution will ind uce a time ev olution on ρ of the form ρ t = N ( t, x, λ ) ϕ. F ur thermore, since (9) is inv arian t under the group, b ot h K and N will dep end on the d if ferential in v ariant s of the f lo w. These equations are def ined b y the horizon tal comp onent of the p ullbac k of the Maurer–Cartan form of the group, ω = dg g − 1 b y the moving frame ρ , th at is K dx + N dt . Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 7 If w e no w ev aluate the structur e e quation f or the Maurer–Cartan form, i.e., dω + 1 2 [ ω , ω ] = 0, along ρ x and ρ t , we get K t = N x + [ N , K ] whic h is exactly the in v arian tization of the f lo w (9); therefore it is indep enden t of λ . Hence, a λ -dep enden t geometric realization of an inte grable system pr o vides an AKNS representat ion of the system. See [7] for more inform ation. Example 4. The AKNS representat ion f or K d V is v ery w ell kno wn. It is giv en by the system ϕ x =  − λ − 1 − q λ  ϕ, ϕ t =  − 1 2 q x − λq + 2 λ 3 − q + 2 λ 2 1 2 q xx + λq x + q ( − q + 2 λ 2 ) 1 2 q x + λq − 2 λ 3  ϕ. Comparing it to (6) and (8 ) w e see that q = 1 2 S ( u ) + λ 2 and h ence u w ill dep end on λ . F ur thermore, h = q − 2 λ 2 pro vides λ -dep enden t R P 1 geometric realizatio ns for Kd V, n amely u t = u x  − 1 2 S ( u ) − 3 λ 3  . A complete description of this example can b e found in [7]. (Notice that the KdV equation they represent is dif feren t, bu t equiv alen t, to our introdu ctory example. This is merely du e to a dif feren t c hoice of inv arian t.) In this p ap er we will not f o cus on the study of solutions (see [7] instead) but rather on the in teraction b etw een geometry and in tegrable s ystems. Hence we will largely ignore the sp ectral parameter λ and its role. 3 Hamiltonian stru ctures generated b y group-based mo ving frames Consider the group of lo ops L G = C ∞ ( S 1 , G ) and its Lie algebra L g = C ∞ ( S 1 , g ). Assume g is semisimple. On e can def ine t w o natural P oisson brac k ets on L g ∗ (see [45] for more information), namely , if H , F : L g ∗ → R are tw o functionals def ined on L g ∗ and if L ∈ L g ∗ , we d ef ine {H , F } 1 ( L ) = Z S 1  δ H δ L ( L )  x + ad ∗  δ H δ L ( L )  ( L ) , δ F δ L ( L )  dx, (10) where h , i is the natural coupling b etw een g ∗ and g , and wh ere δ H δL ( L ) is the v ariational deriv ativ e of H at L identif ied, as usu al, with an element of L g . One also has a compatible family of second brac k ets, namely {H , F } 2 ( L ) = Z S 1  ad ∗  δ H δ L ( L )  ( L 0 ) , δ F δ L ( L )  dx, (11) where L 0 ∈ g ∗ is an y constan t elemen t. Since g is semisimple we can identi fy g with its dual g ∗ and w e will do so from now on. F rom no w on w e w ill also assume that our curves on homogeneous manif olds ha v e a gr oup mono dr omy , i.e., there exists m ∈ G suc h that u ( t + T ) = m · u ( t ) , 8 G. Mar ´ ı Bef fa where T is th e p erio d. Un der these assumptions, the Serr et–F renet equations will b e p erio dic. One could, instead, assum e that u is asymptotic at ±∞ , so that the in v ariants will v anish at inf in it y . W e would th en w ork with the analogous of (10) an d (11 ). The question we w ould lik e to inv estig ate next is w hether or n ot these t w o brac k ets can b e reduced to the space of dif f eren tial inv arian ts, or the sp ace of dif feren tia l inv arian ts asso ciated to sp ecial t yp es of f lo ws. W e will d escrib e af f ine and symmetric cases separately . 3.1 Af f ine manif olds Assume M = ( G ⋉ R n ) /G is an af f ine manifold, G semisimple. In this case a moving f rame can b e represented as ρ =  1 0 ρ u ρ G  (12) acting on R n as ρ · u = ρ G u + ρ u . A left inv arian t mo ving frame with ρ · o = u will hold ρ u = u and , in view of T h eorem 2, ρ G will ha v e in its columns a classical moving fr ame. In this case K = ρ − 1 ρ x is giv en by K =  0 0 ρ − 1 G ( ρ u ) x ρ − 1 G ( ρ G ) x  . In [32] it was sh o wn that ρ − 1 G ( ρ u ) x con tains all f irst order dif feren tial inv arian ts. I t was also explained ho w one could mak e this term constan t by c ho osing a s p ecial parametrization if necessary . Let’s call that constan t ρ − 1 G ( ρ u ) x = Λ. Our main to ol to f ind Poisson b rac k ets is via red u ction, and as a p revious step, we need to wr ite the space of dif f eren tial inv arian ts as a quotien t in L g ∗ . The pro of of the follo wing Th eorem can b e found in [32]. Theorem 4 ([32]) . L et N ⊂ G b e the isotr op y sub gr oup of Λ . Assume that we cho ose moving fr ames as ab ove and let K b e the sp ac e of Serr et–F r enet e quations determine d b y these moving fr ames for curves in a neighb orh o o d of a generic curve u . Then, ther e exists an op en set of L g ∗ , let’s c al l it U , such that U / L N ∼ = K , wher e L N acts on L g ∗ using the g auge (or Kac–Mo o dy) tr ansformation a ∗ ( n )( L ) = n − 1 n x + n − 1 Ln. (13) In view of this theorem, our next theorem comes as no surp rise. Theorem 5 ([32]) . The Hamiltonian structur e (10) r e duc es to U / L N ∼ = K to define a Poisson br acket in the sp ac e of differ ential invariants of c u rves. Example 5. If we choose G = S O (3) and M the Euclidean space, for appropr iate c hoice of normalization constan ts our left m o ving frame is giv en by ρ =  1 0 u T N B  , where { T , N , B } is the classical Euclidean Serret–F r enet frame. Its Serret–F renet equations will lo ok lik e K = ρ − 1 ρ x =     0 0 0 0 ( u 1 · u 1 ) 0 − κ 0 0 κ 0 − τ 0 0 τ 0     . Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 9 In this case, if we choose to parametrize our cu rv e by arc-length, Λ = e 1 where, as u sual, w e denote b y e k the standard basis of R n . The matrix K can b e clearly id en tif ied with its o (3) blo c k and hence K can b e considered as a sub space of L o (3) ∗ . The isotropy subgroup N is giv en b y matrices of the form  1 0 0 Θ  with Θ ∈ S O (2). Using this information we can f ind the reduced brac k et algebraically . F or this we tak e any functional h : K → R . Let’s call H an extension of h to L o (3) ∗ , constan t on the gauge leav es of L N . Its v ariational deriv at iv e at K needs to lo ok lik e δ H δ L ( K ) =    0 δh δκ α − δh δκ 0 δh δτ − α − δh δτ 0    for some α to b e determined. Since H is constant on the gauge lea v es of L N  n − 1 n x + n − 1 K n, δ H δ L ( K )  = 0 , for an y n ∈ L N . This is equiv alent to  δ H δ L ( K )  x +  K, δ H δ L ( K )  ∈ L n o , where n is the Lie algebra of N and n o is its annih ilator. F rom here    0 ( δh δκ ) x − ατ α x − κ δh δτ + τ δh δκ ∗ 0 ( δh δτ ) x + κα ∗ ∗ 0    =   0 ∗ ∗ ∗ 0 0 ∗ 0 0   , where ∗ indicates entries that are not, at least for n o w, r elev an t. Hence α = − 1 κ ( δh δτ ) x . The brac k et is th us giv en by { f , h } R 1 ( K ) = Z S 1 tr       0 ∗ ∗ − ( δh δκ ) x − τ κ ( δh δτ ) x 0 0  1 κ  δh δτ  x  x + κ δh δτ − τ δh δκ 0 0       0 δf δκ − 1 κ ( δf δτ ) x − δf δκ 0 δf δτ 1 κ ( δf δτ ) x − δf δτ 0       dx = − 2 Z S 1  δf δκ δf δτ  R δh δκ δh δτ ! dx, where R is R = D τ κ D D τ κ − D − D 1 κ D 1 κ D ! . The second Hamiltonian stru cture (11) can also b e redu ced to K with the general choic e L 0 =   0 a b − a 0 c − b − c 0   . T he reduced brac k et is found when app lying (11) to the v ariational d er iv ativ es of extensions that, as b efore, are constan t on the L N lea v es. Th us, it is straightforw ard to c hec k that the second reduced b r ac k et is give n by { f , h } R 2 ( K ) = 2 Z S 1  δf δκ δf δτ  ( a A + b B + c C ) δh δκ δh δτ ! dx, 10 G. Mar ´ ı Bef fa where A = 0 0 0 1 κ D − D 1 κ ! , B =  0 1 − 1 0  , C = 0 1 κ D D 1 κ 0 ! . These are all Hamiltonian s tructures and they app ea red in [38]. In fact, the structur e P shown in the in tro duction can b e written as P = − RC − 1 R and, h ence, P is in the Hamiltonian hierarc h y generated by R and C . A study of in tegrable systems asso ciated to these br ac k ets, and th eir geometric realizations, wa s done in [38]. See also [24]. Our f ir s t redu ced br ac k et is directly related to geometric r ealizati ons. In f act, b y c ho osing a sp ecia l parameter x we can obtain geometric r ealizati ons of systems that are Hamiltonian with resp ect to the reduced b rac k et. W e exp lain this n ext. Let ρ b e giv en as in (12) and assume u is a solution of the inv arian t equ ation u t = ρ G r , (14) where r = ( r i ) is a dif feren tial in v ariant ve ctor, that is, r i are all fu nctions of the en tries of K and its der iv ativ es. If the parameter has b een f ixed s o as to guarante e that ρ − 1 G ρ u = Λ is constan t, then r has to b e mo dif ied to guarante e that the evo lutions (14) pr eserv e the parameter. (In the runn in g example ρ G = ( T , N , B ) and ρ G r = r 1 T + r 2 N + r 3 B with r 2 = r ′ 1 κ once the arc-length is c hosen as parameter.) Theorem 6 ([32]) . If ther e exists a Hamiltonian h : K → R and a lo c al extension H c onsta nt on the le aves of N such that δ H δ L ( K )Λ = r x + K r , (15) then the invariantization of e v olution (14) is Hamiltonian with r esp e ct to the r e d uc e d br acket { , } R 1 and its asso ciate d Hamiltonian is h . If (after choosing a sp ecial parameter if n ecessary) relation (15) can b e solv ed for r giv en a certain Hamiltonian h , then th e Theorem guaran tees a geometric realization for the r ed uced Hamiltonian system. S uc h is the case for the VF f low, Saw ada– Koterra [32, 41], m o dif ied KdV and others [38]. One can f ind many geometric realizations of in tegrable sys tems in af f ine manifolds (see, for example, [2, 11, 24, 25, 26, 28, 46, 47, 49, 51]). Many of th ese are r ealizat ions of mo dif ied KdV equations (or its generalizations), sine-Gordon and Sc hr¨ odinger f lo w s . T hese systems ha v e geometric realizations also in n on-af f ine manifolds (see [1, 25, 26, 47, 48, 49]). Nev ertheless, a common feature to the generation of these realizatio ns is the existence of a classical moving frame that r esem bles the classic al natur al movi ng frame, that is, the deriv at iv es of the n on- tangen tial v ectors of the classical frame all ha v e a tangen tial direction. Thus, it seem to b e the case that the existence of geometric r ealizat ions for these sy s tems is linked to the existence of a natural frame. This close relationship b et ween geometry and th e t yp e of inte grable system is p erhap s clearer in our next s tudy , that of symmetric m an if olds . Before mo ving on, w e ha v e one f inal commen t in this line of thought. There are other manifolds w h ose geometry is giv en b y a linear (rather than af f ine) action of the group. W e can still follo w a similar appr oac h , reduce the b rac k ets and stud y Hamiltonian s tr uctures on the sp ace of dif f eren tial in v ariants. F or example, in the case of centro-a f f ine geometry one considers th e linear action of SL( n ) on R n and the asso ciated geometry is that of star-shap ed curv es. If w e assume that curve s are parametrized by the centro-a f f ine arc-length, w e can reduce b oth b rac k ets and obtain a p encil of Po isson brac k ets. Th is p encil coincides with the Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 11 bi-Hamiltonian structure of KdV. I n deed, a geometric realization for KdV w as f ou n d b y Pink all in [44]. This r ealizat ion is the one guarant eed b y the r eduction, as explained in [7]. The in teresting asp ect of th e cent ro-af f ine case is the follo wing: there is a natur al identif icatio n of a star-shap ed curve with a p ro jectiv e cur v e. The identif ication is giv en by th e in tersection of the curv e with the lines going through the origin. If the star-shap ed curve is nonde gener ate (that is, d et( γ , γ x , . . . , γ ( n − 1) ) 6 = 0. F or example, in the planar case the cur ve is nev er in the radial direction), the identif icatio n is well- def in ed. F urthermore, if we parametrized star-shap ed cu rv es with centroa f f ine arc-length (that is, if d et( γ , γ x , . . . , γ ( n − 1) ) = 1), the iden tif ication is 1-to-1 and th e geometries are Poi sson-equiv alent, the Po isson isomorphism giv en by the iden tif ication. In fact, Pink all’s geometric realization is the star-shap ed v ersion of the Sc h w arzian Kd V u nder this r elation (see [7] for more details). T h e existence of a geometric r ealizati on f or KdV seems to imply the existence of a b ackg round p ro jectiv e geometry . 3.2 Symmetric manifolds Assume that M is a sy m metric manifold which is lo cally equiv alen t to G/H with: (a) G semisim- ple; (b) its (Cartan) connection giv en b y the Maurer–Cartan form (i.e. the manifold is f lat); (c) the Lie algebra g has a gradation of length 1, i.e. g = g − 1 ⊕ g 0 ⊕ g 1 (16) with h = g 0 ⊕ g 1 , where h is the Lie algebra of H . If M is a symmetric manifold of this t yp e, G splits lo cally as G − 1 · G 0 · G 1 with H giv en by G 0 · G 1 . The sub grou p G 0 is called the isotr opic sub gr oup of G and it is the comp onen t of G that acts linearly on G/H (for m ore in formation see [4] or [42]). That means G 0 is the comp onen t of the group acting on f irst order fr ames. According to Theorem 2, the ρ 0 factor of a left moving frame ρ will determine a classical moving fr ame (see also [31]). As in the previous case, the basis for the def inition of a P oisson brac k et on the space of in v ariant s is to express that sp ace as a quotien t in L g ∗ . This is the r esult in the follo wing Theorem. F or a complete description and pro ofs see [31]. Notice that, if ρ is a (right) mo ving frame along a curve in a symmetric manifold w ith ρ · u = o , then ρ · u 1 is alw a ys constan t. In general ρ · u 1 is describ ed by f irst ord er in v arian ts, b ut cur v es in symmetric manifolds do n ot ha v e non-constant f ir st order dif f eren tial inv arian ts (inv arian ts are third order or higher), and hence ρ · u 1 m ust b e constan t. Theorem 7 ([31]) . L et M = G/H b e a symmetric manifold as ab ove. Assume that for ev e ry curve in a neighb orho o d of a generic curve u in M we c ho ose a left moving f r ame ρ with ρ · o = u and ρ − 1 · u 1 = ˆ Λ c onsta nt. Assume that we cho ose a se ction of G/H so we c an lo c al ly identify the manifold with G − 1 and its tangent with g − 1 . L et Λ ∈ g − 1 r epr esent ˆ Λ and let K b e the manifold of Serr et–F r enet e quations for ρ along curves in a neighb orho o d of u . Cle arly K ⊂ L g ∗ . Denote by h Λ i the line ar subsp ac e of C ∞ ( S 1 , g ∗ ) given by h Λ i = { α Λ , α ( x ) > 0 } . Then the sp ac e K c an b e describ e d as a quotient U / N , wher e U is an op en set of h Λ i⊕L g 0 ⊕L g 1 and wher e N = N 0 ·L G 1 ⊂ L G 0 ·L G 1 acts on U via the Kac–Mo o dy action (13) . The su b gr oup N 0 is the isotr opy sub gr oup of h Λ i i n L G 0 . As b efore, after writing K as a qu otien t, one gets a redu ction theorem. Theorem 8 ([31]) . The Poisson br acket (10) c an b e r e duc e d to K and ther e exi sts a wel l-define d Poisson br acket { , } R 1 define d on a ge ner ating set of indep endent differ ential invariants. Example 6. As we sa w b efore, the (left) Serret–F r enet equations f or the R P 1 case are giv en by K =  0 1 k 0  12 G. Mar ´ ı Bef fa where k = − 1 2 S ( u ). The splitting of the Lie algebra s l (2) is giv en by  0 β 0 0  +  α 0 0 − α  +  0 0 γ 0  ∈ g − 1 + g 0 + g 1 . In this case Λ =  0 1 0 0  ∈ g − 1 . Th e isotropic subgroup of h Λ i in G 0 is G 0 itself, and so N = L G 0 · L G 1 or subsp ace of lo w er tr iangular matrices. T o redu ce the br ack et (10) w e need to ha v e a fun ctional h : K → R and to f ind an extension H : M → R su c h that  δ H δ L ( K )  x +  K, δ H δ L ( K )  ∈ n 0 (17) for any K ∈ K , wh ere n = L g 0 ⊕ L g 1 and n 0 is its annihilator (wh ic h we can identify with L g 1 = L g ∗ − 1 ). Also, if H is an extension of h , its v ariational d eriv ativ e at K will b e given b y δ H δ L ( K ) =  a δh δk ( k ) b − a  , where a and b are to b e determined. On the other hand condition (17) translates in to a x + b − k δh δk ( k )  δh δk ( k )  x − 2 a b x + 2 k a − a x − b + k δh δk ( k ) ! =  0 0 ∗ 0  . F rom her e a = 1 2  δh δk ( k )  x and b = k δh δk ( k ) − 1 2  δh δk ( k )  xx . W e are now ready to f ind the reduced b rac k et. If f , h : K → R are tw o f unctionals and F and H are extensions v anishin g on the N -lea v es, th e r educed brac k et is giv en by { f , h } R 1 ( K ) = Z S 1 tr  δ F δ L ( K )  δ H δ L ( K )  x +  K, δ H δ L ( K )  dx = Z S 1 tr  ∗ δf δk ( k ) ∗ ∗  0 0 k  δh δk ( k )  x + ( k δh δk ( k )) x − 1 2  δh δk ( k )  xxx 0 !! dx = Z S 1 δ f δ k ( k )  − 1 2 D 3 + k D + D k  δ h δ k ( k ) dx. The dif f er en ce in co ef f icien ts as compared to the introd uctory example is du e to the fact that k = − 1 2 S ( u ) an d not S ( u ). As it h app ens , the companion brack et also redu ces for L 0 = Λ ∗ =  0 0 1 0  . Ind eed, it is giv en b y { f , h } R 2 ( K ) = Z S 1 tr  δ F δ L ( K )  0 0 1 0  , δ H δ L ( K )  = Z S 1 tr      1 2  δf δk ( k )  x δf δk ( k ) ∗ − 1 2  δf δk ( k )  x   − δh δk ( k ) 0  δh δk ( k )  x δh δk ( k ) !    dx = 2 Z S 1 δ f δ k ( k ) D δ h δ k ( k ) dx. It is not true in general that (11) is also r ed ucible to K . In fact, one f inds that f or M = R P n and G = PS L( n + 1) the second brack et (11) is also reducible to K wh en L 0 = Λ ∗ ∈ g ∗ . The Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 13 resulting t w o brac k ets are th e f ir st and second Hamiltonian structure for Ad ler–Gel’fand–Dikii f lo ws. Bu t if M is the s o-calle d Lagrangian Grassmannian, G = Sp(4), the second b rac k et is nev er reducible to K [34]. One interesti ng commen t on the conn ection to AKNS r epresen tations: as b efore, the reduc- tion of the brac k et (10) is dir ectly link ed to geometric realizations. But the reduction of (11) indicates the existence of an AKNS representat ion and an integrable sys tem. In th e KdV ex- ample w e describ ed ho w the Serret–F renet equations (6) were gauge equ iv alen t to λ = 0 using the gauge (7). If we gauge the x -ev olution of the KdV AKNS representati on in Example 4 by that same elemen t we get that the m atrix A c hanges in to A λ =  0 1 q − λ 2 0  = A − λ 2 L 0 . Therefore, up to a constant gauge, L 0 indicates the p osition of the sp ectral p arameter in the KdV AKNS represent ation in Example 4. In fact, it go es further. On e can pro v e that the co ef f icien t h (as in (3)) of the λ -dep enden t r ealizati ons for K dV determined by this AKNS represent ation (that is, h = q − 2 λ 2 ) is giv en by the v ariational d eriv ativ e of the Hamiltonian functional used to write Kd V as Hamiltonian sys tem in the p encil { , } R 1 − λ 2 { , } R 2 . In the same fashion, the NLS , when written in terms of κ and τ as in (2), has a second Hamiltonian structure obtained w hen reducing (11) w ith the c hoice L 0 =   0 0 0 0 0 1 0 − 1 0   . One can see th at this system has an AKNS repr esentati on with x ev olution giv en b y ρ x =   0 κ 0 − κ 0 τ − λ 0 − τ + λ 0   ρ and w here the t comp onent is determined by the ev olution indu ced on the right mo ving fr ame ρ b y a u evo lution whose in v arian t coef f icien ts h and g as jn (1) are giv en by the v ariational deriv ative of the Hamiltonian used to write this Euclidean representa tion of NLS as Hamiltonian with resp ect to the p encil { , } R 1 − λ { , } R 2 . Th is must b e a kno wn fact on integ rable systems, but we could not f ind it in the literature. F or a complete d escription of this relation see [7 ]. Bac k to symmetric sp aces. Recall that g = g − 1 ⊕ g 0 ⊕ g 1 with h = g 0 ⊕ g 1 , so that w e can iden tify T x M w ith g − 1 . R ecall also that d Φ ρ ( o ) = ( T 1 . . . T n ) is a classical mo ving fr ame. Theorem 9 ([31]) . Assume ρ is as in the statement of The or em 7 . Assume u ( t, x ) is a solution of the evolution u t = d Φ ρ ( o ) r = r 1 T 1 + · · · + r n T n , (18) wher e r = ( r i ) is a ve ctor of differ ential invariants. Then, if ther e exists a Hamiltonian functional h : K → R and an extension of h , H : U → R , c onstant of the le aves of N and such that  δ H δL ( L )  − 1 = r (the subindex − 1 indic ates the c omp onent in g − 1 ), then the evolution induc e d on the gener ating system of differ ential invariants define d by K is Hamiltonian with asso ciate d Hamiltonian h . Let us call the generating d if ferential inv arian ts k . Given a Hamiltonian system k t = ξ h ( k ) = P ( k ) δh δ k , its asso ciated geometric ev olution (18) will b e its geometric realization in M . Therefore they alw a ys exist, the previous theorem guaran tees their existence. In deed, one only n eeds to extend the fu nctional h preserving the lea v es. As explained in [31] and describ ed in our r u nning examples, one can f ind δ H δL ( K ) along K explicitly u s ing a s im p le algebraic pro cess. Then, the 14 G. Mar ´ ı Bef fa co ef f icien ts r i of the r ealizati on are given by  δ H δL ( K )  − 1 , as ident if ied w ith th e tangen t to the manifold using our section of G/H . Some examples are giv en in [31, 33, 34, 35]. As w e ha v e previously p oin ted out, th e s econd br ack et d o es n ever redu ce to the sp ace of in v ariant s w hen G = Sp(4), the L agrangian Grassmannian. S till, one can select a subman if old of in v ariants and stud y r eductions of the b rac k ets on the su bmanifold where the other inv arian ts v anish. T his is equiv alen t to stud ying Hamiltonian ev olutions of sp ecial t yp es of f lows on Sp(4) /H . In particular, the author def ined dif ferentia l inv arian ts of pro jectiv e t yp e in [33]. She then sho w ed h o w b oth brac k ets (10) and (11 ) (for some choic e of L 0 ) could b e red u ced on f lo ws of curv es with v anish ing n on-pro jectiv e dif feren tial in v ariants. Th e reductions pro du ced Hamiltonian stru ctur es an d geomet ric realiz ations for in tegrable systems of KdV-t yp e. Th is, and its implication for the geometry of G/H is describ ed next. 4 Completely int egrable systems of KdV t yp e asso ciated to dif feren tial inv arian ts of p ro jectiv e t yp e There are some dif feren tial inv arian ts of cur v es in symmetric spaces that one might call of pr o- je ctive typ e . They are generated by th e action of the group on second f rames (hence they cannot b e found us ing a classical m o ving frame) and most of them closel y resem ble th e S c h w arzian deriv ative . In terms of the gradation g = g − 1 ⊕ g 0 ⊕ g 1 , if w e c ho ose an app ropriate moving frame, th ese inv arian ts w ill app ea r in K 1 , w here K = ρ − 1 ρ x = K − 1 + K 0 + K 1 is the graded splitting of the Serret–F renet equation asso ciated to the mo ving frame ρ . Of cours e, the simplest example of d if ferential in v ariants of pro jectiv e-t yp e are pro jectiv e dif f er ential inv arian ts. A s it wa s sh o wn in [31], a mo ving fr ame can b e c hosen so that all dif f er ential inv arian ts app ear in the g 1 comp onent of the Serret–F ren et equations, while all en tries outside g 1 are constan t. M ore examples of dif feren tia l inv arian ts of pro jectiv e typ e app ear in [33, 34, 35] and [37 ]. In this series of pap ers the author show ed how in m an y symmetric sp aces one can f in d geometric realizations ind ucing ev olutions of Kd V t yp e on the dif feren tial in v ariants of pro jectiv e t yp e. Indeed, if G/H is a symmetric sp ace as ab ov e, it is kn o wn [27] th at g is the direct s um of the follo wing simple Lie algebras: 1. g = s l ( p + q ) with p , q ∈ Z + . If q = 1 th en G/H ≡ R P n . In general G/H is th e Gr assmannian . 2. g = s p (2 n ), the manif old G/H is called the L agr an gian Gr assmannian and it can b e iden tif ied w ith the manifold of Lagrangian planes in R 2 n . 3. g = o ( n, n ), the manifold G/H is called the manifold of r e duc e d pur e spinors . 4. g = o ( p + 1 , q + 1) with p, q ∈ Z + . I f q = 0 the manifold G/H is isomorp hic to the M¨ obius spher e , th e lo cal mo d el f or f lat conformal manifolds. 5. T wo exceptional cases, g = E 6 and g = E 7 . W e will n ext describ e the situation for eac h one of the 1–4 cases ab o v e. The Grassmannian case (1) (other than q = 1) and the exceptional cases (5) hav e n ot y et b een studied. 4.1 Pro jective case Let G = P S L( n + 1). If g ∈ G then, lo cally g = g − 1 g 0 g 1 =  I u 0 1   Θ 0 0 (detΘ) − 1   I 0 v T 1  Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 15 with u, v ∈ R n and Θ ∈ GL( n ). If w e def ine H by th e choice u = 0, then G/H ∼ = R P n , the n -pro jectiv e space. Th is factorization corresp ond s to the splitting given by the gradation (16). A section for G/H can b e tak en to b e the g − 1 factor. As with an y other homogeneous space, the action of G on this section is completely determined by the relation g  I u 0 1  =  I g · u 0 1  h for some h ∈ H . The corresp ond ing splitting of the Lie algebra is giv en by V = V − 1 + V 0 + V 1 =  0 a 0 0  +  A 0 0 − tr A  +  0 0 b T 0  ∈ g − 1 ⊕ g 0 ⊕ g 1 . W e can iden tify the f irst term ab o ve with the tangen t to the manifold. The follo wing theorem d escrib es the type of mo ving frame and in v ariant manifold we will c ho ose. It is essentia l to choose a simple enough representat ion for K so that one can readily recognize th e result of the reduction. Naturally , an y c hoice of m o ving fr ame will pro d uce a c hoice for K and a Hamiltonian structure. But sho wing the equiv alence of Po isson str u ctures is a non- trivial problem, and hence its recognition is an imp ortan t part of the p roblem. As we said b efore, all dif feren tial inv arian ts are of p ro jectiv e-t yp e. Theorem 10 (reformulatio n of Wilczynski [50]) . Ther e exists a left moving fr am e ρ along nonde gener ate curves in R P n such that its Serr et–F r enet e quations ar e define d by matric es of the form K =        0 1 0 . . . 0 0 0 1 . . . 0 . . . . . . . . . . . . . . . 0 0 . . . 0 1 k 1 k 2 . . . k n 0        , wher e k i , i = 1 , . . . , n ar e, i n gener al , a gener ating c ombination of the Wilczynski pro jectiv e in v ariant s and their derivatives. F or the pr ecise relation b et ween these inv arian ts and Wylczynski’s in v ariant s, see [7 ]. The follo wing result was originally obtained b y Drinfel’d and Sok olo v in [12]. Th eir description of the quotien t is not the same as our s, but [31] sho w ed that our redu ction and th eirs are equiv alent . Theorem 11 ([12 ]) . Assume K i s r epr esente d by matric es of the form ab ove. Then, the r e duc- tion of (10 ) to K is gi ven by the A d ler–Gel’fand–Dikii (AGD) br acket or se c ond Hamiltonian structur e for gener alize d KdV. The br acket (11) r e duc es for the choic e L 0 = e ∗ n and its r e duction is the first Hamiltonian structur e for gener alize d KdV e quations. Finally , the follo wing theorem iden tif ies geometric realizations for any f lo w Hamiltonian with resp ect to the redu ction of (10); in particular, it provides geometric r ealizat ions for generalized KdV equations or A GD f lo w s . The case n = 2 was originally pro v ed in [18]. Theorem 12 ([36 ]) . Assume u : J ⊂ R 2 → PS L( n + 1) /H is a solution of u t = h 1 T 1 + h 2 T 2 + · · · + h n T n , wher e T i form a pr oje ctive classic al moving fr ame. 16 G. Mar ´ ı Bef fa Then, k = ( k i ) satisfies an e quation of the form k t = P h , wher e k = ( k 1 , . . . , k n ) T , h = ( h 1 , . . . , h n ) T and wher e P is the P oisson tensor defining the A d ler–Gel’fand–Dikii Hamiltonian structur e . In p articular, we obtain a pr oje ctive ge o- metric r e aliza tion for a gener alize d KdV system of e quations. In our next section w e lo ok at some cases for w hic h n ot all d if ferential in v ariants of curv es are of pro jectiv e t yp e. 4.2 The Lagrangian Grassmannian and the manifold of reduced pure Spinors These t w o examples are dif f eren t, but their dif feren tial inv arian ts of pro jectiv e t yp e b eha v e similarly and so w e will pr esen t them in a join t section. L agr angian Gr assmannian. Let G = Sp(2 n ). If g ∈ G then, lo cally g = g − 1 g 0 g 1 =  I u 0 I   Θ 0 0 Θ − T   I 0 S I  with u and S symmetric n × n matrices and Θ ∈ GL( n ). Again, this factorizati on corresp onds to the sp litting giv en by th e gradation (16). The subgroup H is lo cally def in ed by th e choic e u = 0 and a lo cal section of the quotien t can b e represente d by g − 1 . As u sual, the action of the group is determined by the relation g  I u 0 I  =  I g · u 0 I  h for some h ∈ H . The corresp on d ing sp litting of the algebra is giv en by V = V − 1 + V 0 + V 1 =  0 S 1 0 0  +  A 0 0 − A T  +  0 0 0 S 2  , where S 1 and S 2 are symmetric matrices and A ∈ gl ( n ). The manifold G/H is usually called the L agr angian Gr assmanian in R 2 n and it is iden tif ied with the manifold of Lagrangian p lanes in R 2 n . The follo wing theorem describ es a repr esen tation of the manifold K for curves of Lagrangian planes in R 2 n under th e ab o v e action of Sp(2 n ). Theorem 13 ([34]) . Ther e exists a left moving fr ame ρ along a generic curve of L agr angian planes such that its Serr et–F r enet e q uations ar e gi ven b y K = ρ − 1 ρ x =  K 0 I K 1 K 0  , wher e K 0 is skew-symmetric and c ontains al l differ ential invariants of or der 4 , and wher e K 1 = − 1 2 S d . The matrix S d is diagonal and c ontains in its diagonal the eigenvalues of the L agr angian Schwarzian derivative (Ovsienko [43] ) S ( u ) = u − 1 / 2 1  u 3 − 3 2 u 2 u − 1 1 u 2  ( u − 1 / 2 1 ) T . The entries of K 0 and K 1 ar e functional ly indep endent differ ential invaria nts for cu rves of L agr angian plan es in R 2 n under the action of Sp(2 n ) . They gener ate al l other differ ential in- variants. The n differ e ntial invariants that app e ar in K 1 ar e the invariants of pr oje ctive typ e. Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 17 No w w e describ e some of the geometric f lows that pr eserv e the v alue K 0 = 0. Therefore, geometric f lo ws as b elo w will af fect only inv arian ts of pro jectiv e type, if prop er initial conditions are c hosen. Theorem 14 ([34 ]) . Assume u : J ⊂ R 2 → Sp(2 n ) /H is a flow solution of u t = Θ T u 1 / 2 1 h u 1 / 2 1 Θ , (19) wher e Θ( x, t ) ∈ O ( n ) is the matrix diagonalizing S ( u ) (i.e., Θ S ( u )Θ T = S d ) and wher e h is a symmetric matrix of differ ential invariants. Assume h is diagonal. Then the flow pr eserves K 0 = 0 . Finally , our next theorem giv es inte grable PDEs with ge ometric realizatio ns as geometric f lo ws of Lagrangian planes. Theorem 15 ([34]) . L et K 1 b e the submanifold of K given by K 0 = 0 . Then, the r e duc e d br acket on K r estricts to K 1 to induc e a de c oup le d system of n se c ond Hamiltonian structur es for KdV. Br acket (11) also r e duc es to K 1 (even though it do es not in gener al r e duc e to K for any value of L 0 ). The r e duction for the choic e L 0 =  0 0 I 0  =  0 I 0 0  ∗ ∈ g 1 is a de c ouple d system of n first KdV Hamitonian structur es. F urthermo r e, assume u ( t, x ) is a flow solution of (19) with h = S d . Then (19) b e c omes the L agr angian Schwarzian KdV evolution u t = u 3 − 3 2 u 2 u − 1 1 u 2 . If we cho ose initial c onditions for which K 0 = 0 , then the differ ential invariants S d of the flow satisfy the e quation ( S d ) t =  D 3 + S d D + ( S d ) x  h , wher e D is the diagonal matrix with d dx down its diagonal. Accordingly , if we c ho ose h = S d , then S d is th e solution of a de c ouple d system of n KdV e quations ( S d ) t = ( S d ) xxx + 3 S d ( S d ) x . R e d uc e d pur e Spinors. A parallel d escription can b e giv en for a dif ferent case, that of G = O ( n, n ). In this case, if g ∈ G , lo cally g = g − 1 g 0 g 1 =  I − u − u u I + u  1 2  Θ − 1 + Θ T Θ T − Θ − 1 Θ T − Θ − 1 Θ − 1 + Θ T   I − Z Z − Z I + Z  , where u and S are no w skew-symmetric matrices and where Θ ∈ GL( n, R ). T he corresp onding gradation of the algebra is giv en by V = V − 1 + V 0 + V 1 ∈ g − 1 ⊕ g 0 ⊕ g 1 with V − 1 = V − 1 ( y ) =  − y − y y y  , V 0 = V 0 ( C ) =  A B B A  , V 1 = V 1 ( z ) =  − z z − z z  , y and z sk ew symmetric an d C = A + B given by the symmetric ( B ) and sk ew-symmertric ( A ) comp onent s of C . Assume no w that G = O (2 m, 2 m ). This case has b een wo rk ed out in [33]. The h omogeneous space is lo cally equiv alent to the manifold of reduced p ure sp inors in the sense of [4]. Th e o d d dimensional sp inor case is work ed out in [37]. Although someho w similar it is more cumbersome to describ e, so we refer the reader to [37]. 18 G. Mar ´ ı Bef fa Theorem 16 ([33]) . L et u b e a generic curve i n O (2 m, 2 m ) /H . Ther e exists a left moving fr ame ρ such that the lef t Serr et–F r enet e quatio ns asso ciate d to ρ ar e define d by K = V − 1 ( J ) + V 0 ( R ) + 1 8 V 1 ( D ) , wher e J =  0 I m − I m o  and R is of the form R =  R 1 R 2 R 3 − R T 1  ∈ S p(2 m ) with R 2 and R 3 symmetric, R 1 ∈ g l ( m ) . The matrix R c ontains in the entries off the diagonals of R i , i = 1 , 2 , 3 , a gener ating set of indep endent fourth or der differ ential invariants. The diagonals of R i , i = 1 , 2 , 3 c ontain a set of 3 m indep endent and gener ating differ ential invariants of or der 5 for m > 3 and of or der 5 and higher if m ≤ 3 . The matrix D is the skew-symmetric diagonaliza tion of the Spinor Schwarzian derivative. The S pinor Sc h w arzian deriv at iv e [33] is describ ed as f ollo ws: if u is a generic curve repr e- sen ted by skew-symmetric matrices, u 1 is non degenerate and can b e b rought to a normal form using µ ∈ g l (2 m ). The matrix µ is determined up to an elemen t of the symplectic group S p(2 m ) (see [33]) and µu 1 µ T = J. W e def in e th e Spinor Sc hwarzian derivative to b e S ( u ) = µ  u 3 − 3 2 u 2 u − 1 1 u 2  µ T , again, unique up to the action of S p(2 m ). One can then p ro v e [33] that, for a generic curve, the Sc h w arzian deriv ativ e can b e d iagonalize using an elemen t of th e s ymplectic group . That is, there exists θ ∈ Sp(2 m ) suc h that θ S ( u ) θ T = D =  0 d − d 0  with d diagonal. The matrix D is the one app earing in the Serr et–F renet equations and it con tains in its entries m dif feren tial inv arian ts of pr oje ctive typ e . The Spinor case seems to b e d if ferent f rom others and the b eh a vior of the P oisson brac k- ets (10) and (11) is not completely un dersto o d y et. Still we d o kno w how the in v ariants of pro jectiv e t yp e b eha v e u n der the analogous of th e K dV S c h w arzian ev olution. T hat is describ ed in the follo wing theorem. The Spinor KdV Schwarzian evolution is def ined b y the equation u t = u 3 − 3 2 u 2 u − 1 1 u 2 . Theorem 17 ([33]) . L et ρ is a moving fr am e for which normalizat ion e qu ations of fourth or der ar e define d by c on stants c 4 . A ssu me that, as the fourth or der invariants vanish, [ R, b R ] = b b R + blo ck diagonals , wher e b R and b b R ar e any matric es whose only non-zer o entries ar e in the same p osition as the nonzer o norm alize d entries in R . A ssume also that [ R, [ R, b R ]] d = 0 for b R as ab ove, wher e d indic ates the diagonals in the main four b lo cks. Then, if we cho ose i ni tial c onditions with vanishing f ourth or der invariants, these r emain zer o u nder the KdV Schwarzian flow, D t and ( R d ) t de c oupl e, and D evolves as D t J = D xxx J + 3 DD x , fol low ing a de c ouple d system of KdV e qu ations. Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 19 Although the hypothesis of this theorem seem to b e very r estrictive , they aren’t. In fact, they are easily ac hiev ed wh en we construct m oving frames in a dim en sion larger that 4, although they are m ore restrictiv e for the lo w er dimensions. The need for these conditions w as explained in [33]. This is the f irs t case where the manifold of v anishing non-pro jectiv e inv arian ts is not pre- serv ed. Not only th e v anishing of f ifth ord er inv arian ts is not pr eserv ed, b ut the system blows u p when we approac h the su bmanifold of v anishing f ifth order in v ariants. This situation (and the analogous one for th e o dd dim en sional case) is not w ell un dersto o d. I t is p ossible that c ho osing normalization equations f or wh ic h c 4 in v olv e deriv ative s of the third order d if ferential in v ariants will p r o duce a b etter b ehav ed moving frame. Or p er h aps the f ifth order dif feren tial in v ariants are also of pr o jectiv e t yp e and the Hamiltonian b eha vior is m ore complicated th at app ears to b e. The main problem un d erstanding this case is the choi ce of a mo ving frame that simplif ies the stu d y of the reduced ev olution. In the spinorial case suc h a c hoice is highly not trivial and w e d o not know whether there is n o s uc h c hoice or it is just v ery in v olv ed. It is also not kno wn whether or not (10) and (11) redu ce to K 1 . F or more information see [33] and [37]. The Lagrangian and Sp inorial examples describ e a certain t yp e of b eha vior (ev olving as a d ecoupled system of KdVs) of inv arian ts of pro jectiv e typ e th at is certainly dif f eren t from our f irst examp le, that of R P n . Still, th ere is a third b eha vior that is dif feren t from these t w o. These ev olutions of K dV t yp e app ear in conformal manifolds . In the conformal case we ha v e only tw o dif f er ential inv arian ts of pro jectiv e t yp e, and their Hamiltonian b eha v ior is that of a complexly coupled system of KdVs. That is what we describ e in the next sub section. 4.3 Conformal case In th is section we study the case of G = O ( p + 1 , q + 1) acting on R p + q as describ ed in [42]. T h e case q = 0 w as originally stud ied in [35]. Using th e gradation app earing in [27] we can lo cally factor an element of the group as g = g 1 g 0 g − 1 (this is a factorizatio n for a righ t mo ving frame, not a left one, so it is sligh tly d if ferent fr om the p revious examples), with g i ∈ G i where g − 1 ( Y ) =     1 − 1 2 | | Y | | 2 − Y T 1 − 1 2 | | Y | | 2 Y T 2 Y 1 I p Y 1 0 1 2 | | Y | | 2 Y T 1 1 + 1 2 | | Y | | 2 − Y T 2 Y 2 0 Y 2 I q     , g 0 ( a, b, Θ) =     a 0 b 0 0 Θ 11 0 Θ 12 b 0 a 0 0 Θ 21 0 Θ 22     , g 1 ( Z ) =     1 − 1 2 | | Z | | 2 Z T 1 1 2 | | Z | | 2 Z T 2 − Z 1 I p Z 1 0 − 1 2 | | Z | | 2 Z T 1 1 + 1 2 | | Z | | 2 Z T 2 Z 2 0 − Z 2 I q     . The splitting Z =  Z 1 Z 2  and Y =  Y 1 Y 2  is into p and q comp onen ts, | | X | | 2 is give n b y the f lat metric of signature ( p, q ) and also Θ =  Θ 11 Θ 12 Θ 21 Θ 22  ∈ O ( p, q ), a 2 − b 2 = 1. Without lo osing generalit y w e will assume that the f lat m etric is giv en by | | X | | 2 = X T J X , where J =  I p 0 0 − I q  . The corresp ond ing splitting in the algebra is giv en by V − 1 ( y ) =     0 − y T 1 0 y T 2 y 1 0 y 1 0 0 y T 1 0 − y T 2 y 2 0 y 2 0     , V 0 ( α, A ) =     0 0 α 0 0 A 11 0 A 12 α 0 0 0 0 A 21 0 A 22     , 20 G. Mar ´ ı Bef fa V 1 ( z ) =     0 z T 1 0 z T 2 − z 1 0 z 1 0 0 z T 1 0 z T 2 z 2 0 − z 2 0     , where y =  y 1 y 2  , z =  z 1 z 2  are the p and q comp onen ts and where A = ( A ij ) ∈ o ( p, q ). The algebra structur e can b e describ ed as [ V 0 ( α, A ) , V 1 ( z )] = V 1 ( J AJ z + αz ) , [ V 0 ( α, A ) , V − 1 ( y )] = V − 1 ( Ay − αy ) , [ V 1 ( z ) , V − 1 ( y )] = 2 V 0  z T y , J z y T J − y z T  , [ V 0 ( α, A ) , V 0 ( β , B )] = V 0 (0 , [ A, B ]) . With this factorizat ion one c ho oses H to b e def ined by Y = 0 and uses G − 1 as a lo cal section of G/H (as such Y = − u , n ot u ). As b efore, the f ollo w ing th eorem describ es conv enient c hoices of mo ving frames. Theorem 18 ([33]) . Ther e exists a lef t moving fr ame ρ suc h that the Serr et–F r enet e quation for ρ is given by ρ − 1 ρ x = K = K 1 + K 0 + K − 1 wher e K − 1 = V − 1 ( e 1 ) , K 1 = V 1 ( k 1 e 1 + k 2 e 2 ) and K 0 = V 0 (0 , ˆ K 0 ) with ˆ K 0 =  A 0 B 0 B T 0 0  , and A 0 =        0 0 0 0 . . . 0 0 0 − k 3 − k 4 . . . − k p 0 k 3 0 0 . . . 0 . . . . . . . . . . . . . . . 0 0 k p 0 . . . 0 0        , B 0 =        0 0 . . . 0 − k p +1 − k p +2 . . . − k p + q 0 0 . . . 0 . . . . . . . . . . . . 0 0 . . . 0        . In this case there are only t w o generating dif feren tial inv arian ts of pro jectiv e typ e, n amely k 1 and k 2 . Again, the b eha vior of the P oisson b rac k ets (10) and (11) with resp ect to this s ubman- ifold is sp otless. Theorem 19 ([33]) . L et K 1 b e the submanifold of K given by K 0 = 0 . Then, the r e duction of (10) to K r estricts to K 1 to induc e the se c ond H amiltonian structur e for a c omp lexly c ouple d KdV system. Br acket (11) also r e duc e d to K 1 to pr o duc e the first Hamiltonian structur e for this system. And, again, the geometric r ealizat ion f or complexly coupled KdV is foun d. Theorem 20 ([33 ]) . Assume u : J ⊂ R 2 → O ( p + 1 , q + 1) /H is a solution of u t = h 1 T + h 2 N , wher e T and N ar e c onform al tangent and normal (se e [33] ) and h 1 , h 2 ar e any two functions of k 1 , k 2 and their derivatives. Then the flow has a limit as K 0 → 0 . A s K 0 → 0 , the e v olution of k 1 and k 2 b e c omes  k 1 k 2  t =  − 1 2 D 3 + k 1 D + D k 1 k 2 D + D k 2 k 2 D + D k 2 1 2 D 3 − k 1 D − D k 1   h 1 h 2  . If we cho ose h 1 = k 1 and h 2 = k 2 , then the evolution is a c ompl exly c oup le d system of KdV e quations. Geometric Realizati ons of Bi-Hamiltonian Integrable Systems 21 5 Discussion The aim of this pap er is to review some of the kno wn evidence linkin g the charact er of dif feren tial in v ariant of curv es in homogeneous sp aces and the geometric realizations of inte grable systems in those manifolds. I n particular, w e h av e describ ed how pro jectiv e geometry and geo metric realizatio ns of KdV-t yp e ev olutions seem to b e v ery closely related. A s im ilar case can p er- haps b e made f or Schr¨ o dinger f lo ws, mKdV and sine-Gordon f lo ws as link ed to Riemannian geometry . As we said b efore, seve ral authors [1, 25, 26, 28, 29, 38, 46, 48, 49], ha v e d escrib ed geometric realizations of th ese ev olutions on manifolds that h a v e what amoun ts to b e a classical natur al mo ving fr ame, i.e., a frame whose deriv ative s of n on tangen tial v ecto rs hav e a tangen- tial direction. T h is f rame app ears in R iemann ian manifolds and is generated by the action of the group in f irst ord er f r ames. Thus, one could call the inv arian ts they generate inv arian ts of Riemannian-t yp e. In fact, the most interesting question is ho w the geometry of the m an if old itself generates these geomet ric realizat ions. And, further, if a manifold hosts a geometric realization of an in tegrable system of a certain type, do es th at fact hav e any imp lications for the geometry of the manifold? I n the case of pro jectiv e geometry , the follo wing conjecture due to M. East w oo d p oints us in this direction. Conjecture. In this typ e of symmetric sp ac es ther e exists a natur al pr oje ctive structur e along curves that gener ates Hamiltonian structur es of KdV typ e along some flows. In the conformal case G = O ( p + 1 , q + 1) the tw o in v ariants of pr o jectiv e t yp e are directly connected to inv arian t d if feren tial op erators that app ear in the w ork of Baile y and East w oo d (see [5, 6]). The authors d ef ined conformal circles as solutions of a dif feren tial equation. The equation def ines the curves together with a preferred parametrization. The parametrizations endo w conformal circles w ith a p r o jectiv e structur e (theirs is an explicit pro of of Cartan’s ob- serv ation that a cur ve in a conformal manifold inher its a natural pr o jectiv e structure, see [10]). W e n o w k n o w that the v anishing of the dif feren tial equation in [5] implies the v anishing of b oth dif f er ential inv arian ts of pr o jectiv e t yp e found in [33]. Therefore, the complexly coupled sys- tem of KdV equations could b e generated by the p ro jectiv e structur e on conformal cu rv es th at Cartan originally describ ed. 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