Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

In tegrable generalizatio ns of Sc hr¨ odinger maps and Heisen b erg spin mo dels from Hamiltonian ﬂo ws of curv es and su rfaces Stephen C. Anco Dep artment of Mathematics, Br o ck University, St. C atharines, ON C a nada ∗ R. Myrzakulo v Dep artment of Gener al and The or etic al Physics, Eur asian National University, Astana, 010008 , Kazakhstan † Abstract A mo ving fr ame form ulation of non-stretc hin g geometric curve ﬂo ws in Euclidean space is us ed to derive a 1+1 dimensional hierarc hy of in tegrable S O (3)-in v a riant v ector mo dels con taining the Heisen b erg ferromagnetic spin mo del as well as a mo del giv en b y a spin-vect or v ersion of the mKdV equation. These mo dels describe a geometric realizat ion of the NLS hierarc h y of soliton equations wh ose b i-Hamilto nian structure is sho wn to b e enco ded in the F renet equations of the mo ving frame. Th is deriv atio n yields an explicit b i-Hamilt onian str ucture, recursion op erator, and constan ts of motion for eac h mo del in the hierarc hy . A generalization of these results to geometric surface ﬂo ws is presen ted, where the surfaces are non-stretc hing in one direction w h ile stretc hing in all transv erse directions. Through th e F renet equations of a movi n g frame, suc h su rface ﬂows are sho wn to enco de a hierarch y of 2+1 dimen s ional int egrable S O (3)-in v arian t vec tor models, along with their bi-Hamiltonian stru cture, r ecursion op erator, and constan ts of motion, describ ing a geometric realizatio n of 2+1 dimensional b i-Hamilto nian NLS and mKdV soliton equations. Based on the w ell-kno wn equiv alence b et we en th e Heisen b erg mo del and the Sc hr ¨ odinger map equation in 1+1 dimensions, a geometrical f orm ulation of these hierarc hies of 1+1 and 2+1 v ector m o d els is give n in terms of dynamical maps in to the 2-sphere. In particular, this formulati on yields a new in tegrable generalization of the Schr¨ odinger map equation in 2+1 dimen sions as wel l as a m K dV analog of this map equation corresp onding to the mKd V spin mo del in 1+1 and 2+1 d imensions. ∗ Electronic address: sanco @bro cku.ca † Electronic addr ess: cnlpmyra@mail.ru 1 I. INTR ODUCTI ON AND SUMMAR Y Spin systems are an imp ort a n t class o f dynamical vec tor mo dels from b oth ph ysical and mathematical p oints of view. In ph ysics suc h mo dels describ e nonlinear dynamics of magnetic materials, while in mathematics they giv e rise to asso ciated geometric ﬂows of curv es where the unit ta ngen t v ector along a curv e is iden tiﬁed with a dynamical spin v ector. A main example [14] is the Heisen berg mo del fo r the dynamics of an isotropic ferromag net spin system in 1+1 dimensions. The geometric curv e ﬂow describ ed by this S O (3)-inv ariant mo del corresp onds to the equations of motio n of a non- stretc hing v ortex ﬁlament in Eu- clidean space. Remark ably , the v ortex ﬁlamen t equations are an in tegrable Hamiltonian system tha t is equiv alen t to the 1+1 dimensional fo cusing nonlinear Sc hr¨ odinger equation (NLS) through a c hange of dynamical v ariables kno wn as a Hasimoto transformation [1 3 ]. The v ort ex ﬁlamen t equations are one example in an inﬁnite hierarc h y of non-stretc hing geometric ﬂows of space curve s whose equations of motion ha v e a w ell-understo o d in tegra- bilit y: e.g. a Lax pair and an asso ciat ed isosp ectral linear eigen v alue problem; an inﬁnite set of symmetries a nd constan ts of motion; and exact solutions with solitonic prop erties. This integrabilit y structure tur ns out to hav e a simple geometric orig in. In pa rticular, a ll of these equations of motion are g enerated through a recursion op erator tha t can b e deriv ed geometrically [10, 23] from the Serret-F renet structure equations giv en by a S O (3 ) moving frame formulation for a rbitrary non-stretc hing curv e ﬂows in Euclidean space, with the com- p onen ts of the frame connection matrix pro viding the dynamical v ariables that app ear in the equations of motion. More recen tly , these S O (3 ) frame structure equations hav e b een found to g eometrically enco de a pair of compatible Hamiltonian op erators that yield a concrete bi-Hamiltonian structure fo r the equations of motion of each integrable curv e ﬂow in the hierarc h y [17]. The explicit bi-Hamiltonian form of the resulting equations of motion dep ends on a c hoice of the S O (3) moving frame for the underlying space curv e, whic h determines the form of the frame connection matrix a nd hence yields the dynamical v ariables in terms of the curve. In the case of the v ortex ﬁlamen t equations, the dynamical v ariables consist of the curv at ure in v ariant, κ , a nd the torsion inv ariant, τ , of the space curv e, corresp onding to the c hoice of a classical F renet fra me [12] give n by t he unit tangen t v ector, unit normal a nd bi-normal 2 v ectors, a long the curve . Other geometrical c hoices of a mov ing frame can b e made [2], since there is a S O (3) ga uge freedom relat ing a n y t w o ort honormal frames along an arbitrar y curv e in Euclidean space. In particular, the Hasimoto transformation arises geometrically as a ga uge transformation fro m a F renet frame to a parallel frame [7], where the fr ame v ectors in the normal space of the curv e are ch osen suc h that their deriv ativ e with resp ect to the arclength s along the curv e lies in the tangen t space of the curv e. This choice of frame is unique up to rigid S O (2) rotations acting on the normal v ectors b y the same angle at all p oin ts along the curv e, while leav ing in v ariant the tangen t v ector. Th e corresp onding pair of dynamical v ariables (deﬁned b y the connection matrix of a pa rallel frame) are natura lly equiv alent to a single complex-v alued v ariable u = κ exp( i R τ ds ) that is determined b y the curv e only up to constan t phase rotations u → e iφ u (where φ is indep enden t of arclength s ). This dynamical v ariable u thus has the geometrical meaning [5] of a U (1) ≃ S O (2) co v ariant of the space curv e. Imp ortan tly , the resulting Hamiltonian structure for the equations o f motion lo o ks simplest in t erms of the cov ariant u , whic h directly incorp orates the Hasimoto transformation, rather than using the classical in v ariants κ and τ . The purp ose o f the presen t pap er will b e to give some new applications of these ideas to the study of in tegrable v ector mo dels in 1+1 a nd 2+1 dimensions. Firstly , fro m the hierarc h y of non-stretc hing geometric space curv e ﬂo ws that con tains the vortex ﬁlamen t equations, w e deriv e the complete hierarc hy o f corr esp onding integrable S O (3)- in v arian t v ector mo dels in 1+1 dimensions, along with their bi-Hamiltonian in tegra- bilit y structure in explicit form. In addition to the Heisen b erg mo del, this hierarc h y will b e seen to contain a mo del that describes a spin-v ector ve rsion of t he mKdV equation. Our results pro vide a new deriv a tion of the Hamiltonian structure, recursion op erator, and constan ts of motion for these mo dels. Secondly , w e extend the deriv atio n to a geometrically a nalogous class of surface ﬂows where the surface is non-stretching in one co ordinate direction while stretc hing in all trans- v erse directions. Suc h surfaces a rise in a natural fa shion from a spatial Hamiltonian ﬂo w of non-stretch ing space curv es. This generalization will b e show n to give rise to a class of 2+1 dimensional NLS and mKdV soliton equations with an explicit bi- Ha miltonian struc- ture, yielding a hierarch y of in tegrable S O (3)- inv aria n t v ector mo dels in 2+1 dimensions. In pa r t icular, this hierarc hy includes 2+1 generalizations of the Heisen b erg spin mo del and the mKdV spin mo del, which w ere found in earlier work by one of us [15, 18–21]. Our 3 deriv ation here, in contrast, yields t he explicit bi- Hamiltonian structure, recursion op erator, and constants of motion, whic h are new results for these mo dels. W e also write out the cor- resp onding surface ﬂows explicitly in terms of g eometric v a riables giv en by [1 2] the geo desic and normal curv atures and the relative torsion o f the non-stretc hing co ordinate lines o n the surface. The surface ﬂow arising from the 2+1 in tegrable Heisen b erg mo del will b e seen to describe a sheet of non-stretc hing vortex ﬁlamen ts in Euclidean space. Lastly , w e a lso deriv e an in teresting geometric formulation o f t hese results by viewing the spin v ector as a dynamical map into the 2-sphere in Euclidean space. This fo rm ula - tion is based on t he w ell-know n geometrical equiv alence b et w een the Heisen b erg mo del and the Sc hr¨ odinger map equation in 1+1 dimensions [25]. When applied to the 1+1 and 2+1 dimensional hierarc hies of S O (3)- in v arian t v ector mo dels, our deriv a t ion yields a new inte- grable g eneralizatio n of the Schr¨ odinger map equation in 2+1 dimensions as w ell as a new mKdV analog o f this map equation corr esponding t o the mKdV spin-ve ctor mo del in 1+1 and 2+1 dimensions. The rest of the pap er is organized as f o llo ws. In section I I, we review from a uniﬁed p oint of view the mathematical relationships amongst 1+1 dimensional ve ctor mo dels, dynamical maps into the 2-sphere, non-stretc hing curv e ﬂo ws in Euclidean space, F renet and pa rallel frames, and the Hasimoto transformation. In section I I I, w e deriv e the NLS hierar ch y of soliton equations in terms of the g eometrical cov ariant u giv en b y the F renet equations of a mo ving parallel f r ame for non-stretching space curv e ﬂow s. This approach directly yields the explicit bi-Hamiltonian structure of these soliton equations, including a f o rm ula for the Hamiltonians. As examples, the para llel-frame F renet equations are used to sho w, ﬁrstly , ho w the NLS equation itself corresp onds geometrically to the Heisen b erg spin mo del and the Sc hr¨ odinger map equation; a nd secondly , ho w the mKdV spin mo del and the mKdV map equation arise geometrically from the next soliton equation in the NLS hierarch y . Section IV contains sev era l main results. W e w ork out the equations of mo t io n for the space curve s corresp onding to the NLS hierarc h y and write do wn t he induced ﬂo ws on t he curv ature and torsion inv aria n ts κ, τ . Next w e deriv e the resulting geometrical hierarc hies of v ector mo dels and dynamical map equations, along with their bi- Hamiltonian structure, recursion op erators, and constan ts of motion. This new deriv ation inv olv es only the pa rallel- frame F renet equations plus the bi- Hamiltonian structure of the NLS hierarc h y . The explicit bi-Hamiltonian form of the Sc hr¨ odinger map equation and Heisen b erg mo del, including a 4 geometric expression fo r the Hamiltonians, are pr esen ted as examples. In section V, we consider surfaces generated by a spatial Hamiltonian ﬂow of curv es with a parallel framing in Euclidean space. The underlying Hamiltonian structure is shown to arise naturally fr om the F renet equations of the induced fr ame along the surface. This formulation is then used in section VI to study surface ﬂows expressed in terms of the co v ariant v ariable u geometrically asso ciated with t he non-stretc hing space curv es t ha t foliate the surface, where the surface is stretc hing in all directions transv erse to these curv es. W e show that the bi- Hamiltonian structure for 1+1 ﬂo ws on u has a natural extension to 2 +1 ﬂo ws based on the observ ation that the Hamiltonian o p era t ors in v olve only the co ordinat e in the non-stretc hing direction on the surface. This leads to a hierarch y of 2+1 ﬂo ws on u , with the start ing ﬂo w giv en geometrically by translations in the co o r dina t e in the transv erse direction, whic h yields a 2+1 generalization of the NLS hierarch y . The ﬁnal t w o sections of the pap er con tain our main new results. In section VI I, w e use the surface F renet equations to deriv e the complete hierarc hies of integrable 2+ 1 v ector mo dels and dynamical maps arising from the 2+1 generalization of the NLS hierar ch y . The deriv ation yields the explicit bi-Hamiltonia n structure of these tw o hierarchie s, in a ddition to their resp ectiv e recursion op erators a nd constan ts of motion. As examples, the integrable generalizations of the Heisen b erg mo del and the mKdV spin mo del in 2+1 dimensions are written dow n in detail, as w ell as t he corresp onding new 2+1 dimensional in tegra ble gen- eralizations of the Sc hr¨ odinger map equation and mKdV map equation. In section VI I I, w e work o ut the equations of motio n f o r the surface ﬂows that corresp ond to the previ- ous hierarc hies. These equations are obtained b y means of a diﬀeren t f r a ming deﬁned in a purely geometrical fashion b y the non- stretc hing co o r dinate direction o n the surface a nd the orthog onal direction of the surface normal in Euclidean space. W e also discuss asp ects of b oth the intrinsic and extrinsic geometry of t he resulting surface motions. In particular, w e obtain a recursion op erator , constan ts of motion, and explicit ev olution equations fo r - m ulated in terms of geometric v ariables giv en b y the geo desic curv ature, normal curv ature, and relativ e torsion o f the non-stretc hing co ordinate lines on the surface. Some concluding remarks on future extensions of this w ork are giv en in section IX. 5 I I. VECTOR MODELS AND SP A CE CUR VE F LO WS W e start from an a r bitrary S O (3 ) v ector mo del in 1+1 dimensions, S t = f ( S, S x , S xx . . . ) , | S | = 1 (2.1) where S ( t, x ) = ( S 1 , S 2 , S 3 ) is a dynamical unit v ector in Euclidean space, f is a v ector function ⊥ S , and x b elongs to some one-dimensional domain C . A running example will b e the Heisen b erg spin mo del S t = S ∧ S xx = ( S ∧ S x ) x (2.2) with C b eing R or S 1 . There are t w o diﬀeren t wa ys t o asso ciate a curv e ﬂow to equation ( 2 .1). One for m ulat io n consists of intrins ically iden tifying S with a map γ into the unit sphere S 2 ⊂ R 3 . T hen S t and S x corresp ond to γ t and γ x ; ∂ x + S ( S x · ) corresp onds to the cov ariant deriv ative ∇ x on the sphere with resp ect t o the ta ngen t direction γ x ; and S ∧ corresp onds to the Ho dge dual ∗ = J (i.e. a complex structure on the sphere). Under these iden tiﬁcations, eac h v ector mo del (2 .1) describ es a curve ﬂow γ t = F ( γ x , ∇ x γ x , . . . ) (2.3) for γ ( t, x ) on S 2 . Ex. the Heisen b erg mo del (2.2) corresp onds to γ t = J ∇ x γ x (2.4) whic h is t he Sc hr¨ odinger map equation on S 2 . Alternativ ely , in an extrinsic formulation, S can b e iden tiﬁed with the unit t a ngen t v ector T along a non-stretc hing space curv e giv en by a p osition v ector ~ r in Euclidean space, S = T = ~ r x (2.5) where x is the arclength a long the curv e ~ r ( x ). Then the equation of motion of ~ r is ~ r tx = f ( ~ r x , ~ r xx , . . . ) , | ~ r x | = 1 , (2.6) or equiv alently ~ r t = Z x f ( ~ r x , ~ r xx , . . . ) dx, | ~ r x | = 1 , (2.7) 6 under whic h the arclength o f t he curv e is preserv ed, i.e. R C | ~ r x | dx = ℓ is a constan t of the motion. Ex. the Heisen b erg mo del (2.2) corresp onds to ~ r t = ~ r x ∧ ~ r xx (2.8) with | ~ r x | = 1. This is the equation o f motion of a non- stretc hing v ortex ﬁlamen t studied by Hasimoto [13]. T o pro ceed we ﬁrst in tro duce a F renet f rame E along ~ r ( x ). It is expressed in matrix column nota tion by E =      T N B      (2.9) where T = ~ r x , N = | T x | − 1 T x = | ~ r xx | − 1 ~ r xx , B = T ∧ N = | ~ r xx | − 1 ~ r x ∧ ~ r xx . (2.10) Here N is the unit normal and B is the unit bi-normal of t he space curv e ~ r ( x ). Note w e ha v e the relations S = T , S x = κN , S ∧ S x = κB , (2.11) where κ = T x · N = | S x | (2.12) is the curv ature of ~ r ( x ), and τ = N x · B = | S x | − 2 S xx · ( S ∧ S x ) (2.13) is the torsion of ~ r ( x ). The Serret-F renet equations of this frame (2.9) are given by E x = KE (2.14) with K =      0 κ 0 − κ 0 τ 0 − τ 0      ∈ so (3 ) . (2.15) F rom the equation of motion (2.1) fo r S w e obta in the frame evolution equation E t = AE , A =      0 a 2 a 3 − a 2 0 a 1 − a 3 − a 1 0      ∈ so (3 ) (2.16) 7 where a 1 = f x · B / | S x | = f x · ( S ∧ S x ) / | S x | 2 , a 2 = f · N = f · S x / | S x | , a 3 = f · B = f · ( S ∧ S x ) / | S x | , are determined by taking the t -deriv ative of (2.11) and substituting (2.1), follo w ed by ap- plying resp ectiv e pro jections orthogonal to T , N , B . This ev olution of the frame E induces ev olution equations for κ and τ through the zero- curv ature relatio n K t = A x + [ A , K ]. Ex. the Heisen b erg mo del (2.2) giv es the v ortex ﬁlamen t equations in t erms o f the curv ature and torsion [14]: κ t = − κτ x − 2 κ x τ = − ( κ 2 τ ) x κ , τ t = κ xxx κ − κ xx κ x κ 2 − 2 τ τ x + κκ x = ( κ xx κ − τ 2 + 1 2 κ 2 ) x . (2.17) Next w e p erform a S O (2) gauge transformation on t he normal v ectors in the F renet frame (2.9): e E 1 = E 1 = T , e E 2 = E 2 cos θ + E 3 sin θ , e E 3 = − E 2 sin θ + E 3 cos θ (2.18) with the rotat io n angle θ deﬁned b y θ x = − τ (2.19) so t hus e E 1 x = κ cos θ e E 2 − κ sin θ e E 3 ⊥ T , (2.20) e E 2 x = − κ cos θ e E 1 k T , e E 3 x = κ sin θ e E 1 k T . (2.21) This is called a p ar al lel fr ami n g [7 ] of the space curv e ~ r ( x ). The fra me v ectors (2 .1 8) are c haracterized b y the geometrical prop erty that along ~ r ( x ) their deriv a t ives lie completely in the no rmal space (2.20) or in the tangent space (2.21). Such a frame is unique up to a rigid ( x -indep enden t) rota t io n θ → θ + φ, φ = const. (2.22) acting on the pair of normal v ectors. 8 In matrix notation the Serret-F renet equations of a parallel frame ar e give n by e E x = U e E (2.23) with e E =      T cos θ N + sin θ B − sin θ N + cos θ B      , U =      0 u 2 u 3 − u 2 0 u 1 − u 3 − u 1 0      ∈ so (3) (2.24) where u 1 = 0 , u 2 = κ cos θ = κ cos( R τ dx ) , u 3 = − κ sin θ = κ sin( R τ dx ) (2.25) are the comp onents of the principal normal T x of ~ r ( x ). The evolution o f this frame e E t = W e E (2.26) is describ ed b y the matrix W =      0  2  3 −  2 0  1 −  3 −  1 0      ∈ so (3) (2.27) whic h is r elated to U through the zero-curv a ture equation U t − W x + [ U , W ] = 0 . (2.28) Note W can b e determined directly f rom the mo del (2.1) via t he relatio ns ( 2.9) and (2.11). It now b ecomes conv enien t to w ork in terms of a complex v ariable formalism  =  2 + i 3 , (2.29) u = u 2 + iu 3 = κe − iθ = κ exp( i R τ dx ) , (2.30) enco ding the w ell-kno wn Hasimoto transformation [13]. Ex. in the Heisen b erg mo del (2.2), the vortex ﬁlament equations on κ and τ transform in to the NLS equation on u : − i u t = u xx + 1 2 | u | 2 u. (2.31) Th us, Hasimoto’s transformation has t he geometrical interpretation [10] of a S O (2) gauge transformation on t he no r ma l fra me of the curve ~ r ( x ), relating a F renet frame t o a parallel frame. 9 Remark: Since the form (2 .25) of a pa r allel frame is preserv ed b y S O (2) r o tations (2.22), the complex scalar v ariable (2.30) give n b y the Ha simoto tra nsformation is uniquely determined by the curv e ~ r ( x ) up to rigid phase rotations u → e − iφ u , depending on an arbitrary constant φ . Therefore, u has the geometrical meaning of a c ovariant of the curve [5] relative to the group S O (2) ≃ U (1), while | u | = κ and (arg u ) x = τ are in v ariants of the curv e. I I I. BI-HAMIL TONIAN FLOWS AND O PERA TORS F or a general v ector mo del (2.1) t he zero-curv ature equation (2.28) gives an ev olution equation o n u , u t =  x − i 1 u, (3.1) plus an auxiliary equation relating  1 to u ,  1 x = Im( ¯  u ) . (3.2) F rom (3.2) we can eliminate  1 = D − 1 x Im( ¯  u ) in terms of u and  , and then w e see (3.1) yields u t = D x  − iuD − 1 x Im( ¯  u ) = H (  ) (3.3) where  is determined from (2 .1) via the frame evolution equation (2.26 ). Prop osition 1. H = D x − iuD − 1 x Im( u C ) (3.4) is a Hamiltoni an op er ator with r esp e ct to the ﬂow variab le u ( t, x ) , whe n c e the evo l ution e quation (3.3) h as a Ham i l toni a n structur e u t = H ( δ H /δ ¯ u ) (3.5) iﬀ  = δ H /δ ¯ u (3.6) holds for some Hamiltonian H = Z C H ( x, u, ¯ u , u x , ¯ u x , u xx , ¯ u xx , . . . ) dx. (3.7) 10 Here C is the complex conjuga tion op erator, and C = R or S 1 is the domain of x . In the presen t setting, an o p erat or D is Hamiltonian if it deﬁnes a n asso ciated P oisson brac ke t { H , G } = Z C Re( D ( δ H /δ ¯ u ) δ G /δ u ) dx (3.8) ob eying sk ew-symmetry { H , G } = −{ G , H } and the Jacobi identit y { F , { H , G }} + cyclic = 0, for a ll real- v alued functionals F , G , H on the x -j et space of the ﬂow v ariable u . Prop osition 2. (i) The Hami l toni a n op er ator H is invariant with r esp e ct to U (1) phase r otations e iλ H e − iλ = H | u → e iλ u . (ii) A se c ond Hami l toni a n op er ator is given by I = − i (3.9) which is similarly U (1) - i nvariant, e iλ I e − iλ = I . (ii i ) The op er ators H a n d I a r e a c om p atible Hamiltonian p a ir (i.e . every line ar c ombin ation is again a Hamiltonian op er ator), and thei r c omp ositions deﬁne U ( 1 ) -invariant h e r e ditary r e cursion op e r ators R = HI − 1 = i ( D x + uD − 1 x Re( u C )) , R ∗ = I − 1 H = iD x − uD − 1 x Re( iu C ) . (3.10) (iv) Comp osition o f R and H yields a thir d U (1) -invarian t Hamiltonian op er ator E = RH = iD 2 x + D x ( uD − 1 x Im( u C )) + iuD − 1 x Re( uD x C ) (3.11) = iD 2 x + i | u | 2 + u x D − 1 x Im( u C ) − iuD − 1 x Re( u x C ) satisfying e iλ E e − iλ = E | u → e iλ u . In p articular, E , H , I f o rm a c omp atible Hami l ton i a n triple. These Prop o sitions are a sp ecial case of group-inv ariant bi-Hamiltonian op erators deriv ed from non- stretc hing curve ﬂow s in constant-curv ature spaces and general symmetric spaces in recen t w ork [3–5, 24]. Moreo v er, in the presen t complex v aria ble formalism, Prop osition 2 pro vides a substan tial simpliﬁcation o f some main results in [17] o n Hamiltonian op erators connected with non-stretchin g curv e ﬂo ws in Euclidean space. Because phase rotation on u is a symmetry of b oth H and I , the recursion op erator R generates a hierarch y of comm uting Hamiltonian ve ctor ﬁelds giv en by i ( n ) ∂ /∂ u = R n ( iu ) ∂ /∂ u , n = 0 , 1 , 2 , . . . (3.12) where  ( n ) = δ H ( n ) /δ ¯ u = R ∗ n ( u ) , n = 0 , 1 , 2 , . . . (3.13) 11 are Hamiltonian deriv atives , starting with  (0) = u, H (0) = ¯ uu = | u | 2 (3.14) whic h corresp onds to phase-rota t ion iu∂ /∂ u . Next in the hierarc hy comes  (1) = iu x , H (1) = i 2 ( ¯ uu x − u ¯ u x ) = Im( ¯ u x u ) , (3.15) follo we d by  (2) = − ( u xx + 1 2 | u | 2 u ) , H (2) = | u x | 2 − 1 4 | u | 4 , (3.16) corresp onding to resp ectiv e Hamiltonian v ector ﬁelds − u x ∂ /∂ u whic h is x -translation a nd − i ( u xx + 1 2 | u | 2 u ) ∂ /∂ u whic h is of NLS for m. Through Prop ositions 1 and 2, this hierarch y pro duces in tegrable ev o lutio n equations on u ( t, x ) with a t r i- Hamiltonian structure. An explicit for m ulat io n of this result has not app eared previously in the literature. Theorem 1. The r e is a hier ar chy of in te gr able bi-Hamiltonian ﬂows on u ( t, x ) given by u t = H ( δ H ( n ) /δ ¯ u ) = I ( δ H ( n +1) /δ ¯ u ) , n = 0 , 1 , 2 , . . . (3.17) (c al le d the + n ﬂow ) in terms of Hamiltonia ns H ( n ) = R C H ( n ) dx whe r e H ( n ) = 2 1 + n D − 1 x Im( ¯ u ( i H ) n +1 u ) n = 0 , 1 , 2 , . . . (3.18) ar e lo c al Hamiltonian densities. Mor e over, al l the ﬂows for n 6 = 0 have a tri-Hamil toni a n structur e u t = E ( δ H ( n − 1) /δ ¯ u ) , n = 1 , 2 , . . . . (3.19) Remarks: Each ﬂow n = 0 , +1 , +2 , . . . in the hierarch y is U (1)- in v ariant under the phase ro t a tion u → e iλ u a nd has scaling w eigh t t → λ 1+ n t under the NLS scaling symmetry x → λx , u → λ − 1 u , where t he scaling w eigh t of H ( n ) is − 2 − n . Additionally , these ﬂow s on u ( t, x ) eac h admit constants of motion (under suitable b oundary conditions) D t Z C | u | 2 dx = 0 , D t Z C i ¯ uu x dx = 0 , D t Z C | u x | 2 − 1 4 | u | 4 dx = 0 , . . . (3.20) and symmetries − u x ∂ /∂ u, − i ( u xx + 1 2 | u | 2 u ) ∂ /∂ u , ( u xxx + 3 2 | u | 2 u ) ∂ /∂ u , . . . (3.21) 12 resp ectiv ely comprising a ll of the Hamiltonians (3.1 8) in the hierarch y and all of the corre- sp onding Hamilto nian ve ctor ﬁelds (3.1 2 ). A t the b ottom of the hierarch y , the 0 ﬂow is give n b y a linear trav eling w a ve equation u t = u x , and next the + 1 ﬂow pro duces the NLS equation (2.31 ). The +2 ﬂow yields the complex mKdV equation − u t = u xxx + 3 2 | u | 2 u (3.22) whic h corresp onds to an mKdV analog of t he v ort ex ﬁlamen t equations, − κ t = ( κ xx + 1 2 κ 3 ) x − 3 2 ( τ 2 κ 2 ) x κ = κ xxx + 3 2 ( κ 2 − 2 τ 2 ) κ x − 3 κτ τ x , (3.23) − τ t = ( τ xx + 3 ( τ κ x ) x κ + 3 2 τ κ 2 − τ 3 ) x (3.24) = τ xxx + 3 τ xx κ x κ + τ x (6 κ xx κ − 3 κ 2 x κ 2 − 3 τ 2 + 3 2 κ 2 ) + τ (3 κ xxx κ − 3 κ xx κ x κ 2 + 3 κκ x ) , as o btained through the Hasimoto transformation u = κ exp( − iθ ). The ev olut io n equations describing the 0 , +1 , +2 , . . . ﬂows on u eac h arise fro m geomet- ric space curve ﬂo ws corresp onding to S O (3 )-in v ariant v ector mo dels (2.1). T o mak e this corresp ondence explicit, it is con ve nient to introduce a complex frame notation E k = e E 1 = T , E ⊥ = e E 2 + i e E 3 = e − iθ ( N + iB ) (3.25) satisfying E k ∧ E ⊥ = − iE ⊥ = e E 3 − i e E 2 = e − iθ ( B − iN ) (3.26) and E k · E k = 1 , E ⊥ · ¯ E ⊥ = 2 , E k · E ⊥ = 0 = E ⊥ · E ⊥ . (3.27) The F renet equations (2.14) b ecome E k x = R e( ¯ uE ⊥ ) , E ⊥ x = − u E k , (3.28) while from (2.27 ), (2.29), (3.2), the evolution of the frame is given b y the equations E k t = R e( ¯  E ⊥ ) , E ⊥ t = iD − 1 x Im(  ¯ u ) E ⊥ −  E k . (3.29) Then an y ﬂo w b elonging to the general class  =  ( u , ¯ u, u x , ¯ u x , u xx , ¯ u xx , . . . ) (3.30) 13 will determine a v ector mo del (2.1) via S = E k , f = Re( ¯  E ⊥ ) , (3.31) where f is expressed in terms of S , S x , S xx , etc. through the F renet equations (2.23)–(2 .24). Ex. 1 : The +1 ﬂow  = iu x yields E k t = − Re( i ¯ u x E ⊥ ) . (3.32) By rewriting ¯ u x E ⊥ = ( ¯ uE ⊥ ) x + ¯ uuE k w e obtain Re( i ¯ uE ⊥ ) = − E k ∧ Re( ¯ uE ⊥ ) = − E k ∧ E k x , Re( i ¯ uuE k ) = Re( i | u | 2 ) E k = 0 , and hence E k t = ( E k ∧ E k x ) x . (3.33) The iden tiﬁcations (3.31 ) then directly give the S O (3 ) Heisen b erg mo del (2.2), whic h cor- resp onds t o the non-stretc hing space curv e ﬂow (2.17) or equiv alently ~ r t = κB , | ~ r x | = 1 (3.34) expresse d as a geometric ﬂow. Ex. 2 : The +2 ﬂow  = − ( u xx + 1 2 | u | 2 u ) yields − E k t = R e(( ¯ u xx + 1 2 | u | 2 ¯ u ) E ⊥ ) = Re( ¯ u xx E ⊥ ) + 1 2 | u | 2 E k x . (3.35) Here we can rewrite the ﬁrst term as Re( ¯ u xx E ⊥ ) = Re( ¯ uE ⊥ ) xx − Re( ¯ uE ⊥ x ) x − Re( ¯ u x E ⊥ x ) = E k xxx + ( | u | 2 E k ) x + 1 2 ( ¯ uu ) x E k , with | u | 2 = | E k x | 2 , and thu s − E k t = E k xxx + 3 2 ( | E k x | 2 E k ) x . (3.36) Hence, E k = S giv es − S t = S xxx + 3 2 ( | S x | 2 S ) x (3.37) 14 whic h can b e view ed as a n S O (3) mKdV mo del. The corresp onding no n- stretc hing space curv e ﬂo w lo oks lik e − ~ r t = ~ r xxx + 3 2 | ~ r xx | 2 ~ r x , | ~ r x | = 1 . (3.38) This describes a geometric ﬂo w [16] − ~ r t = 1 2 κ 2 T + κ x N + κτ B , | ~ r x | = 1 (3.39) whic h is equiv alent to the ev olution (3.23) and (3.24 ) on the curv ature and torsion of ~ r ( x ). Remark: A diﬀeren t geometric deriv ation of the mKdV mo del (3 .37) app ears in w ork [2] on non- stretc hing ﬂo ws of curve s in three-dimensional manifolds with constant curv a t ure, i.e. S 3 , H 3 , R 3 , where the spin v ector S is iden tiﬁed with the comp onents of the unit tangen t v ector in a mo ving frame deﬁned by parallel transp ort along the curv e. The mKdV mo del a lso has b een deriv ed in [1 1 ] as a higher-order symmetry of the Heisen b erg mo del b y non-geometric metho ds. All of these S O (3) v ector mo dels describe dynamical maps γ o n the unit sphere S 2 ⊂ R 3 b y means of the iden tiﬁcations: S t ↔ γ t , S x ↔ γ x , ∂ x + S ( S x · ) ↔ ∇ x , S ∧ ↔ J = ∗ (3.40) and th us ∇ x γ x ↔ S xx + | S x | 2 S, ∇ 2 x γ x ↔ S xxx + | S x | 2 S x + 3 2 ( | S x | 2 ) x S, (3.41) J γ x ↔ S ∧ S x , J ∇ x γ x ↔ S ∧ S xx , (3.42) g ( γ x , γ x ) = | γ x | 2 g ↔ | S x | 2 = S x · S x (3.43) where g denotes the Riemannian metric on the sphere S 2 (giv en b y restricting the Euclidean inner pro duct in R 3 to the tangent space of S 2 ⊂ R 3 ). In particular, the S O (3) Heisen b erg mo del yields the Sc hr¨ odinger map equation (2.4) on S 2 , while the S O (3) mKdV mo del (3.37) is iden t iﬁed with − γ t = ∇ 2 x γ x + 1 2 | γ x | 2 g γ x (3.44) whic h is a mKdV map equation on S 2 (i.e. a dynamical map version of the p oten tial mKdV equation). Th us, Theorem 1 prov ides a g eometric realization o f the hierarc hies of integrable v ector mo dels and dynamical maps con taining the Heisen b erg mo del and the Sch r¨ odinger map as w ell as their mKdV counterparts. 15 IV. GEOMETRIC HIE RAR CHY O F IN TEGRABLE VEC TOR MODELS AND D YNAMICAL MAPS In general, a n y non-stretchin g space curv e ﬂow (2.7) can b e written in terms o f a F renet frame (2.9) by an equation of motion of the form ~ r t = aT + bN + cB (4.1) suc h that D x a = κb. (4.2) This relatio n b et w een the tangen tial and normal comp onents of the motion arises due to the non-stretc hing prop erty | ~ r x | = 1 (4.3) b y whic h the motion preserv es the lo cal arclength ds = | ~ r x | dx of t he space curve if (and only if ) ~ r x · ~ r tx = 0. As a consequence, through the Serret-F renet equations (2 .14)–(2.15), the ta ngen t ve ctor T = ~ r x along the space curv e ob eys the equation o f motion T t = ˆ f 1 N + ˆ f 2 B = f ⊥ T (4.4) giv en b y a linear combination o f the normal and bi-normal vec tors with co eﬃcien ts ˆ f 1 = D x b − τ c + κa, ˆ f 2 = D x c + τ b. (4.5) No w w e consider a Hasimoto tra nsfor ma t io n ( 2.18)–(2.19) from the F renet frame (2.9 ) to a parallel frame (3.25) . The equation of motion (4 .1 ) on ~ r tak es the form ~ r t = h k E k + Re( ¯ h ⊥ E ⊥ ) = h k T + Re( h ⊥ e iθ ) N + Im( h ⊥ e iθ ) B (4.6) in terms of the t a ngen tia l and normal comp onen ts given b y h ⊥ = ( b + ic ) e − iθ , h k = a, (4.7) with these comp onen ts satisfying the relation (4.2) giv en b y D x h k = R e( ¯ uh ⊥ ) (4.8) where u = κe − iθ . Corresp ondingly , from the F renet equations (3.28) of the para llel frame, the equation of motion (4.4) for the tangent v ector T = ~ r x has the form T t = Re( ¯  E ⊥ ) = Re(  e iθ ) N + Im(  e iθ ) B (4.9) 16 in terms of  = D x h ⊥ + h k u (4.10) whic h enco des the normal and bi-normal comp onen ts ˆ f 1 + i ˆ f 2 =  e iθ . (4.11) The ev o lution of T is thus sp eciﬁed by the v ariable  , while the underlying ev olution of ~ r is sp eciﬁed in terms of the v ariable h ⊥ , with h k giv en by the no n-stretc hing condition (4.8). F ro m equation (4.10) these v ar iables are related b y  = D x h ⊥ + uD − 1 x Re( ¯ uh ⊥ ) = J ( h ⊥ ) . (4.12) The op erator here J = D x + uD − 1 x Re( u C ) (4.13) is related to the Ha miltonian op erato r I = − i b y the prop erties − J = R ∗ I − 1 = I − 1 R and − J − 1 = R − 1 I = I R ∗− 1 , (4.14) where R and R ∗ are the r ecursion op erato rs (3.10). Consequen tly , J − 1 is a f ormal Hamil- tonian op erato r compatible with I . Prop osition 3. Th e evo l ution (4.1) of a non-str e tching sp ac e c urve ~ r ( x ) c an b e e xpr esse d in terms of a ge ometric a l variable that determine s the c orr esp onding evolution (4.4) of the tangent ve ctor T = ~ r x thr ough the r elation T t = ( ~ r t ) x . I n p a rticular, h ⊥ = J − 1 (  ) = R − 1 ( i ) = i R ∗− 1 (  ) (4.15) yields the normal c om p onents of the evo lution ve ctor ~ r t in a p ar al lel fr ame, wher e  r epr e- sents the fr ame c o m p onents of T t . The curvatur e κ and torsio n τ of ~ r ( x ) c orr esp ondingly have the evolution κ t = D x ˆ f 1 − τ ˆ f 2 = Re( e iθ D x  ) (4.16) τ t = D x ( κ − 1 D x ˆ f 2 + τ κ − 1 ˆ f 1 ) + κ ˆ f 2 = D x ( κ − 1 Im( e iθ D x  )) + κ Im( e iθ  ) (4.17) which c an b e expr esse d in terms of the F r ene t fr ame c o eﬃcie n ts a, b, c of ~ r t thr ough the r elations ( 4.5). 17 Conditions will no w b e stated within the general class of ﬂo ws (3.3 0) on u suc h tha t the v ario us ev olutions (4.16)– ( 4 .17), (4.9), (4.6), (4.4), (4.1) , and (3.5)–(3.6) each deﬁne a geometric ﬂow. Theorem 2. F or a non-str etching ﬂow of a sp ac e curve ~ r ( x ) in R 3 , the fol lowing c on d itions ar e e quivale n t: (i) Its tangent ve ctor T = ~ r x = S ob eys a S O (3) -invariant ve ctor mo del iﬀ ˆ f 1 and ˆ f 2 ar e functions of s c alar invaria nts forme d out of S and its x derivatives (mo dulo diﬀer ential c onse quenc es of S · S = 1 ), i.e. ˆ f 1 + i ˆ f 2 = ˆ f ( S x · S x , S x · S xx , S xx · S xx , . . . , S xx · ( S ∧ S x ) , S xxx · ( S ∧ S x ) , S xxx · ( S ∧ S xx ) , . . . ) . (4.18) (ii) Its p ri n cip al norma l c om p onent u = T x · E ⊥ in a p ar al lel fr ame (3.25 ) satisﬁes a U (1) - invariant evolution e quation iﬀ  is an e quivariant function of u , ¯ u , and x derivatives of u a nd ¯ u , under the action of a rigid ( x -indep endent) U (1) r otation gr oup u → e − iφ u ( with φ = c onst.), i.e.  = u ˆ f ( | u | , | u | x , | u | xx , . . . , (arg u ) x , ( arg u ) xx , . . . ) . (4.19 ) (iii) Its curvatur e κ and torsion τ satisfy ge o metric evolution e quations in terms of inva ri a n ts and diﬀer ential invariants of ~ r ( x ) iﬀ  e iθ is a function of κ , τ , a n d their x derivatives, i.e.  = e − iθ κ ˆ f ( κ, κ x , κ xx , . . . , τ , τ x , τ xx , . . . ) . ( 4 .20) (iv) Its e quation of motion is invari a n t under the Euclide a n iso m etry gr oup S O (3) ⋊ R 3 iﬀ a, b, c ar e sc al a r functions of the curvatur e κ , torsion τ , and their x -derivatives, subje ct to the non -str e tchi n g c ondition (4.2). The pro of of this prop osition a moun t s to enum erating the Euclidean (diﬀeren tial) inv ari- an ts of a space curv e ~ r ( x ) with an a rclength parameterization x , as sho wn in app endix A. W e are no w able to deriv e the entire hierarc hy of S O (3) - in v ariant v ector mo dels and geometric space curv e motions t hat corresp ond to all of the U (1)-inv ariant ﬂo ws on u in Theorem 1. F rom equation (4.9) combined with the hierarch y (3.6), the ev olution of the spin ve ctor S = T = ~ r x can b e written as S t = R e( ¯ E ⊥ R ∗ n ( u )) , n = 0 , 1 , 2 , . . . (4.21) 18 as generated via t he recursion op erator R ∗ = iD x − uD − 1 x Re( iu C ). The main step is now to establish the op erato r iden tity Re( ¯ E ⊥ R ∗ ) = S R e( E ⊥ C ) (4.22) where S is a spin v ector op erator corresp onding to R ∗ , and Re( E ⊥ C ) is t he op erator that pro duces a vec tor f in the p erp space of S in R 3 when applied to the comp onen ts of f with resp ect to E ⊥ (i.e. f = R e( E ⊥ C ˆ f ) if f · S = 0 , where ˆ f = f · E ⊥ ). Through the F renet equations (3 .28) and the o r t ho normalit y relations (3.2 7 ) on E ⊥ and E k , w e straigh tforw ardly ﬁnd Re( ¯ E ⊥ u ) = E k x = S x (4.23) and Re( ¯ E ⊥ iD x ˆ f ) = E k ∧ Re( ¯ E ⊥ D x ˆ f ) = S ∧ D x Re( E ⊥ C ˆ f ) = S ∧ D x f D − 1 x Re( iu C ˆ f ) = D − 1 x Re(( E k ∧ E k x ) · E ⊥ C ˆ f ) = D − 1 x Re(( S ∧ S x ) · f ) for a ny v ector f ( x ), orthogonal to S in R 3 , with comp onen ts ˆ f ( x ) = E ⊥ · f ( x ). Hence, this yields the v ector o p era t or S = S ∧ D x − S x D − 1 x ( S ∧ S x ) · (4.24) Theorem 3. The bi-Hamiltonian ﬂows (3.17) on u ( t, x ) c orr esp ond to a hie r ar chy of inte- gr able S O (3) -in variant ve ctor mo dels S t =  S ∧ D x − S x D − 1 x ( S ∧ S x ) ·  n S x = f ( n ) , n = 0 , 1 , 2 , . . . (4.25) gener ate d b y the r e cursion op er ator ( 4 . 24). These mo dels have the e quivalent ge o m etric al formulation γ t =  J ∇ x − γ x D − 1 x g ( J γ x , )  n γ x = F ( n ) , n = 0 , 1 , 2 , . . . (4.26) expr esse d as dynami c al maps γ ( t, x ) into the 2-sph e r e S 2 ⊂ R 3 . Mor e over, the Hamiltoni a ns (3.18) for al l the ﬂows on u ( t, x ) c orr esp ond to a set of c onstants of motion H (0) = R C H (0) dx , H (1) = R C H (1) dx , H (2) = R C H (2) dx , etc. for e a c h v e ctor mo del (4.25) and e ach dyna m ic al map e quation (4.26) . This entir e se t has the explicit form given by the dens ities (m o dulo total x -derivatives) (1 + n ) H ( n ) = D − 1 x ( S x · D x f ( n ) ) = D − 1 x g ( γ x , ∇ x F ( n ) ) , n = 0 , 1 , 2 , . . . (4.2 7) 19 which ar e sc alar p olynomials forme d o ut of S O (3) -invaria nt we dge p r o ducts an d dot pr o ducts of S, S x , S xx , . . . in terms o f the e quations of motion fo r S ( t, x ) , or e quivalently, s c alar inne r pr o d ucts of γ x , J γ x , ∇ x γ x , J ∇ x γ x , . . . forme d in terms of the e quations of motion for γ ( t, x ) . The v ector mo dels (4.25) hav e b een deriv ed previously in [6] b y non-geometric metho ds (based on a Lax pair represen tation). As one new result, Theorem 3 deriv es the equiv a- len t dynamical map equations (4.26 ), along with their recursion op erato r, a nd prov ides an explicit expression for the constan ts of motion f or all o f these integrable generalizations of the Heisen b erg spin mo del (2.2) and the mKdV spin mo del (3 .37). In particular, from the hierarc h y (4.25), higher-deriv ativ e v ersions of the Heisen b erg spin mo del n = 1 as given b y n = 3 , 5 , . . . are seen t o describ e higher-deriv ativ e Sc hr¨ odinger maps; similarly , higher- deriv ativ e v ersions of the mKdV spin mo del n = 2 as give n by n = 4 , 6 , . . . are found to describe higher-deriv ativ e mKdV maps. Ex: n = 3 and n = 4 respective ly yield a 4th or der Heisen b erg v ector mo del S t = S ∧ S xxxx + 5 2 ( | S x | 2 S ∧ S x ) x (4.28) and a 5th order mKdV vec tor mo del − S t = S xxxxx − 5 2 ( | S xx | 2 S + | S x | 2 S x − 7 4 | S x | 4 S ) x + 5 2 ( | S x | 2 S ) xxx (4.29) whic h are described g eometrically by a 4th o r der Sch r¨ odinger map equation − γ t = J ∇ 3 x γ x + 1 2 ∇ x ( | γ x | 2 g J γ x ) − 1 2 g ( J γ x , ∇ x γ x ) γ x (4.30) and a 5th order mKdV map equation γ t = ∇ 4 x γ x + 3 4 ( | γ x | 2 g ) x ∇ x γ x + ( 5 4 ( | γ x | 2 g ) xx − 3 2 |∇ x γ x | 2 g + 3 8 | γ x | 4 g ) γ x . (4.31) Remark: The equations o f motion of these dynamical maps γ ( t, x ) do not lo cally preserv e the arclength ds = | γ x | g dx in the x direction along γ on S 2 , namely | γ x | g has a no ntrivial time ev olution for an y of the dynamical map equations (4.26), and additionally , the total arclength R C | γ x | g dx is time dep enden t. The spin v ector recursion op erator (4.24) has t he factorizatio n S = ( D x + S S x · + S x D − 1 x S x · )( S ∧ ) (4.32) 20 where, as show n b y the results in [6], the op erators D x + S S x · + S x D − 1 x S x · and ( S ∧ ) − 1 = − S ∧ constitute a compat ible Hamiltonian pa ir with resp ect to the spin ve ctor v ariable S ( t, x ). By comparison, using the presen t geometric framew ork, w e now directly deriv e the explicit bi- Hamiltonian structure of the hierarch y of v ector mo dels (4 .2 5) thro ugh the bi- Hamiltonian ﬂo w equation (3.17) on u ( t, x ) in Theorem 1. The starting p oint is the v ariational iden tit y δ H ( n ) ≡ δ S · ( δ H ( n ) /δ S ) ≡ 2R e( ¯  ( n ) δ u ) (4.33) holding mo dulo to tal x -deriv a t iv es and mo dulo (diﬀeren tial consequences of ) S · δ S = 0, with the Hamilto nian densities give n in the equiv alent forms ( 3 .18) and (4.27), and with  ( n ) giv en by the hierarc hy (3.1 3) generated throug h the recursion op erator R ∗ . T o b egin w e derive an explicit expression f o r δ u in terms o f δ S = δ E k b y t a king the F reche t deriv ative of the F renet equation (3.28) for the normal v ectors in a pa rallel fra me with resp ect to u : D x δ E ⊥ = − δ uE k − uδ E k . (4.34) The ¯ E ⊥ comp onen t of (4.34) giv es D x ( ¯ E ⊥ · δ E ⊥ ) = − u ¯ E ⊥ · δ E k − ¯ uE k · δ E ⊥ = − 2 i Re ( i ¯ uE ⊥ · δ E k ) via E k · δ E ⊥ = − E ⊥ · δ E k due to the o rthogonalit y of E ⊥ , E k . Similarly , the E k comp onen t of (4 .3 4) yields δ u = D x ( E ⊥ · δ E k ) + 1 2 u ¯ E ⊥ · δ E ⊥ . Com bining these t w o expressions, w e obtain δ u = D x ( E ⊥ · δ E k ) − iuD − 1 x Re( i ¯ uE ⊥ · δ E k ) (4.35) whence Re( ¯  ( n ) δ u ) = R e( ¯  ( n ) D x ( E ⊥ · δ E k )) + Im( ¯  ( n ) u ) D − 1 x Re( i ¯ uE ⊥ · δ E k ) . (4.36) In tegration b y parts on b oth terms in (4.36) then yields Re( ¯  ( n ) δ u ) = − Re( E ⊥ · δ E k D x ¯  ( n ) ) − Re( i ¯ uE ⊥ · δ E k D − 1 x Im( ¯  ( n ) u )) ≡ − δ E k · Re( ¯ E ⊥ H (  ( n ) )) (4.37) 21 mo dulo total x -deriv ativ es. Next, after use of the r elat io ns i R ∗ = −H and i ¯ E ⊥ = E k ∧ ¯ E ⊥ whic h follow f rom (3.10) and (3.26), w e g et Re( ¯  ( n ) δ u ) ≡ δ E k · ( E k ∧ Re( ¯ E ⊥  ( n +1) )) . (4.38) Hence, equating (4.38 ) and (4.33) yields δ S · ( δ H ( n ) /δ S − 2 S ∧ Re ( ¯ E ⊥  ( n +1) )) ≡ 0 from whic h we obtain the v ariatio nal relation − S ∧ ( δ H ( n ) /δ S ) = 2Re( ¯ E ⊥  ( n +1) ) = 2Re( ¯ E ⊥ R ∗ n +1 ( u )) = 2 f ( n +1) (4.39) where the ﬁnal equality comes from t he equation of motio n (4.21) comb ined with the hier- arc h y (4 .2 5). This result (4 .39), together with the factorization of the recursion op erato r (4.32), leads to the following Hamiltonia n structure. Theorem 4. In terms of the Hamiltoni a n densities (4.27), the hier ar chy o f ve ctor mo dels (4.25) for S ( t, x ) h a s two Hamiltonian s tructur es S t = − S ∧ ( δ H ( n − 1) /δ S ) = f ( n ) , n = 1 , 2 , . . . (4.40) and, for n 6 = 1 , S t = D x ( δ H ( n − 2) /δ S + S D − 1 x ( S x · δ H ( n − 2) /δ S )) = f ( n ) , n = 2 , 3 , . . . (4.41) wher e S ∧ and D x + S S x · + S x D − 1 x S x · (4.42) ar e a c omp atible p a i r of Hamiltonian op er a tors. Co rr esp o n dingly, the hier ar chy of dynamic a l maps (4.26 ) on γ ( t, x ) has the Hamiltonia n structur es γ t = − J ( δ H ( n − 1) /δ γ ) = F ( n ) , n = 1 , 2 , . . . (4.4 3 ) = ∇ x ( δ H ( n − 2) /δ γ ) + γ x D − 1 x g ( γ x , δ H ( n − 2) /δ γ ) , n = 2 , 3 , . . . (4.44) in terms of the c omp atible Hamiltonian op er ators J and ∇ x + γ x D − 1 x g ( γ x , ) (4.45) as given by the ge ometric al ide n tiﬁc ations (3.40). M or e over, these bi- Ham i l toni a n p airs (4.42) and (4.45) a r e ge ometric al ly e quivalent to the p air of c omp atible symple ctic ( i nverse Hamiltonian) op er ators −I − 1 and J = −R ∗ I − 1 with r esp e ct to the ﬂow variable u ( t, x ) in the hier ar chy (3.17)–(3.18 ) (c f . Theorems 1 and 3 ). 22 These results pro vide an explicit geometrical fo rm ula t ion of t he abstract symple ctic struc- ture g iv en in [28] for the Schr¨ odinger map equation (2.4), i.e. ( n = 1) γ t = J ∇ x γ x = − J ( δ H (0) /δ γ ) , H (0) = 1 2 g ( γ x , γ x ) , (4.46) and its higher-order generalizations ( n = 3 , 5 , . . . ); the Hamiltonian structure of the mKdV map, i.e. ( n = 2) γ t = −∇ 2 x γ x − 1 2 | γ x | 2 g γ x = ∇ x ( δ H (0) /δ γ ) + γ x D − 1 x g ( γ x , δ H (0) /δ γ ) (4.47) = − J ( δ H (1) /δ γ ) , H (1) = − 1 2 g ( ∇ x γ x , J γ x ) , (4.48) and its higher-order g eneralizations ( n = 4 , 6 , . . . ) has not app eared previously in the litera- ture. Note the n = 1 and n = 2 v ector mo dels respectiv ely yield the we ll-known Hamiltonia n structure of the Heisen b erg spin mo del (2.2), S t = S ∧ S xx = − S ∧ ( δ H (0) /δ S ) , H (0) = 1 2 | S x | 2 , (4.49) and the mKdV spin mo del (3 .3 7), S t = − S xxx − 3 2 ( | S x | 2 S ) x = D x ( δ H (0) /δ S + S D − 1 x ( S x · δ H (0) /δ S )) (4.50) = − S ∧ ( δ H (1) /δ S ) , H (1) = 1 2 S x · ( S ∧ S xx ) . (4.51) Remark: There is a second Hamiltonian structure for b oth the Sc hr¨ odinger map equation and the Heisen b erg spin mo del. This structure, in contrast to t he ﬁrst Hamiltonia n structure (4.46) and (4.49), turns out to in v olv e a non-p olynomial Hamiltonian densit y deﬁned as follo ws. Let ξ ( γ ) b e a v ector ﬁeld on S 2 suc h that its div ergence is constan t at all p oin ts γ ∈ S 2 . (This prop ert y geometrically characterize s ξ ( γ ) as a ho mo t hetic v ector with resp ect to t he metric-normalized v olume form ǫ g on S 2 , i.e. L ξ ǫ g = cǫ g for some constan t c 6 = 0.) Then the Hamiltonia n densit y given by H ( − 1) = g ( ξ ( γ ) , J ∇ γ x ) , div g ξ ( γ ) = 1 (4.52) can b e sho wn to satisfy (see App endix B) δ H ( − 1) ≡ g ( δ γ , J ∇ γ x ) (4.53) mo dulo total x -deriv a tiv es, so thus γ t = ∇ x ( δ H ( − 1) /δ γ ) + γ x D − 1 x g ( γ x , δ H ( − 1) /δ γ ) = ∇ x ( J γ x ) (4.54) 23 yields a second Hamiltonian structure for the Sc hr¨ o dinger map (2.4). The corresp onding second Hamiltonian structure for the Heisen b erg spin mo del (2.2) is giv en b y S t = D x ( δ H ( − 1) /δ S + S D − 1 x ( S x · δ H ( − 1) /δ S )) = ( S ∧ S x ) x , H ( − 1) = ξ ( S ) · ( S ∧ S x ) (4.55) in terms of a v ector function ξ ( S ) suc h that S · ξ ( S ) = 0 , ∂ ⊥ S · ξ ( S ) = 1 , (4.56) where the op erator ∂ ⊥ S = ∂ S − S ( S · ∂ S ) is the orthogonal pro jection of a gradien t with resp ect to the comp onen ts of S . F rom the prop erties S · ∂ ⊥ S = 0 , ∂ ⊥ S ( S · S ) = 0 , (4.57) the Hamilto nian densit y can b e sho wn to satisfy δ H ( − 1) ≡ δ S · ( S ∧ S x ) (4.58) mo dulo total x -deriv ativ es (see App endix C). Thus , b oth the Heisen b erg spin mo del and the Sc hr¨ odinger map equation are Hamiltonia n equations o f mo t io n with resp ect to the corresp onding bi- Hamiltonian pair s (4.42 ) and (4.45). T o conclude, a coun terpart of Theorem 3 will no w b e stated for the underlying space curv e motions on ~ r . Theorem 5. The S O (3) -invariant ve ctor mo dels (4.25) o n S ( t, x ) c orr esp on d to a hier ar chy of in te g r able ﬂows of non-str etchin g sp ac e curves ~ r t = a ( n − 1) T + b ( n − 1) N + c ( n − 1) B , | ~ r x | = 1 , n = 1 , 2 , . . . (4.59) with ge ometric c o eﬃc ients c ( n ) = R e( Q n κ ) , b ( n ) = − Im ( Q n κ ) , (4.60) a ( n ) = − D − 1 x ( κ Im( Q n κ )) (4.61) given in terms of the r e cursion op er ator Q = iD x − τ − κD − 1 x ( κ Im) = e iθ R ∗ e − iθ , (4.62) wher e κ, τ a r e the curvatur e and torsi o n o f ~ r . The tangential c o eﬃci e n ts (4.61) yield a set of non-trivial c on s tants of motion R C a (1) dx = − R C 1 2 κ 2 dx , R C a (2) dx = R C τ κ 2 dx , etc. for e ach curve ﬂow (4.59) in the hier ar chy (unde r suitable b oundary c onditions), wher e C = R or S 1 is the c o or dinate domai n of x . 24 The equations of motion (4.59) arise from writing t he ﬂow equation (4 .6) on ~ r in terms of the hierarc hy (3.6) by means of the relation (4.15). This yields a ( n ) = D − 1 x Re( i ¯ u ( n − 1) ) = − D − 1 x Im( ¯ u ( n − 1) ) , (4.63) b ( n ) = R e( ie iθ  ( n − 1) ) = − Im( e iθ  ( n − 1) ) , (4.64) c ( n ) = Im ( ie iθ  ( n − 1) ) = Re( e iθ  ( n − 1) ) . (4.65) The geometric form (4.60)–(4 .6 1) for these co eﬃcien ts is then obtained through the iden tity e iθ  ( n − 1) = e iθ R ∗ n − 1 u = ( e iθ R ∗ e − iθ ) n − 1 e iθ u (4.66 ) com bined with the Hasimoto transforma t io n κ = e iθ u, τ = − θ x . (4.67) V. SURF A CES AND SP A TIAL HAMIL TONI AN CUR VE FLOWS The results in Theorem 1 for 1+1 ﬂo ws can b e generalized in a natural w ay to 2+1 ﬂo ws b y considering surfaces ~ r ( x, y ) tha t are foliated b y space curv es with a parallel framing in R 3 . Here y will denote a co ordinate assumed to b e transv erse to these curv es, and x will denote the arclength co ordinate along the curve s, so th us | ~ r x | = 1 . (5.1) In this setting, a parallel fr ame consists of a triple of unit v ectors whose deriv ativ es a lo ng eac h y = const . co ordinate line on the surface ~ r ( x, y ) lie completely in either the tangen t space or the normal plane of this line in R 3 . The explicit form of suc h a frame is giv en b y the vec tors e E 1 = T = ~ r x , (5.2) e E 2 = R e( e − iθ ( N + iB )) = | ~ r xx | − 1 (cos θ ~ r xx + sin θ ~ r x ∧ ~ r xx ) , (5.3) e E 3 = Im( e − iθ ( N + iB )) = | ~ r xx | − 1 ( − sin θ ~ r xx + cos θ ~ r x ∧ ~ r xx ) , (5.4) with θ x = −| ~ r xx | − 2 ( ~ r x ∧ ~ r xx ) · ~ r xxx (5.5) 25 where T , N , B resp ectiv ely denote t he unit tangent ve ctor, unit normal and bi-nor mal vec tors of the y = const . co ordinate lines. In matrix notation these frame vec tors satisfy the F renet equations e E x = U e E , e E y = V e E (5.6) giv en b y e E =      e E 1 e E 2 e E 3      , U =      0 u 2 u 3 − u 2 0 0 − u 3 0 0      ∈ so (3 ) , V =      0 v 2 v 3 − v 2 0 v 1 − v 3 − v 1 0      ∈ so (3 ) , (5 .7) where u 2 + iu 3 = u = ( e E 2 + i e E 3 ) · e E 1 x , v 2 + iv 3 = v = ( e E 2 + i e E 3 ) · e E 1 y (5.8) are complex scalar v ar ia bles, and v 1 = 1 2 ( e E 3 + i e E 2 ) · ( e E 2 y + i e E 3 y ) = ¯ v 1 (5.9) is a real scalar v ariable. Note, similarly to the case of space curv es, here w e ha ve ( e E 3 + i e E 2 ) · ( e E 2 x + i e E 3 x ) = 0 (5.10) whic h characterizes e E ( x, y ) as a par a llel fr a me adapt ed to the fo lia tion of the surface ~ r ( x, y ) b y x -co ordinat e lines, i.e. e E 1 x ⊥ T , e E 2 x k T , e E 3 x k T . F rom ( 5 .5) we see this c hoice of fr a ming is geometrically unique up to rigid rota t ions that a ct on the no rmal vec tors (5.3)–(5.4): θ → θ + φ, e E 2 + i e E 3 → exp( − iφ )( e E 2 + i e E 3 ) , with φ =const. . (5.11) Under suc h r o tations, v 1 is inv ariant, while u and v undergo a rigid phase rotation, u → e − iφ u, v → e − iφ v . (5.12) T o pro ceed w e need to write do wn the tangent vec tors of ~ r ( x, y ) in terms of the parallel framing, ~ r x = e E 1 , ~ r y = q 1 e E 1 + q 2 e E 2 + q 3 e E 3 . (5.1 3) The resp ectiv e pro jections of ~ r y orthogonal and par a llel to ~ r x are giv en b y the scalar v a riables q 2 + iq 3 = q = ( e E 2 + i e E 3 ) · ~ r y , q 1 = e E 1 · ~ r y , (5.14) 26 where, under rigid rotations (5.11) on the normal vectors in the parallel frame, q 1 is in v ariant while q transforms b y a rig id phase rotation, q → e − iφ q . (5.15) Remark: D ue to the tra nsfor ma t io n prop erties (5.1 2) and (5 .15), the v ariables u , v , q represen t U (1 )- c ovariants of ~ r ( x, y ) a s geometrically deﬁned with resp ect to the x -co ordinate lines, where U ( 1 ) is the equiv a lence g r o up of rigid rotations ( 5 .11) that preserv es the fo rm of the framing (5.2)–(5 .5 ) for the surface ~ r ( x, y ). These v ariables will b e seen later (cf. Prop ositions 7 and 8 ) to enco de b oth the in trinsic and extrinsic surfa ce geometry of ~ r ( x, y ) in R 3 . W e will now sho w that all surfaces ~ r ( x, y ) with the x co ordinate satisfying the non- stretc hing prop ert y (5.1) hav e a natura l geometrical interpretation as a spatial Hamiltonian curv e ﬂow with resp ect the y co ordinate. This in terpretation arises directly from the struc- ture equations satisﬁed b y the parallel frame (5.2)–(5.4 ) and the tangen t v ectors (5 .1 3) adapted to these co ordinates. Firstly , the fr a me connection matrices given in (5.7) o b ey the zero-curv ature condition U y − V x + [ U , V ] = 0, whic h yields the structure equations u y − v x + iv 1 u = 0 , v 1 x − Im( ¯ v u ) = 0 . (5.16) Secondly , the f r ame expansions of the tangen t v ectors giv en b y (5.13) ob ey the zero-torsion condition ( ~ r x ) y = ( ~ r y ) x , leading to the structure equations v − q x − q 1 u = 0 , q 1 x − Re( ¯ q u ) = 0 . (5.17) Prop osition 4. Thr ough the fr ame structur e e quations (5.16) and (5 . 1 7), the U (1) - c ovariants u, v , q of ~ r ( x, y ) ar e r elate d by the Hamiltonian e quations u y = H ( v ) , v = J ( q ) , (5.18) with J = −I − 1 R = −R ∗ I − 1 , wher e H = D x − iuD − 1 x Im( u C ) and I = − i ar e a p air of c omp atible Hamiltonian op e r ators with r es p e ct to u , and R = HI − 1 = i ( D x + u D − 1 x Re( u C )) and R ∗ = I − 1 H = iD x − uD − 1 x Re( iu C ) ar e the asso ci a te d r e cursion op er ators. This leads to a main preliminary geometric result. 27 Lemma 1 . L et u b e an arbitr ary c omplex-value d function of x and y , and deﬁne v = H − 1 ( u y ) = i R − 1 ( u y ) , q = J − 1 ( v ) = −R − 2 ( u y ) , (5.19) v 1 = D − 1 x Im( ¯ v u ) , q 1 = R e( ¯ q u ) (5.20) in terms of the function u and the formal invers e op e r ators H − 1 and J − 1 , wher e R − 1 is the formal inve rse of the r e cursion op er ator R = H i = i J . Then the matrix e quations ( 5.6) and the ve ctor e quations (5.13) c on stitute a c onsistent line ar system of 1st or d e r PDEs that determine a surfac e ~ r ( x, y ) with a p ar al lel fr aming of the y = const . c o or d i n ate li n es for which x is the a r clength. F or a g iven function u ( x, y ), the resulting surface ~ r ( x, y ) and frame e E ( x, y ) are unique up t o Euclidean isometries. VI. SURF A C E F LO WS AND 2+1 SO LITON EQUA TIONS The deriv a tion of bi-Hamiltonian soliton equations from non-stretc hing space curv e ﬂo ws in Theorem 1 and their geometrical corresp ondence to in tegrable vector mo dels and dynam- ical map equations in Theorems 3 and 4 will no w b e generalized to geometrically analogous surface ﬂows in R 3 . This will mean we consider surfaces that a re non-stretc hing along one co ordinate direction y et stretc hing in all transv erse directions. (No assumptions will b e place on the top ology of the surfa ce.) Suc h surfa ce ﬂo ws ~ r ( t, x, y ) can b e written naturally in terms of a parallel fra me (5.2)–(5.4) adapted to t he non-stretch ing co ordinate lines, for whic h x will b e the arclength co or dinat e and y will b e a transv erse co ordinate, b y an equation of motion ~ r t = h 1 e E 1 + h 2 e E 2 + h 3 e E 3 , | ~ r x | = 1 , (6.1) with h 1 = D − 1 x Re( ¯ uh ) , h 2 + ih 3 = h. (6.2) Here h 1 and h resp ectiv ely determine the comp onents of t he ﬂo w that are ta ngen tia l and orthogonal to the non-stretc hing x -co ordinate lines on the surface. The relatio n (6.2) imp oses the non- stretching prop ert y b y which the ﬂow (6.1) preserv es the lo cal ar clength ds = | ~ r x | dx along these co ordinate lines, with u = u 2 + iu 3 giv en by t he comp onen ts of the para llel frame connection matrix with resp ect to x fro m (5.7). 28 The corresp onding evolution of the parallel fra me is g iven by the same matrix equations (2.26)–(2.2 7 ) tha t gov ern a pa r a llel framing of non- stretching ﬂows of space curv es. Equiv- alen tly , in the complex v ariable notation (2.29) for the comp onen ts of the ev olution matrix (2.27), the ev olutio n equations on the frame v ectors consist of e E 1 t = R e(  ( e E 2 − i e E 3 )) , e E 2 t + i e E 3 t = −  e E 1 +  1 ( e E 3 − i e E 2 ) , (6.3) where  = D x h +  1 u,  1 x = Re( ¯ uh ) (6.4) are g iv en in terms of h and u . These equations hav e the follow ing Hamiltonian interpre tatio n. Prop osition 5. The r esp e ctive evolution e quations ( 6.1) and (6.3) for the surfac e and the p ar a l lel fr ame ar e r elate d by h = J − 1 (  ) = R − 1 ( i ) = i R ∗− 1 (  ) (6.5) wher e J − 1 is a form a l Hami l toni a n op e r ator c omp atible with the Hamiltonian p air H and I , and wher e R − 1 , R ∗− 1 ar e inverse r e cursion op e r ators (cf. Prop osition 4 ). Hence, surface ﬂo ws of the type (6.1 )–(6.2) can b e expressed in terms of the complex scalar v ariable  . This v ariable also determines the resulting ev olution of t he frame connection matrices (5.7) thro ugh the pair of zero-curv a t ure equations U t − W x + [ U , W ] = 0 (6.6) V t − W y + [ V , W ] = 0 (6.7) whic h express the compatibility b etw een the F renet equations (5.6) and the evolution equa- tions of the parallel frame in matrix f orm (2 .26)–(2.27). F rom (6.6) we see u satisﬁes the same ev olutio n equation (3.3 ) in terms of  a s holds for non- stretc hing space curv e ﬂo ws in Prop osition 1. W e then ﬁnd tha t the ev o lution equation obtained fro m (6 .7 ) ho lds iden ti- cally as a consequence of Lemma 1. This result leads to the following Hamiltonian structure deriv ed from surface ﬂows (6.1) – (6.2). Lemma 2. Th e Hamiltonian structur e (3.5)–(3.7) for 1+ 1 ﬂows gen er alizes to 2+1 ﬂows on u ( t, x, y ) given by Hamiltonians of the form H = Z Z C H ( x, y , u, ¯ u, u x , ¯ u x , u y , ¯ u y , u xx , ¯ u xx , u xy , ¯ u xy , u y y , ¯ u y y , . . . ) dxdy (6.8) 29 (wher e C denotes the c o or din a te domain of ( x, y ) ). In p articular,  = δ H /δ ¯ u (6.9) yields a 2+1 Ham i l toni a n evolution e quation u t = H (  ) , H = D x − iuD − 1 x Im( u C ) , (6.10) c orr es p onding to a surfac e ﬂow given by (6.1), (6.2), (6 . 5). Since the Hamiltonian op erator H do es not con tain the y co ordinate, it ob viously has y - translation symme try . Hence starting fro m − u y ∂ /∂ u , there will b e a hierarc hy of commu ting Hamiltonian v ector ﬁelds i ( n ) ∂ /∂ u = −R n ( u y ) ∂ /∂ u, n = 0 , 1 , 2 , . . . (6.11) where  ( n ) = δ H ( n ) /δ ¯ u = R ∗ n ( iu y ) , n = 0 , 1 , 2 , . . . (6.12) are deriv atives of Hamiltonian densities H ( n ) . An explicit expression for these densities can b e derive d by applying the scaling symmetry metho ds in [1, 5]. Theorem 6. T h e r e cursion op er ator R = HI − 1 = i ( D x + uD − 1 x Re( u C )) pr o duc es a hi e r ar chy of in te g r able b i - Ham iltonian 2 +1 ﬂows u t = H ( δ H ( n − 1) / ¯ u ) = I ( δ H ( n ) /δ ¯ u ) , n = 1 , 2 , . . . (6.13) (c al le d the + n ﬂo w ) in terms of the c omp atible Hamiltonian op er ators H and I , with the Hamiltonians H ( n ) = R R C H ( n ) dxdy given by H ( n ) = − 2 1 + n D − 1 x Re( ¯ u R n +1 ( u y )) , n = 0 , 1 , 2 , . . . (6.14) (mo dulo total x, y -d erivatives). A t the b ottom of this hierarc h y , H (0) = Re( i ¯ uu y ) , δ H (0) /δ ¯ u = iu y =  (0) (6.15) yields the +1 ﬂo w − iu t = u xy + ν 0 u, ν 0 x = | u | | u | y . (6.16) 30 This is a 2 + 1 nonlo cal bi-Hamiltonian NLS equation whic h w as ﬁrst derive d f rom Lax pair metho ds by Zhak a ro v [29] and Strac han [26]. Next in the hierarc h y is H (1) = Re( ¯ u x u y ) − 1 2 ν 0 | u | 2 , δ H (1) /δ ¯ u = − ( u xy + ν 0 u ) =  (1) . (6.17) This yields the +2 ﬂo w − u t = u xxy + | u | 2 u y + ν 0 u x + iν 1 u, ν 1 x = Im( ¯ u y u x ) (6.18) whic h is a 2+1 nonlo cal bi-Hamiltonia n mKdV equation. It can b e written in the equiv a lent form − u t = u xxy + ( ν 0 u ) x + iν 2 u, ν 2 x = Im( ¯ uu xy ) (6.19) studied in work of Calogero [8] and Strachan [27]. These 2+1 ﬂow equations hav e the following in tegrability prop erties. Prop osition 6. The hie r ar chy (6.13)–(6 .14) displ a ys U (1) -in varianc e under p hase r otations u → e iλ u and homo gen eity under sc alings x → λx , y → λy , u → λ − 1 u , with t → λ 2+ n t for the + n ﬂow, wher e the sc aling weight of H ( n ) is − 3 − n . Each of the evolution e quations (6.13) in this hier ar chy ad m its the c onstants of motion D t Z Z C i ¯ uu y dxdy = 0 , D t Z Z C ¯ u x u y − 1 2 ν 0 | u | 2 dxdy = 0 , . . . (6.2 0 ) and D t Z Z C | u | 2 dxdy = 0 , D t Z Z C i ¯ uu x dxdy = 0 , D t Z Z C | u x | 2 − 1 4 | u | 4 dxdy = 0 , . . . (6.21) c omprisin g , r esp e ctively, al l of the 2+1 Hamiltonian s (6.14) plus the 1+1 Hamiltonians (3.18) extende d to two sp atial dim ensions (under suitable b ound a ry c onditions dep ending on the c o or d i n ate do m ain C of ( x, y ) ). A dditional ly, these evolution e quations (6.13) e ach admit the c orr esp ond i n g Hamiltonian symme tries − u y ∂ /∂ u, − i ( u xy + ν 0 u ) ∂ /∂ u , . . . (6.22) plus iu∂ /∂ u, − u x ∂ /∂ u, − i ( u xx + 1 2 | u | 2 u ) ∂ /∂ u , . . . . (6.23) 31 W e note the constan ts of motion (6.2 1) and symmetries (6 .2 3) are inherited from the 1+1 inte gra bilit y prop erties in Theorem 1 as a consequence of the fa ct that the Hamilto - nian phase-rotatio n v ector ﬁeld iu ∂ /∂ u (whic h generates the hierarch y of 1 +1 ﬂow s (3.17)) comm utes with the Hamiltonian y -translation v ector ﬁeld − u y ∂ /∂ u (whic h generates the hierarc h y of 2+1 ﬂo ws (6.13)). VI I. 2+1 VECTOR MODELS AND D YNAMIC AL MAPS Eac h ev olution equation (6 .13) in the hierarc h y presen ted in Theorem 6 determines a surface ﬂo w ~ r ( t, x, y ) and a corresp onding 2+1 v ector mo del for S ( t, x, y ) = ~ r x = e E 1 through the frame ev olution equations (6.3 ) in a similar manner to the deriv ation of ﬂo ws of space curv es and 1+1 ve ctor mo dels. The + 1 ﬂow yields the geometric S O (3) ve ctor mo del known a s the M-I equation [19, 20, 22] S t = S ∧ S xy + v 1 S x = ( S ∧ S y + v 1 S ) x , v 1 x = − S · ( S x ∧ S y ) , (7.1) whic h is a 2+1 integrable generalization of the S O (3) Heisen b erg mo del. It corresp onds to the surface ﬂow ~ r t = ~ r x ∧ ~ r xy + v 1 ~ r x , v 1 x = − ~ r x · ( ~ r xx ∧ ~ r xy ) , | ~ r x | = 1 . (7.2) This ﬂo w equation describes the motion of a sheet of non-stretc hing ﬁlaments in Euclidean space, in analogy with the form of the vortex ﬁlamen t equations (2.8). Some prop erties of the mo del (7.1) ha ve b een studied recen tly in [9, 3 0 ]. The +2 ﬂow pro duces a 2 +1 integrable generalization of the geometric S O (3) mKdV mo del, − S t = S xxy + (( S x · S y + v 0 ) S − v 1 S ∧ S x ) x , v 0 x = | S x || S x | y , (7.3) (the so- called M-XXIX equation [22]) whic h describ es the surface ﬂow − ~ r t = ~ r xxy + ( ~ r xx · ~ r xy + v 0 ) ~ r x − v 1 ~ r x ∧ ~ r xx , v 0 x = | ~ r xx || ~ r xx | y , | ~ r x | = 1 . (7.4) Eac h of these surface ﬂows ~ r ( t, x, y ) in R 3 geometrically corr esp onds to a dynamical map γ ( t, x, y ) o n the unit sphere S 2 ⊂ R 3 through extending the iden tiﬁcations ( 3.41) and (3 .42) as f ollo ws: S y ↔ γ y , ∂ y + S ( S y · ) ↔ ∇ y , (7.5) 32 S xy + ( S x · S y ) S ↔ ∇ x γ y = ∇ y γ x , (7.6) S ∧ S xy ↔ J ∇ x γ y = J ∇ y γ x . (7.7) The +1 and +2 ﬂo ws thereb y yield, resp ective ly , γ t = J ∇ x γ y + v 1 γ x , v 1 x = g ( γ x , J γ y ) , (7.8) and − γ t = ∇ 2 x γ y + | γ x | 2 g γ y − v 1 J ∇ x γ x − v 2 γ x , v 2 x = g ( γ y , ∇ x γ x ) , (7.9) whic h are new nonlo cal 2+1 in tegrable generalizations of the Sc hr¨ odinger map equation (2.4) on S 2 and the mKdV map equation (3.4 4) on S 2 . The complete hierarc h y of vec tor mo dels and dynamical map equations in 2+1 dimensions can b e written do wn in the same manner as in 1+1 dimensions (cf. Theorems 3 a nd 4) b y means of the spin v ector recursion op erator (4.24) and its Ha milto nian factorization (4 .3 2). In particular, the obv ious y -tra nslation inv ariance of this op erator provides t he geometric origin fo r the 2+1 g eneralization of the Heisen b erg spin mo del and the mKdV spin mo del. Theorem 7. (i) The bi- Ham iltonian ﬂows (6.1 3) c orr e s p ond to a hier ar chy of inte gr able S O (3) -in variant ve ctor mo dels in 2+ 1 dimensio n s S t =  S ∧ D x − S x D − 1 x ( S ∧ S x ) ·  n S y = f ( n ) , n = 1 , 2 , . . . (7.10) which ar e g e ometric al ly e quival e nt to 2 + 1 dimensional dynamic a l maps γ into the 2-sp her e S 2 ⊂ R 3 γ t =  J ∇ x − γ x D − 1 x g ( J γ x , )  n γ y = F ( n ) , n = 1 , 2 , . . . . (7.11) (ii) Each ve ctor mo del and dynamic al map in the hier ar chy (7.10)– ( 7 .11) p osse sses a set of p olynomial c onstants of m otion that c orr esp ond to al l of the Hamiltonians (6.14 ) for the +1 , +2 , . . . ﬂows (6.13), i. e. H (0) = RR C H (0) dxdy , H (1) = RR C H (1) dxdy , etc., as obtaine d fr om the Hamiltonia n densities (mo dulo total x, y -derivatives ) (1 + n ) H ( n ) = − D − 1 x ( S x · D x f ( n +1) ) = − D − 1 x g ( γ x , ∇ x F ( n +1) ) , n = 0 , 1 , 2 , . . . (7.12) given in terms of the e quations of motion (7.10) for S ( t, x ) and ( 7.11) for γ ( t, x ) . In addi- tion, the ve ctor m o dels (7.10) and dynamic al maps (7.11 ) e ach p o s sess two non-p olynomial 33 c onstants o f mo tion H ( − 1) = R R C H ( − 1) dxdy and H ( − 2) = R R C H ( − 2) dxdy explic i tly given by H ( − 2) = 1 2 ξ ( S ) · ( S ∧ S y ) = 1 2 g ( ξ ( γ ) , J γ y ) , (7.13) H ( − 1) = 1 2 S x · S y + 1 2 v 1 ξ ( S ) · ( S ∧ S x ) = 1 2 g ( γ x , γ y ) + 1 2 v 1 g ( ξ ( γ ) , J γ x ) , (7.14) wher e ξ ( γ ) is a ve ctor ﬁeld with c ovariantly-c on s tan t d iver genc e div g ξ ( γ ) = 1 at al l p oints γ ∈ S 2 , an d wh e r e ξ ( S ) is an ana lo gous ve ctor function satisfying S · ξ ( S ) = 0 , ∂ ⊥ S · ξ ( S ) = 1 , i n terms of the c omp onent-wise gr adient op er ator ∂ ⊥ S = ∂ S − S ( S · ∂ S ) with pr op erties (4.57). Thes e Hamiltonian d ensities (7.13)–(7.1 4) c orr esp ond to two c omp atible nonlo c al Hamiltonian structur es f o r the +0 ﬂow u t = E ( δ H ( − 2) / ¯ u ) = H ( δ H ( − 1) /δ ¯ u ) = − u y (7.15) with (c f . Lemma 1 ) − δ H ( − 1) / ¯ u = R ∗− 1 ( iu y ) = v , − δ H ( − 2) / ¯ u = R ∗− 2 ( iu y ) = − iq . (7.16) (iii) In terms of the Hamiltonian densi ties (7.12 ) , (7.1 3) and (7.14), a l l the 2+1 ve ctor mo dels (7.10) a n d dynamic al map e quations (7.1 1) have the bi-Hamiltonian structur e S t = − S ∧ ( δ H ( n − 2) /δ S ) = D x ( δ H ( n − 3) /δ S + S D − 1 x ( S x · δ H ( n − 3) /δ S )) (7.17) = f ( n ) , n = 1 , 2 , . . . and γ t = − J ( δ H ( n − 2) /δ γ ) = ∇ x ( δ H ( n − 3) /δ γ ) + γ x D − 1 x g ( γ x , δ H ( n − 3) /δ γ ) (7.18) = F ( n ) , n = 1 , 2 , . . . given by the r esp e ctive p airs ( 4 .42) and (4.45) of c o m p atible Hamiltonian op er ators. Remark: Explicit bi-Hamiltonian formulations for the 2 + 1 generalization of the Heisen- b erg spin mo del (7.1) a nd the g eometrically corresp onding new 2+1 in tegrable Sc hr¨ o dinger map (7.8) are giv en by S t = − S ∧ ( − S xy + v 1 S ∧ S x ) = D x ( S ∧ S y + S D − 1 x ( S x · ( S ∧ S y ))) (7.19 ) and γ t = − J ( −∇ y γ x + v 1 J γ x ) = ∇ x ( J γ y ) + γ x D − 1 x g ( γ x , J γ y ) (7.20) 34 where δ H ( − 2) = δ S · ( S ∧ S y ) = g ( δ γ , J γ y ) (7.21) δ H ( − 1) = δ S · ( − S xy + v 1 S ∧ S x ) = g ( δ γ , −∇ y γ x + v 1 J γ x ) (7.22) yield the resp ectiv e deriv atives of the non-p olynomial Hamiltonian densities ( 7 .13) a nd (7.14). These tw o densities in addition to all the p olynomial densities ( 7 .12) give a set of constan ts of motion for the Hamiltonian equations (7.19 ) a nd (7.20). In pa r t icular, the ﬁrst f o ur constan ts of motion are explicitly given by the integrals D t Z Z C ξ ( S ) · ( S ∧ S y ) dxdy = D t Z Z C g ( ξ ( γ ) , J γ y ) dxdy = 0 , (7.23) D t Z Z C S x · S y + v 1 ξ ( S ) · ( S ∧ S x ) dxdy = D t Z Z C g ( γ x , γ y ) + v 1 g ( ξ ( γ ) , J γ x ) dxdy = 0 , (7 .24) D t Z Z C − S · ( S x ∧ S xy ) + v 1 | S x | 2 dxdy = D t Z Z C g ( ∇ y γ x , J γ x ) + v 1 | γ x | 2 g dxdy = 0 , (7.25) D t Z Z C S xx · S xy − | S x | 2 ( S x · S y + 1 2 v 0 ) − v 1 S · ( S x ∧ S xx ) dxdy = D t Z Z C g ( ∇ y γ x , ∇ x γ x ) − 1 2 v 0 | γ x | 2 g − v 1 g ( J γ x , ∇ x γ x ) dxdy = 0 (7.26) under suitable b oundary conditions (where C denotes t he co ordinate domain of ( x, y )). The v ector ﬁeld ξ ( γ ) on S 2 , or equiv alently the v ector function ξ ( S ), in the non-p olynomial constan t of motio n (7.2 3 ) has the geometrical meaning o f a homothetic v ector with resp ect to the metric-normalized v olume fo r m ǫ g on S 2 , i.e. L ξ ǫ g = ǫ g . Theorem 7 is established as follo ws. In parts (i) and (ii), the deriv ation of the equations of motion (7.10) and Hamiltonians (7.12) for S ( t, x ) in v olv es com bining the ev olutio n equation (6.3) for the frame vec tor e E 1 = S with the hierarc hy ( 6 .11) for the v ariable  b y means o f the identities u ( e E 2 − i e E 3 ) = S x − iS ∧ S x , (7.27) u y ( e E 2 − i e E 3 ) = − ( S ∧ S ( S y ) + i S ( S y )) , (7.28) in addition to Re( ¯ u ˆ f ) = S x · f , Im( ¯ u ˆ f ) = ( S ∧ S x ) · f , (7.29) Re( ¯ u y ˆ f ) = S ( S y ) · ( S ∧ f ) , Im( ¯ u y ˆ f ) = S ( S y ) · f (7.30) 35 holding f o r v ectors f in R 3 , with comp onen ts ˆ f = ( e E 2 + i e E 3 ) · f , suc h that f · S = 0; here S is t he recursion op erator (4.24). Similarly , the deriv ation o f the Hamiltonian structures (7.17) in part ( iii) fo r n 6 = 1 relies on applying the previous iden tities to the bi-Hamiltonian structure (6.1 3 ) for the ﬂow equations on u ( t, x ). The n = 1 case reduces to computing the Hamiltonian deriv ativ es (7 .2 1)–(7.22), whic h is carried out in app endices B and C. Finally , all of the corresp onding results for γ ( t, x ) are an immediate consequence of the geometric iden tiﬁcations (7.5)–(7 .7) in addition to (3 .41)–(3.42). VI I I. GEOMETRIC FORMULA TIO N There is a natura l geometric formulation for the surface ﬂow s ( 6.1) corresp onding t o the 2+1 v ector mo dels (7.10) and 2+1 dynamical maps (7.11) in Theorem 7. W e b egin b y writing down the the intrinsic and extrinsic surface geometry of ~ r ( x, y ) in terms of the v ariables u, v , q , v 1 , q 1 app earing in the structure equations of the pa r a llel framing fo r the non-stretc hing x -co ordinate lines. Prop osition 7. L et ~ r ( x, y ) in R 3 b e a surfac e with a p ar al lel fr aming (5.1)–(5 . 5 ) adapte d to the x c o o r dinate lines, satisfying the structur e e quations (5.16) and (5.17) . Then , on the surfac e ~ r ( x, y ) , the inﬁni tesim al ar clength is give n by the line element ds 2 = ( dx + q 1 dy ) 2 + | q | 2 dy 2 , (8.1) and the inﬁnitesimal surfac e ar e a i s given by the ar e a element dA = | q | d x ∧ dy . (8.2) All other asp ects of the intrinsic surface geometry can b e deriv ed from the line elemen t (8.1). In particular, the 1st f undamental form (i.e. the surface metric tensor) is simply dxdx + 2 q 1 dxdy + | q | 2 dy d y , f rom whic h the Gauss curv ature can b e directly computed in terms of the x, y co o rdinates [1 2]. The extrinsic surface geometry can b e determined through the surface normal ve ctor ~ n = ~ r x ∧ ~ r y = Im( ¯ q ( e E 2 + i e E 3 )) , | ~ n | = | q | , (8.3) as giv en by the expression (5.1 3) f or the surface t a ngen t vec tor s ~ r x and ~ r y in terms of the parallel frame along the x -co ordinate lines. This normal vector (8.3) dep ends on a 36 c hoice of the transv erse co ordinate y due t o its normalizatio n factor | q | . T o pro ceed, w e use the follo wing nat ura l g eometric framing [12] that is deﬁned en tirely by the no n-stretc hing direction o n the surface and the ort ho g onal direction of t he surface unit-norma l in R 3 : e k = ~ r x = e E 1 , (8.4) e ⊥ = | ~ n | − 1 ~ n = Im( e iψ ( e E 2 + i e E 3 )) , (8.5) ∗ e k = | ~ n | − 1 ~ n ∧ ~ r x = R e( e iψ ( e E 2 + i e E 3 )) , (8.6) where ψ = arg( q ) . (8.7) (Here ∗ denotes the Ho dge dual acting in the tangent plane at each p oint on the surface.) Note this f r a me (8.4)–(8 .6 ) diﬀers fr o m a parallel frame b y a rotat io n through the angle (8.7) applied to the frame v ectors in t he normal plane relativ e to the x -co o rdinate lines, with e k and ∗ e k b eing an orthog onal pair of unit-ta ngen t v ectors on the surface ~ r ( x, y ), and e ⊥ b eing a unit-normal v ector for the surface. The F r enet equations of the frame e k , ∗ e k , e ⊥ directly enco de the extrinsic geometry of the surface ~ r ( x, y ). In matrix notation, the F renet equations with resp ect to the x, y co ordinates are giv en by      e k ∗ e k e ⊥      x =      0 α β − α 0 γ − β − γ 0           e k ∗ e k e ⊥      ,      e k ∗ e k e ⊥      y =      0 µ ρ − µ 0 σ − ρ − σ 0           e k ∗ e k e ⊥      , (8.8 ) where α = ∗ e k · e k x = Re( ue iψ ) , β = e ⊥ · e k x = Im( u e iψ ) , (8.9) γ = e ⊥ · ∗ e k x = | q | − 1 Im( q x e iψ ) = − ψ x (8.10) and µ = ∗ e k · e k y = R e( v e iψ ) = q 1 α + | q | x , ρ = e ⊥ · e k y = Im ( v e iψ ) = q 1 β − | q | ψ x , (8.11) σ = e ⊥ · ∗ e k y = | q | − 1 Im(( q y + iv 1 q ) e iψ ) = v 1 − ψ y (8.12) are obtained through the relations (5.8), (5.9), (5.1 4) together with the structure equations (5.16), (5 .1 7). With resp ect to the x, y co ordinat es on the surface, t he scalars α and µ are kno wn as the ge o desic c urvatur es ; β a nd ρ as the norma l curvatur es ; γ a nd σ as the r elative torsions [12 ]. 37 Prop osition 8. F or a surfac e ~ r ( x, y ) with a p ar a l lel f r aming ( 5.1)–(5.5) adapte d to the x c o o r dinate line s , satisfying the s tructur e e quations (5.16) and (5.17), the 2nd f und a m ental form is given by Π = ( ~ r x dx + ~ r y dy ) · ( e ⊥ x dx + e ⊥ y dy ) = − ( β d xdx + 2 ρdxdy + ( σ − q 1 γ ) | q | d y dy ) . (8.13) The c omp one n ts of Π with r esp e ct to the surfac e tangent fr ame e k , ∗ e k yield the extrinsic curvatur e s c alars k 11 = e k · D k e ⊥ = − β , (8.14) k 12 = e k · D ∗k e ⊥ = k 21 = ∗ e k · D k e ⊥ = − γ , (8.15) k 22 = ∗ e k · D ∗k e ⊥ = −| q | − 1 ( σ − q 1 γ ) , (8.16) wher e D k = e k ⌋ D = D x and D ∗k = ∗ e k ⌋ D = | q | − 1 ( D y − q 1 D x ) denote the pr oje ctions of the total exterior derivative D on the surfac e in the dir e ctions tangential and ortho gonal to the x -c o or dinate line s . All asp ects of the extrinsic surfa ce geometry of ~ r ( x, y ) can b e determined fro m the ex- trinsic curv ature matrix   k 11 k 12 k 21 k 22   . In particular, the mean curv ature of the surface [1 2] H = ( k 11 + k 22 ) / 2 (8.17) is given by the no r ma lized tr a ce of the extrinsic curv ature mat r ix. No w, in terms of the frame v ectors ( 8.4)–(8.6), an y surface ﬂo w (6.1)–(6 .2) in whic h the x -co ordinate lines are non-stretc hing can b e written in the form ~ r t = ae k + b ∗ e k + ce ⊥ , | ~ r x | = 1 (8.18) with a = D − 1 x Re( ¯ uh ) , b = Re( e iψ h ) , c = Im( e iψ h ) . (8.19) Through Prop osition 5, w e then obtain a hierarc h y of ﬂo ws on ~ r corresp onding to the bi- Hamiltonian 2+1 ﬂows on u in Theorem 6, as g iv en by b ( n ) + ic ( n ) = − e iψ R n − 1 ( u y ) , a ( n ) = − D − 1 x Re( ¯ u R n − 1 ( u y )) (8.20) 38 in terms of the recursion op erator R = i ( D x + u D − 1 x Re( u C )). Moreov er, these co eﬃcien ts (8.20) ha v e a geometrical form ulation deriv ed from the op erator iden tit y e iψ R e − iψ = iD x + ψ x + iue iψ D − 1 x Re( ue iψ C ) = iD x − γ − ( β − iα ) D − 1 x Re(( α + iβ ) C ) (8.21) com bined with e iψ u y = ( e iψ u ) y − iψ y e iψ u = α y + β ψ y + i ( β y − αψ y ) . (8.22) This leads to the follo wing geometric coun terpart of Theorem 7. Theorem 8. The inte gr able 2+1 ve ctor mo d e ls (7.10) and inte gr able 2+1 dynami c al maps (7.11) c orr esp ond to a hi e r ar chy of surfac e ﬂows in R 3 , ~ r t = a ( n − 1) e k + b ( n − 1) ∗ e k + c ( n − 1) e ⊥ , | ~ r x | = 1 , n = 1 , 2 , . . . (8.23) with ge ometric c o eﬃc ients b ( n ) + ic ( n ) = −P n − 1 ( α y + β ψ y + i ( β y − α ψ y )) , a ( n ) = D − 1 x ( αb ( n ) + β c ( n ) ) , (8.24) given in terms of the r e cursion op er ator P = iD x − γ − ( β − iα ) D − 1 x Re(( α + iβ ) C ) . (8.25) Her e α an d β ar e the ge o desic curvatur e and norm al curvatur e of the no n-str etching x - c o o r dinate li n es on the surfac e ~ r ( t, x, y ) , γ is the r elative torsion of these line s , a nd ψ = − R γ dx . Remark: The b ottom ﬂow in this hierar ch y can b e written in the alternativ e form b (0) = − ρ, c (0) = µ, a (0) = D − 1 x ( β µ − αρ ) = v 1 (8.26) b y means of the relation P − 1 ( α y + β ψ y + i ( β y − α ψ y )) = ρ − iµ (8.27) expresse d in terms of the geo desic curv ature µ and normal curv ature ρ of the y -co ordinate lines on the surface ~ r ( t, x, y ). This relation (8.27 ) is obtained f r om − ie iψ v = P − 1 ( e iψ u y ) whic h is a straightforw ard consequence o f Lemma 1. 39 Geometric prop erties of the surface ﬂows in the hierarc hy (8.2 3) can b e straigh tforwardly deriv ed fr o m the results in Prop ositions 7 and 8 combine d with the explicit ev olution equa- tions for the v ariables q and v (as determined b y Lemma 1). In terms o f the surface ﬂow equation ( 8 .18)–(8.19) and the op erator iden tity (8.21), these evolution equations are giv en b y iv t = w 1 v + e − iψ ( D y + iσ ) P ( b + ic ) , (8.28) q t = aq x − iq w 1 + e − iψ ( D y − q 1 D x + i ( σ − q 1 γ ))( b + ic ) , (8 .29) with w 1 x = Re(( α − iβ ) P ( b + ic )) , q 1 x = | q | α = − Re(( α − iβ ) P ( ρ − iµ ) ) , (8.30) from whic h w e obta in the ev olution of the geo desic curv atur e, normal curv ature, and relativ e torsion of the x -co ordinate lines: α t = D x ( aα ) + Re( D 1 2 ( b + ic )) + β Im( D 2 ( b + ic )) , (8 .3 1) β t = D x ( aβ ) + Im ( D 1 2 ( b + ic )) − α Im( D 2 ( b + ic )) , (8.32) γ t = D x ( aγ + Im( D 2 ( b + ic ))) − Re(( β + iα ) D 1 ( b + ic )) , (8.33) where D 1 = D k − ik 12 , D 2 = D ∗k − ik 22 (8.34) are a pair of geometric deriv ativ e op erato r s asso ciated with the x -co o r dinate lines on the surface (cf. Prop osition 8). Similarly , w e ﬁnd the area elemen t of the surface has the ev o lution ( dA ) t = a ( dA ) x + Re( D 2 ( b + ic )) d A = ε dA + L X dA (8.35) whic h consists of an inﬁnitesimal change due to the t angen t ia l part of the surface ﬂow X = ae k + b ∗ e k plus a m ultiplicativ e expansion/con traction fa cto r ε = 2 H c related to the mean curv ature (8.17) o f the surface through the no rmal part of the surface ﬂow L e ⊥ dA = 2 H dA . These dev elopmen ts now lead to the fo llo wing geometric results. Theorem 9. Under e a ch ﬂow n = 1 , 2 , . . . in the hier ar chy (8 . 2 3), the surfac e ~ r ( t, x, y ) is non-str etching the x -c o o r dinate di r e ction while str etching in al l tr a nsverse dir e ctions, such that surfac e a r e a lo c al ly exp ands/c ontr acts by the dynamic al factor ε ( n ) = 2 H Im( P n − 1 ( ρ − iµ )) 40 wher e H i s the me a n curvatur e (8.17) of the surfac e. The ge o desic curvatur e α , normal curvatur e β , and r elative torsion γ = − ψ x of the x -c o or dinate lin es in e a ch surfac e ﬂow (8.23) satisfy the inte gr able system of e volution e q uations α t + iβ t = D x ( a ( n − 1) ( α + iβ )) − D 1 2 P n − 1 ( ρ − iµ ) − ( β − iα )Im( D 2 P n − 1 ( ρ − iµ )) (8.36) ψ t = a ( n − 1) ψ x + a ( n ) + Im( D 2 P n − 1 ( ρ − iµ )) (8.37) in terms of the r e cursion op er ator (8.25) and the p ai r of ge om etric op er ators (8.3 4 ), with the ge o desic curvatur e µ and normal curvatur e ρ of the y -c o or dinate li n es g iven by (8.2 7 ), and wher e a ( n ) = − D − 1 x Re(( α − iβ ) P n ( ρ − iµ )) (8.38 ) yields (mo d ulo total x, y -derivatives ) a set of non-trivial c onstants of motion R R C a (2) dxdy = 1 2 R R C αβ y − β α y − ψ y ( α 2 + β 2 ) dxdy , R R C a (3) dxdy = 1 2 R R C α x α y + β x β y + ψ x ψ y ( α 2 + β 2 ) + ψ x ( α y β − β y α ) + ψ y ( α x β − β x α ) dxdy , etc. for the system (8.36) – (8.37). (Her e C denotes the c o or dinate doma in of ( x, y ) ). W e conclude by p oin ting o ut that the surface ﬂows (8.23) in Theorem 8 and the corre- sp onding integrable systems (8.3 6)–(8.37) in Theorem 9 pro vide a geometric realization for the 2 + 1 v ector mo dels (7.10) and 2+1 dynamical maps (7.11) in Theorem 7. Ex. 1 : The +1 surface ﬂow is giv en b y ~ r t = v 1 ~ r x + ρ ~ r x ∧ ˆ n + µ ˆ n, | ~ r x | = 1 , (8.39) with v 1 x = β µ − αρ , where ˆ n = p | ~ r y | 2 − ( ~ r x · ~ r y ) 2 − 1 ~ r x ∧ ~ r y denotes the surface unit-normal. This ﬂo w is a geometric realizatio n o f the 2+1 Heisen b erg mo del (7.1), cor r esp onding to the new in tegrable 2+1 g eneralization of the Sc hr¨ oding er map (7.8). Ex. 2 : Similar ly , a geometric realization of the 2+ 1 mKdV v ector mo del (7.3) and t he corresp onding mKdV map (7.9) is provided by the +2 surface ﬂow − ~ r t = ν 0 ~ r x + ( α y + ψ y β ) ~ r x ∧ ˆ n + ( β y − ψ y α ) ˆ n, | ~ r x | = 1 , (8.40) with ν 0 x = α α y + β β y . The corresp onding geometric in tegrable systems on t he geo desic curv ature α , normal curv ature β , a nd relativ e tor sion γ = − ψ x of the non-stretc hing x -co ordinate lines in these 41 surface ﬂows (8.39) and (8.40) a r e resp ectiv ely giv en b y α t + iβ t = v 1 ( α x + iβ x ) − D 1 2 ( ρ − iµ ) + ( α + iβ )( µβ − ρα + i Im ( D 2 ( ρ − iµ ))) (8 .41) ψ t = v 1 ψ x − ν 0 + Im( D 2 ( ρ − iµ )) (8.42) and α t + iβ t = − ν 0 ( α x + iβ x ) − D 1 2 ( α y + β ψ y + i ( β y − α ψ y )) − ( α + iβ )  1 2 ( α 2 + β 2 ) y − i Im( D 2 ( α y + β ψ y + i ( β y − α ψ y )))  (8.43) ψ t = − ν 0 ψ x + ν 2 + Im( D 2 ( α y + β ψ y + i ( β y − αψ y ))) (8.44) with ν 2 x = α β xy − β α xy − ψ y ( αα x + β β x ) − ψ x ( αα y + β β y ) − ψ xy ( α 2 + β 2 ). IX. CONCLUDIN G REMARKS There are some directions in whic h to extend the g eometrical corresp ondence among in tegrable v ector mo dels, Hamiltonian curv e and surface ﬂo ws, and bi-Hamiltonian soliton equations presen ted in this pa p er. First, it w ould b e of in terest t o generalize this corresp ondence to integrable mo dels for spin v ectors S in Euclidean spaces R N for N ≥ 3. In part icular, t he Heisen b erg mo del S t = S ∧ S xx , S · S = 1, is we ll kno wn to hav e a natural generalization where the v ector w edge pro duct and dot pro duct in R 3 are replaced b y a Lie brack et [ , ] a nd (negativ e deﬁnite) Killing form h , i of an y non-ab elian semisimple Lie a lg ebra o n R N , i.e. S t = [ S, S xx ], −h S, S i = 1. All of our results in this pap er hav e a direct extension t o such a Lie alg ebra setting by applying the metho ds of Ref.[5] to non-stretching curve ﬂo ws in semisimple Lie algebras view ed as ﬂat Klein geometries. This will lead to a large class of in tegrable 2+1 generalizations of t he Heisen b erg mo del for spin v ectors S in semisimple Lie algebras. Second, an in teresting op en pro blem is to ﬁnd a similar geometric deriv ation for non- isotropic spin v ector mo dels in R 3 as describ ed by the Landau- Lifshitz equation S t = S ∧ ( S xx + J S ), S · S = 1, where J = diag( j 1 , j 2 , j 3 ) is a constant matrix whic h measures the deviation from isotrop y . This equation ha s t w o compatible Hamiltonian structures [6], one of whic h uses the same Hamiltonian o p erat o r S ∧ that arises in the isotropic case (i.e. in the Heisen b erg mo del). The second Hamiltonian op erato r, how ev er, in v olv es the anisotropy matrix J, whic h cannot b e derive d f r om the frame structure equations for non-stretchin g 42 curv e ﬂows in Euclidean space. This suggests a non-Euclidean g eometric setting will b e needed instead. Ac kno wledgments S.C.A. is supp orted b y an NSER C researc h grant. App endix A: Pro of of Theorem 2 Let ~ r ( x ) b e a space curv e with x as t he arclength, i.e. | ~ r ( x ) | = 1, and let T , N , B b e its F renet frame (2.10). T o prov e Theorem 2, w e will en umerate the Euclidean inv aria nts o f ~ r ( x ). Firstly , since the curv ature κ = T x · N = | T x | and to r sion τ = N x · B = | T x | − 2 T xx · ( T ∧ T x ) are in v aria n t ly deﬁned in terms of the unit tangen t vec tor T = ~ r x along ~ r ( x ), so are all of their x deriv atives . This establishes part (iii) of the theorem. Secondly , these (diﬀeren tia l) inv ariants generate all p ossible scalar expressions formed out of T , T x , T xx , . . . by dot pro ducts a nd wedge pro ducts, as sho wn from a recursiv e a pplicatio n of the F renet equations (2.14)–(2 .15). Sp eciﬁcally , T x = κN , (A1) T xx = − κ 2 T + κ x N + κτ B , (A2) T xxx = − 3 κκ x T + ( κ xx − κ 3 − κτ 2 ) N + (2 κ x τ + τ x κ ) B , (A3) etc. yields T x · T x = − T xx · T = κ 2 , (A4) T xx · T x = 1 2 ( T x · T x ) x = − 1 3 T xxx · T = κκ x , (A5) T xxx · T x = 1 2 ( T x · T x ) xx − T xx · T xx = − 1 4 T xxxx · T = κκ xx − κ 4 − κ 2 τ 2 , (A6) etc. 43 and T xx · ( T ∧ T x ) = κ 2 τ , (A7) T xxx · ( T ∧ T x ) = ( T xx · ( T ∧ T x )) x = κ 2 τ x + 2 τ κκ x , ( A8) T xxx · ( T ∧ T xx ) = − κτ ( κ xx − κ 3 − κτ 2 ) + κ x (2 κ x τ + τ x κ ) , (A9) etc. whic h th us establishes parts (i) a nd (iv) of the theorem. Finally , on the other hand, since a para llel frame along ~ r ( x ) is unique up to a rigid ( x - indep enden t) rotation on the norma l ve ctors, the corresp onding comp onen ts of the principal normal T x of ~ r ( x ) giv en by u = T x · E ⊥ = κe iθ are in v ar ia n t ly deﬁned only up to rotations θ → θ + φ , with φ =const., on E ⊥ = ( N + iB ) e iθ (and likew ise for the comp onen ts of T xx , T xxx , etc.). Suc h U (1) rot ations comprise all transformations preserving t he parallel prop ert y (2.20)–(2.2 1 ) of this framing. Consequen tly , the actual inv ariants of ~ r ( x ) will corresp ond to U (1 )-in v ariants formed out of u , ¯ u , u x , ¯ u x , . . . via the r elat io ns κ = | u | , τ = (a r g u ) x . This establishes part (ii) of the theorem. App endix B: Hamiltonian structure of t he 1+1 and 2+1 Sc hr¨ odinger maps W e will ﬁrst v erif y the second Hamiltonian for the Schr¨ odinger map equation (4 .5 4) a nd its 2 +1 generalization (7.20 ) . The follow ing preliminaries concerning the tangent space structure of S 2 will b e needed. Here u, v , w will b e any triple of t angen t vec tors. (1) The metric tensor g , complex structure tensor J , and metric-norma lized v olume form ǫ g on S 2 satisfy the iden tities g ( u, J v ) = ǫ g ( u, v ) , (B1) ǫ g ( u, v ) w + ǫ g ( v , w ) u + ǫ g ( w , u ) v = 0 . (B2) (2) In lo cal co ordinates on S 2 , the metric-compatible co v ariant deriv ativ e ∇ (i.e. Riemannian connection) and the asso ciated co v ar ia n t dive rgence op erator div g are giv en b y ∇ v u = ∂ v u + Γ v u, (B3) div g u = div u + tr(Γ u ) , (B4) 44 where Γ denotes the Christoﬀel sym b ol [12] determined from g by the prop erties Γ v u = Γ u v , ( ∂ w g )( u , v ) = g ( v , Γ w u ) + g ( u, Γ w v ) . (B5) (3) F or an arbitrary v ariatio n δ γ of the map γ in to S 2 , geometrically represen ted b y a tangen t v ector ﬁeld, the v ariation of g | γ and ǫ g | γ (as induced by their ev aluation at γ ) is giv en by δ g | γ ( u, v ) = ∂ δγ g ( u, v ) = g ( u, Γ δγ v ) + g ( v , Γ δγ u ) , (B6) δ ǫ g | γ ( u, v ) = ( √ det g − 1 ∂ δγ √ det g ) ǫ g ( u, v ) = tr(Γ δγ ) ǫ g ( u, v ) . (B7) No w, consider the 1+1 Hamiltonian density (4 .52) deﬁned in terms o f a v ector ﬁeld ξ ( γ ) with cov aria n tly- constant div ergence, div g ξ ( γ ) = 1. Through the iden tity (B1), this densit y can b e written more con v enien tly as H ( − 1) = ǫ g ( ξ ( γ ) , γ x ) . (B8) Its v a riation is giv en by δ H ( − 1) = t r (Γ δγ ) ǫ g ( ξ ( γ ) , γ x ) + ǫ g ( ∂ δγ ξ ( γ ) , γ x ) + ǫ g ( ξ ( γ ) , D x δ γ ) . ( B9) In tegration b y parts on the t hir d term in (B9) yields ǫ g ( ξ ( γ ) , D x δ γ ) = D x ( ǫ g ( ξ ( γ ) , δ γ )) − ǫ g ( ∂ γ x ξ ( γ ) , δ γ ) − tr(Γ γ x ) ǫ g ( ξ ( γ ) , δ γ ) . (B10) By combining the middle terms in (B9 ) and (B10) via the iden tity (B2), w e get ǫ g ( ∂ δγ ξ ( γ ) , γ x ) − ǫ g ( ∂ γ x ξ ( γ ) , δ γ ) = ǫ g ( δ γ , γ x )div ξ ( γ ) . (B11) Lik ewise, com bining the ﬁrst term in (B9) with the third term in (B10), w e obtain ǫ g ( ξ ( γ ) , γ x )tr(Γ δγ ) − ǫ g ( ξ ( γ ) , δ γ )tr(Γ γ x ) = ǫ g ( δ γ , γ x )tr(Γ ξ ( γ ) ) . (B12) Hence, mo dulo to t a l x -deriv ativ es, (B11) and (B12) com bine to g iv e δ H ( − 1) ≡ ǫ g ( δ γ , γ x )div g ξ ( γ ) = g ( δ γ , J γ x ) whic h yields the Hamiltonia n deriv ative (4.5 3). Next, consider the 2 + 1 Hamiltonia n densities (7.13 ) and (7.14) . The previous deriv ation applies v erbatim to the density (7 .13), yielding its deriv at ive (7.21). F or t he second densit y (7.14), w e lo ok at its t w o terms H ( − 1) = H 1 + H 2 separately: H 1 = 1 2 g ( γ x , γ y ) , H 2 = 1 2 v 1 ǫ g ( ξ ( γ ) , γ x ) , (B13) 45 with v 1 x = ǫ g ( γ x , γ y ) . (B14) First, the v ariatio n of H 1 is given by δ H 1 = 1 2 g ( γ x , D y δ γ ) + 1 2 g ( D x δ γ , γ y ) + 1 2 g ( γ x , Γ δγ γ y ) + 1 2 g ( γ y , Γ δγ γ x ) = 1 2 g ( γ x , ∇ y δ γ ) + 1 2 g ( γ y , ∇ x δ γ ) (B15) through equations (B3) a nd (B5). Integration by parts on t hese terms yields g ( γ x , ∇ y δ γ ) = D y ( g ( γ x , δ γ )) − g ( δ γ , ∇ y γ x ) , g ( γ y , ∇ x δ γ ) = D x ( g ( γ y , δ γ )) − g ( δ γ , ∇ x γ y ) (B16) where, on scalar expressions, a co v ariant deriv ativ e reduces to an ordinary total deriv ative. Hence, mo dulo to t a l x, y -deriv atives , substitution of (B16) in to (B15) yields δ H 1 ≡ − 1 2 g ( δ γ , ∇ x γ y + ∇ y γ x ) = g ( δ γ , −∇ y γ x ) (B17) after w e use t he comm utativit y identit y ∇ x γ y = ∇ y γ x whic h is a consequence of the ﬁrst prop ert y in (B5). Second, the v ariatio n of H 2 is given by δ H 2 = 1 2 ǫ g ( ξ ( γ ) , γ x ) δ v 1 + 1 2 v 1 D x ( ǫ g ( ξ ( γ ) , δ γ )) + 1 2 v 1 ǫ g ( δ γ , γ x )div g ξ ( γ ) (B18) where the last tw o terms come from (B10)–(B12). T o ev a luate the ﬁr st term in (B1 8), w e start with δ v 1 x = ǫ g ( D x δ γ , γ y ) + ǫ g ( γ x , D y δ γ ) + tr(Γ δγ ) ǫ g ( γ x , γ y ) (B19) and use in tegra t io n by parts to expand the ﬁrst t w o terms, giving ǫ g ( D x δ γ , γ y ) = D x ( ǫ g ( δ γ , γ y )) − ǫ g ( δ γ , D x γ y ) − tr(Γ γ x ) ǫ g ( δ γ , γ y ) , (B20) ǫ g ( γ x , D y δ γ ) = D y ( ǫ g ( γ x , δ γ )) − ǫ g ( D y γ x , δ γ ) − tr(Γ γ y ) ǫ g ( γ x , δ γ ) . (B21) Using the iden tity (B2), w e note the Christoﬀel terms in (B19), (B20), (B21) com bine to giv e 0, while t he middle terms in (B2 0 ) and (B2 1) cancel due to D x γ y − D y γ x = ∇ x γ y − ∇ y γ x = 0 . (B22) Hence, ( B1 9) simpliﬁes to a sum of total x, y -deriv ative s δ v 1 x = D x ( ǫ g ( δ γ , γ y )) + D y ( ǫ g ( γ x , δ γ )) . 46 As a result, the ﬁrst term in the v ariation (B18) b ecomes 1 2 ǫ g ( ξ ( γ ) , γ x ) δ v 1 = 1 2 ǫ g ( δ γ , γ y ) ǫ g ( ξ ( γ ) , γ x ) + 1 2 D − 1 x ( D y ǫ g ( γ x , δ γ )) ǫ g ( ξ ( γ ) , γ x ) . (B23) In tegration b y parts on the second term in ( B2 3) yields 1 2 D − 1 x D y ( ǫ g ( γ x , δ γ )) ǫ g ( ξ ( γ ) , γ x ) ≡ 1 2 ǫ g ( γ x , δ γ ) D − 1 x D y ( ǫ g ( ξ ( γ ) , γ x )) . (B24) No w w e use the relation D y ǫ g ( ξ ( γ ) , γ x ) − D x ǫ g ( ξ ( γ ) , γ y ) = ǫ g ( ∂ γ y ξ ( γ ) , γ x ) − ǫ g ( ∂ γ x ξ ( γ ) , γ y ) + ǫ g ( ξ ( γ ) , D y γ x − D x γ y ) + ǫ g ( ξ ( γ ) , γ x )tr(Γ γ y ) − ǫ g ( ξ ( γ ) , γ y )tr(Γ γ x ) = ǫ g ( γ y , γ x )(div ξ ( γ ) + tr(Γ ξ ( γ ) )) = − v 1 x div g ξ ( γ ) obtained via the iden tities (B2) and (B22). Thus , (B24) reduces to 1 2 D − 1 x D y ( ǫ g ( γ x , δ γ )) ǫ g ( ξ ( γ ) , γ x ) ≡ 1 2 ǫ g ( γ x , δ γ ) ǫ g ( ξ ( γ ) , γ y ) + 1 2 v 1 ǫ g ( δ γ , γ x ) (B25) whic h com bines with the ﬁrst term in (B23) b y use of (B2) to giv e 1 2 ǫ g ( ξ ( γ ) , γ x ) δ v 1 ≡ 1 2 ǫ g ( ξ ( γ ) , δ γ ) ǫ g ( γ x , γ y ) + 1 2 v 1 ǫ g ( δ γ , γ x ) . (B26) Substituting (B26) in to the v ariation (B18), and using (B14), w e get δ H 2 ≡ 1 2 v 1 x ǫ g ( ξ ( γ ) , δ γ ) + 1 2 v 1 D x ( ǫ g ( ξ ( γ ) , δ γ )) + v 1 ǫ g ( δ γ , γ x ) ≡ g ( δ γ , v 1 J γ x ) . (B27) Finally , w e combine the separate v ariations (B27) and ( B1 7) to obtain δ H ( − 1) = δ H 1 + δ H 2 ≡ g ( δ γ , −∇ y γ x + v 1 J γ x ) (B28) whic h yields the Hamiltonia n deriv ative (7.2 2). App endix C : Hamiltonian structure of the 1+1 and 2+1 Heisenberg mo dels W e will next v erify the second Ha miltonian for the 1+1 Heisen b erg mo del (4.55), given b y the densit y H ( − 1) = ξ ( S ) · ( S ∧ S x ) . (C1) Here ξ ( S ) is a vec tor function, deﬁned in terms of the spin vec tor S , suc h that S · ξ ( S ) = 0 (C2) 47 and ∂ ⊥ S · ξ ( S ) = 1 , (C3) where ∂ ⊥ S = ∂ S − S ( S · ∂ S ) is a comp onen t-wise gradient op erator satisfying the prop erties (4.57). W e not e that, due to t hese prop erties, ∂ ⊥ S has a well-deﬁne d action on a n y function of S with S · S = 1. T o pro ceed, the following algebraic preliminaries will b e needed. (1) A v ariation of S consists of an arbitr a ry v ector δ S ⊥ S , i.e. S · δ S = 0. (2) The v ariation of ξ ( S ) induced b y δ S is giv en b y δ ξ ( S ) = ( δ S · ∂ ⊥ S ) ξ ( S ) . (C4) Similarly , the total x -deriv a t iv e o f ξ ( S ) is given by D x ξ ( S ) = ( S x · ∂ ⊥ S ) ξ ( S ) . (C5) (3) Since the subspace of v ectors orthogonal to S in R 3 is tw o-dimensional, δ S ∧ S x lies in the one-dimensional p erp space, so thus δ S ∧ S x = − ( δ S · ( S ∧ S x )) S. (C6) No w, the v ariation of the densit y (C1) is giv en b y δ H ( − 1) = δ ξ ( S ) · ( S ∧ S x ) + ξ ( S ) · ( δ S ∧ S x ) + ξ ( S ) · ( S ∧ D x δ S ) . (C7) In tegration b y parts on the t hir d term in (C7) yields ξ ( S ) · ( S ∧ D x δ S ) = D x ( S · ( δ S ∧ ξ ( S ))) − ξ ( S ) · ( S x ∧ δ S ) − δ S · ( D x ξ ( S ) ∧ S ) (C8) with the middle terms in (C8) a nd (C7) eac h v anishing due to (C6). Hence, mo dulo tota l x -deriv ativ es, (C7) reduces to δ H ( − 1) ≡ δ ξ ( S ) · ( S ∧ S x ) − δ S · ( D x ξ ( S ) ∧ S ) = S · ( S x ∧ ( δ S · ∂ ⊥ S ) ξ ( S ) − δ S ∧ ( S x · ∂ ⊥ S ) ξ ( S )) (C9) via ( C4) and (C5). T o simplify the terms in (C9), w e ﬁrst rewrite S x ( δ S · ∂ ⊥ S ) − δ S ( S x · ∂ ⊥ S ) = ( δ S ∧ S x ) ∧ ∂ ⊥ S = − δ S · ( S ∧ S x )( S ∧ ∂ ⊥ S ) (C10) b y means of standard ve ctor cross-pro duct iden tities in addition to iden tit y (C6). Th us, (C9) b ecomes δ H ( − 1) ≡ ( δ S · ( S ∧ S x )) S · ( − ( S ∧ ∂ ⊥ S ) ∧ ξ ( S )) (C11) 48 and w e aga in apply v ector cross-pro duct iden tities to rewrite the t erm ( S ∧ ∂ ⊥ S ) ∧ ξ ( S ) = ∂ ⊥ S ( S · ξ ( S ) ) − ξ ( S )( ∂ ⊥ S · S ) − S ( ∂ ⊥ S · ξ ( S )) . ( C12 ) Then, since S · ∂ ⊥ S = 0 = S · ξ ( S ), w e ha v e S · ( − ( S ∧ ∂ ⊥ S ) ∧ ξ ( S )) = ∂ ⊥ S · ξ ( S ) = 1 (C13) whence (C11) simpliﬁes to δ H ( − 1) ≡ δ S · ( S ∧ S x ) yielding the Hamilto nian deriv ative (4.5 8 ). The ab ov e deriv a tion carries o v er v erbatim to also v erify the ﬁrst Hamiltonia n (7.13 ) for the 2+1 Heisen b erg mo del ( 7.19). T o v erify the second Hamiltonian (7.14), we will separately consider the tw o terms in the densit y H ( − 1) = H 1 + H 2 giv en b y H 1 = 1 2 S x · S y , H 2 = 1 2 v 1 ξ ( S ) · ( S ∧ S x ) (C14) with v 1 x = S x · ( S ∧ S y ) = S · ( S y ∧ S x ) . (C15) The follo wing iden tity will b e useful: S y ∧ S x = v 1 x S (C16) holding similarly to (C6). First, the v ariatio n of H 1 is simply δ H 1 = 1 2 S x · D y δ S + 1 2 S y · D x δ S ≡ − δ S · S xy (C17) mo dulo total x, y -deriv ativ es. Next, the v ariatio n of H 2 consists of the terms δ H 2 = 1 2 ξ ( S ) · ( S ∧ S x ) δ v 1 + 1 2 v 1 D x ( δ S · ( ξ ( S ) ∧ S ) ) + 1 2 v 1 δ S · ( S ∧ S x ) (C18) as obtained fr om (C8), (C10), (C12), (C13). T o ev a lua t e the ﬁrst term in (C18), we note the v ariation of (C15) is giv en b y δ v 1 x = δ S · ( S y ∧ S x ) + S · ( S y ∧ D x δ S ) + S · ( D y δ S ∧ S x ) ≡ D y ( δ S · ( S x ∧ S )) + D x ( δ S · ( S ∧ S y )) + 2 v 1 x S · δ S 49 through (C16). Since the last term v anishes due to the ort hogonalit y δ S ⊥ S , this yields δ v 1 = δ S · ( S ∧ S y ) − D − 1 x D y ( δ S · ( S ∧ S x )) . Hence, t he ﬁrst term in ( C18 ) b ecomes 1 2 ξ ( S ) · ( S ∧ S x ) δ v 1 ≡ 1 2 δ S · ( S ∧ S y ) ξ ( S ) · ( S ∧ S x ) − 1 2 δ S · ( S ∧ S x ) D − 1 x D y ( ξ ( S ) · ( S ∧ S x )) (C19) after integration by parts. W e simplify the second t erm in (C19) b y using the relatio ns D y ( ξ ( S ) · ( S ∧ S x )) − D x ( ξ ( S ) · ( S ∧ S y )) = S · ( S x ∧ ( S y · ∂ ⊥ S ) ξ ( S ) − S y ∧ ( S x · ∂ ⊥ S ) ξ ( S )) + 2 ξ ( S ) · ( S y ∧ S x ) where, similarly to (C10) and (C12), S x ( S y · ∂ ⊥ S ) − S y ( S x · ∂ ⊥ S ) = S · ( S y ∧ S x )( S ∧ ∂ ⊥ S ) yields S · ( S x ∧ ( S y · ∂ ⊥ S ) ξ ( S ) − S y ∧ ( S x · ∂ ⊥ S ) ξ ( S )) = − v 1 x ∂ ⊥ S · ξ ( S ) , while ξ ( S ) · ( S y ∧ S x ) = 0 holds due to (C16) and (C2). Th us, we ha v e D − 1 x D y ( ξ ( S ) · ( S ∧ S x )) = ξ ( S ) · ( S ∧ S y ) − v 1 , whence (C19) simpliﬁes to 1 2 ξ ( S ) · ( S ∧ S x ) δ v 1 ≡ 1 2 δ S · ( S ∧ S y ) ξ ( S ) · ( S ∧ S x ) − 1 2 δ S · ( S ∧ S x ) ξ ( S ) · ( S ∧ S y ) + 1 2 v 1 δ S · ( S ∧ S x ) . 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Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

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