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

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrodinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.


I. INTRODUCTION AND SUMMARY
Spin systems are an important class of dynamical vector models from both physical and mathematical points of view. In physics such models describe nonlinear dynamics of magnetic materials, while in mathematics they give rise to associated geometric flows of curves where the unit tangent vector along a curve is identified with a dynamical spin vector.
A main example [14] is the Heisenberg model for the dynamics of an isotropic ferromagnet spin system in 1+1 dimensions. The geometric curve flow described by this SO(3)-invariant model corresponds to the equations of motion of a non-stretching vortex filament in Euclidean space. Remarkably, the vortex filament equations are an integrable Hamiltonian system that is equivalent to the 1+1 dimensional focusing nonlinear Schrödinger equation (NLS) through a change of dynamical variables known as a Hasimoto transformation [13].
The vortex filament equations are one example in an infinite hierarchy of non-stretching geometric flows of space curves whose equations of motion have a well-understood integrability: e.g. a Lax pair and an associated isospectral linear eigenvalue problem; an infinite set of symmetries and constants of motion; and exact solutions with solitonic properties.
This integrability structure turns out to have a simple geometric origin. In particular, all of these equations of motion are generated through a recursion operator that can be derived geometrically [10,23] from the Serret-Frenet structure equations given by a SO(3) moving frame formulation for arbitrary non-stretching curve flows in Euclidean space, with the components of the frame connection matrix providing the dynamical variables that appear in the equations of motion. More recently, these SO(3) frame structure equations have been found to geometrically encode a pair of compatible Hamiltonian operators that yield a concrete bi-Hamiltonian structure for the equations of motion of each integrable curve flow in the hierarchy [17].
The explicit bi-Hamiltonian form of the resulting equations of motion depends on a choice of the SO(3) moving frame for the underlying space curve, which determines the form of the frame connection matrix and hence yields the dynamical variables in terms of the curve. In the case of the vortex filament equations, the dynamical variables consist of the curvature invariant, κ, and the torsion invariant, τ , of the space curve, corresponding to the choice of a classical Frenet frame [12] given by the unit tangent vector, unit normal and bi-normal vectors, along the curve. Other geometrical choices of a moving frame can be made [2], since there is a SO(3) gauge freedom relating any two orthonormal frames along an arbitrary curve in Euclidean space. In particular, the Hasimoto transformation arises geometrically as a gauge transformation from a Frenet frame to a parallel frame [7], where the frame vectors in the normal space of the curve are chosen such that their derivative with respect to the arclength s along the curve lies in the tangent space of the curve. This choice of frame is unique up to rigid SO(2) rotations acting on the normal vectors by the same angle at all points along the curve, while leaving invariant the tangent vector. The corresponding pair of dynamical variables (defined by the connection matrix of a parallel frame) are naturally equivalent to a single complex-valued variable u = κ exp(i τ ds) that is determined by the curve only up to constant phase rotations u → e iφ u (where φ is independent of arclength s).
This dynamical variable u thus has the geometrical meaning [5] of a U(1) ≃ SO(2) covariant of the space curve. Importantly, the resulting Hamiltonian structure for the equations of motion looks simplest in terms of the covariant u, which directly incorporates the Hasimoto transformation, rather than using the classical invariants κ and τ .
The purpose of the present paper will be to give some new applications of these ideas to the study of integrable vector models in 1+1 and 2+1 dimensions.
Firstly, from the hierarchy of non-stretching geometric space curve flows that contains the vortex filament equations, we derive the complete hierarchy of corresponding integrable SO(3)-invariant vector models in 1+1 dimensions, along with their bi-Hamiltonian integrability structure in explicit form. In addition to the Heisenberg model, this hierarchy will be seen to contain a model that describes a spin-vector version of the mKdV equation.
Our results provide a new derivation of the Hamiltonian structure, recursion operator, and constants of motion for these models.
Secondly, we extend the derivation to a geometrically analogous class of surface flows where the surface is non-stretching in one coordinate direction while stretching in all transverse directions. Such surfaces arise in a natural fashion from a spatial Hamiltonian flow of non-stretching space curves. This generalization will be shown to give rise to a class of 2+1 dimensional NLS and mKdV soliton equations with an explicit bi-Hamiltonian structure, yielding a hierarchy of integrable SO(3)-invariant vector models in 2+1 dimensions.
In particular, this hierarchy includes 2+1 generalizations of the Heisenberg spin model and the mKdV spin model, which were found in earlier work by one of us [15,[18][19][20][21]. Our derivation here, in contrast, yields the explicit bi-Hamiltonian structure, recursion operator, and constants of motion, which are new results for these models. We also write out the corresponding surface flows explicitly in terms of geometric variables given by [12] the geodesic and normal curvatures and the relative torsion of the non-stretching coordinate lines on the surface. The surface flow arising from the 2+1 integrable Heisenberg model will be seen to describe a sheet of non-stretching vortex filaments in Euclidean space.
Lastly, we also derive an interesting geometric formulation of these results by viewing the spin vector as a dynamical map into the 2-sphere in Euclidean space. This formulation is based on the well-known geometrical equivalence between the Heisenberg model and the Schrödinger map equation in 1+1 dimensions [25]. When applied to the 1+1 and 2+1 In section V, we consider surfaces generated by a spatial Hamiltonian flow of curves with a parallel framing in Euclidean space. The underlying Hamiltonian structure is shown to arise naturally from the Frenet equations of the induced frame along the surface. This formulation is then used in section VI to study surface flows expressed in terms of the covariant variable u geometrically associated with the non-stretching space curves that foliate the surface, where the surface is stretching in all directions transverse to these curves. We show that the bi-Hamiltonian structure for 1+1 flows on u has a natural extension to 2+1 flows based on the observation that the Hamiltonian operators involve only the coordinate in the non-stretching direction on the surface. This leads to a hierarchy of 2+1 flows on u, with the starting flow given geometrically by translations in the coordinate in the transverse direction, which yields a 2+1 generalization of the NLS hierarchy.
The final two sections of the paper contain our main new results. In section VII, we use the surface Frenet equations to derive the complete hierarchies of integrable 2+1 vector models and dynamical maps arising from the 2+1 generalization of the NLS hierarchy. The derivation yields the explicit bi-Hamiltonian structure of these two hierarchies, in addition to their respective recursion operators and constants of motion. As examples, the integrable generalizations of the Heisenberg model and the mKdV spin model in 2+1 dimensions are written down in detail, as well as the corresponding new 2+1 dimensional integrable generalizations of the Schrödinger map equation and mKdV map equation. In section VIII, we work out the equations of motion for the surface flows that correspond to the previous hierarchies. These equations are obtained by means of a different framing defined in a purely geometrical fashion by the non-stretching coordinate direction on the surface and the orthogonal direction of the surface normal in Euclidean space. We also discuss aspects of both the intrinsic and extrinsic geometry of the resulting surface motions. In particular, we obtain a recursion operator, constants of motion, and explicit evolution equations formulated in terms of geometric variables given by the geodesic curvature, normal curvature, and relative torsion of the non-stretching coordinate lines on the surface. Some concluding remarks on future extensions of this work are given in section IX.

II. VECTOR MODELS AND SPACE CURVE FLOWS
We start from an arbitrary SO(3) vector model in 1+1 dimensions, where S(t, x) = (S 1 , S 2 , S 3 ) is a dynamical unit vector in Euclidean space, f is a vector function ⊥ S, and x belongs to some one-dimensional domain C. A running example will be the Heisenberg spin model There are two different ways to associate a curve flow to equation (2.1). One formulation consists of intrinsically identifying S with a map γ into the unit sphere S 2 ⊂ R 3 . Then S t and S x correspond to γ t and γ x ; ∂ x + S(S x ·) corresponds to the covariant derivative ∇ x on the sphere with respect to the tangent direction γ x ; and S∧ corresponds to the Hodge dual * = J (i.e. a complex structure on the sphere). Under these identifications, each vector model (2.1) describes a curve flow for γ(t, x) on S 2 . Ex. the Heisenberg model (2.2) corresponds to which is the Schrödinger map equation on S 2 .
Alternatively, in an extrinsic formulation, S can be identified with the unit tangent vector T along a non-stretching space curve given by a position vector r in Euclidean space, where x is the arclength along the curve r(x). Then the equation of motion of r is 6) or equivalently under which the arclength of the curve is preserved, i.e. C | r x |dx = ℓ is a constant of the motion. Ex. the Heisenberg model (2.2) corresponds to with | r x | = 1. This is the equation of motion of a non-stretching vortex filament studied by Hasimoto [13].
To proceed we first introduce a Frenet frame E along r(x). It is expressed in matrix column notation by Here N is the unit normal and B is the unit bi-normal of the space curve r(x). Note we have the relations is the curvature of r(x), and is the torsion of r(x). The Serret-Frenet equations of this frame (2.9) are given by From the equation of motion (2.1) for S we obtain the frame evolution equation where are determined by taking the t-derivative of (2.11) and substituting (2.1), followed by applying respective projections orthogonal to T, N, B.
This evolution of the frame E induces evolution equations for κ and τ through the zero- Ex. the Heisenberg model (2.2) gives the vortex filament equations in terms of the curvature and torsion [14]: (2.17) Next we perform a SO(2) gauge transformation on the normal vectors in the Frenet frame (2.9): with the rotation angle θ defined by so thus This is called a parallel framing [7] of the space curve r(x). The frame vectors (2.18) are characterized by the geometrical property that along r(x) their derivatives lie completely in the normal space (2.20) or in the tangent space (2.21). Such a frame is unique up to a rigid (x-independent) rotation θ → θ + φ, φ = const. (2.22) acting on the pair of normal vectors.
In matrix notation the Serret-Frenet equations of a parallel frame are given by are the components of the principal normal T x of r(x). The evolution of this frame which is related to U through the zero-curvature equation Note W can be determined directly from the model (2.1) via the relations (2.9) and (2.11).
It now becomes convenient to work in terms of a complex variable formalism determined by the curve 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 covariant of the curve [5] relative to the group SO(2) ≃ U(1), while |u| = κ and (arg u) x = τ are invariants of the curve.

III. BI-HAMILTONIAN FLOWS AND OPERATORS
For a general vector model (2.1) the zero-curvature equation (2.28) gives an evolution equation on u, plus an auxiliary equation relating ̟ 1 to u, x Im(̟u) in terms of u and ̟, and then we see (3.1) yields where ̟ is determined from (2.1) via the frame evolution equation (2.26).
is a Hamiltonian operator with respect to the flow variable u(t, x), whence the evolution equation (3.3) has a Hamiltonian structure holds for some Hamiltonian (iv) Composition of R and H yields a third U(1)-invariant Hamiltonian operator satisfying e iλ Ee −iλ = E| u→e iλ u . In particular, E, H, I form a compatible Hamiltonian triple.
These Propositions are a special case of group-invariant bi-Hamiltonian operators derived from non-stretching curve flows in constant-curvature spaces and general symmetric spaces in recent work [3][4][5]24]. Moreover, in the present complex variable formalism, Proposition 2 provides a substantial simplification of some main results in [17] on Hamiltonian operators connected with non-stretching curve flows in Euclidean space.
Because phase rotation on u is a symmetry of both H and I, the recursion operator R generates a hierarchy of commuting Hamiltonian vector fields given by i̟ (n) ∂/∂u = R n (iu)∂/∂u, n = 0, 1, 2, . . . (3.12) where ̟ (n) = δH (n) /δū = R * n (u), n = 0, 1, 2, . . . (3.13) are Hamiltonian derivatives, starting with which corresponds to phase-rotation iu∂/∂u. Next in the hierarchy comes followed by   At the bottom of the hierarchy, the 0 flow is given by a linear traveling wave equation u t = u x , and next the +1 flow produces the NLS equation (2.31). The +2 flow yields the complex mKdV equation which corresponds to an mKdV analog of the vortex filament equations, as obtained through the Hasimoto transformation u = κ exp(−iθ).
The evolution equations describing the 0, +1, +2, . . . flows on u each arise from geometric space curve flows corresponding to SO(3)-invariant vector models (2.1). To make this correspondence explicit, it is convenient to introduce a complex frame notation and The Frenet equations (2.14) become while from (2.27), (2.29), (3.2), the evolution of the frame is given by the equations Then any flow belonging to the general class will determine a vector model (2.1) via where f is expressed in terms of S, S x , S xx , etc. through the Frenet equations (2.23)-(2.24).
By rewritingū and hence The identifications (3.31) then directly give the SO(3) Heisenberg model (2.2), which corresponds to the non-stretching space curve flow (2.17) or equivalently expressed as a geometric flow.
Here we can rewrite the first term as with |u| 2 = |E x | 2 , and thus which can be viewed as an SO(3) mKdV model. The corresponding non-stretching space curve flow looks like This describes a geometric flow [16] − r t = 1 2 which is equivalent to the evolution (3.23) and (3.24) on the curvature and torsion of r(x).
Remark: A different geometric derivation of the mKdV model (3.37) appears in work [2] on non-stretching flows of curves in three-dimensional manifolds with constant curvature, i.e. S 3 , H 3 , R 3 , where the spin vector S is identified with the components of the unit tangent vector in a moving frame defined by parallel transport along the curve. The mKdV model also has been derived in [11] as a higher-order symmetry of the Heisenberg model by non-geometric methods.
All of these SO(3) vector models describe dynamical maps γ on the unit sphere S 2 ⊂ R 3 by means of the identifications: and thus where g denotes the Riemannian metric on the sphere S 2 (given by restricting the Euclidean inner product in R 3 to the tangent space of S 2 ⊂ R 3 ).
In particular, the SO(3) Heisenberg model yields the Schrödinger map equation (2.4) on which is a mKdV map equation on S 2 (i.e. a dynamical map version of the potential mKdV equation).
Thus, Theorem 1 provides a geometric realization of the hierarchies of integrable vector models and dynamical maps containing the Heisenberg model and the Schrödinger map as well as their mKdV counterparts.

IV. GEOMETRIC HIERARCHY OF INTEGRABLE VECTOR MODELS AND DYNAMICAL MAPS
In general, any non-stretching space curve flow (2.7) can be written in terms of a Frenet frame (2.9) by an equation of motion of the form This relation between the tangential and normal components of the motion arises due to the non-stretching property by which the motion preserves the local arclength ds = | r x |dx of the space curve if (and only if) r x · r tx = 0. As a consequence, through the Serret-Frenet equations (2.14)-(2.15), the tangent vector T = r x along the space curve obeys the equation of motion given by a linear combination of the normal and bi-normal vectors with coefficientŝ Now we consider a Hasimoto transformation (2.18)-(2.19) from the Frenet frame (2.9) to a parallel frame (3.25). The equation of motion (4.1) on r takes the form in terms of the tangential and normal components given by with these components satisfying the relation (4.2) given by where u = κe −iθ . Correspondingly, from the Frenet equations (3.28) of the parallel frame, the equation of motion (4.4) for the tangent vector T = r x has the form in terms of which encodes the normal and bi-normal componentŝ The evolution of T is thus specified by the variable ̟, while the underlying evolution of r is specified in terms of the variable h ⊥ , with h given by the non-stretching condition (4.8). From equation (4.10) these variables are related by The operator here is related to the Hamiltonian operator I = −i by the properties where R and R * are the recursion operators (3.10). Consequently, J −1 is a formal Hamiltonian operator compatible with I.
Proposition 3. The evolution (4.1) of a non-stretching space curve r(x) can be expressed in terms of a geometrical variable that determines the corresponding evolution (4.4) of the tangent vector T = r x through the relation T t = ( r t ) x . In particular, yields the normal components of the evolution vector r t in a parallel frame, where ̟ represents the frame components of T t . The curvature κ and torsion τ of r(x) correspondingly have the evolution which can be expressed in terms of the Frenet frame coefficients a, b, c of r t through the relations (4.5). functions of scalar invariants formed out of S and its x derivatives (modulo differential consequences of S · S = 1), i.e.
(iii) Its curvature κ and torsion τ satisfy geometric evolution equations in terms of invariants and differential invariants of r(x) iff ̟e iθ is a function of κ, τ , and their x derivatives, i.e. The proof of this proposition amounts to enumerating the Euclidean (differential) invariants of a space curve r(x) with an arclength parameterization x, as shown in appendix A.
We are now able to derive the entire hierarchy of SO ( as generated via the recursion operator R * = iD x − uD −1 x Re(iuC). The main step is now to establish the operator identity where S is a spin vector operator corresponding to R * , and Re(E ⊥ C) is the operator that produces a vector f in the perp space of S in R 3 when applied to the components of f with and . Hence, this yields the vector operator Theorem 3. The bi-Hamiltonian flows (3.17) on u(t, x) correspond to a hierarchy of inte- generated by the recursion operator (4.24). These models have the equivalent geometrical expressed as dynamical maps γ(t, x) into the 2-sphere S 2 ⊂ R 3 . Moreover, the Hamiltonians   Ex: n = 3 and n = 4 respectively yield a 4th order Heisenberg vector model and a 5th order mKdV vector model which are described geometrically by a 4th order Schrödinger map equation and a 5th order mKdV map equation Remark: The equations of motion of these dynamical maps γ(t, x) do not locally preserve the arclength ds = |γ x | g dx in the x direction along γ on S 2 , namely |γ x | g has a nontrivial time evolution for any of the dynamical map equations (4.26), and additionally, the total The spin vector recursion operator (4.24) has the factorization where, as shown by the results in [6], the operators D x +SS x · +S x D −1 x S x · and (S∧ ) −1 = −S∧ constitute a compatible Hamiltonian pair with respect to the spin vector variable S(t, x). By comparison, using the present geometric framework, we now directly derive the explicit bi-Hamiltonian structure of the hierarchy of vector models (4.25) through the bi-Hamiltonian flow equation (3.17) on u(t, x) in Theorem 1.
The starting point is the variational identity holding modulo total x-derivatives and modulo (differential consequences of) S · δS = 0, with the Hamiltonian densities given in the equivalent forms (3.18) and (4.27), and with ̟ (n) given by the hierarchy (3.13) generated through the recursion operator R * . To begin we derive an explicit expression for δu in terms of δS = δE by taking the Frechet derivative of the Frenet equation (3.28) for the normal vectors in a parallel frame with respect to u: TheĒ ⊥ component of (4.34) gives via E · δE ⊥ = −E ⊥ · δE due to the orthogonality of E ⊥ , E . Similarly, the E component of (4.34) yields Combining these two expressions, we obtain Integration by parts on both terms in (4.36) then yields modulo total x-derivatives. Next, after use of the relations iR * = −H and iĒ ⊥ = E ∧Ē ⊥ which follow from (3.10) and (3.26), we get Hence, equating (4.38) and (4.33) yields from which we obtain the variational relation   Theorems 1 and 3).
These results provide an explicit geometrical formulation of the abstract symplectic structure given in [28] for the Schrödinger map equation (2.4), i.e. (n = 1) and the mKdV spin model (3.37), modulo total x-derivatives, so thus in terms of a vector function ξ(S) such that To conclude, a counterpart of Theorem 3 will now be stated for the underlying space curve motions on r. with geometric coefficients given in terms of the recursion operator  x Re(iū̟ (n−1) ) = −D −1 x Im(ū̟ (n−1) ), (4.63) b (n) = Re(ie iθ ̟ (n−1) ) = −Im(e iθ ̟ (n−1) ), (4.64) c (n) = Im(ie iθ ̟ (n−1) ) = Re(e iθ ̟ (n−1) ). The results in Theorem 1 for 1+1 flows can be generalized in a natural way to 2+1 flows by considering surfaces r(x, y) that are foliated by space curves with a parallel framing in R 3 . Here y will denote a coordinate assumed to be transverse to these curves, and x will denote the arclength coordinate along the curves, so thus In this setting, a parallel frame consists of a triple of unit vectors whose derivatives along each y = const. coordinate line on the surface r(x, y) lie completely in either the tangent space or the normal plane of this line in R 3 . The explicit form of such a frame is given by the vectors 2) with where T, N, B respectively denote the unit tangent vector, unit normal and bi-normal vectors of the y = const. coordinate lines.
In matrix notation these frame vectors satisfy the Frenet equations given by are complex scalar variables, and is a real scalar variable. Note, similarly to the case of space curves, here we have which characterizes E(x, y) as a parallel frame adapted to the foliation of the surface r(x, y) by x-coordinate lines, i.e. E 1x ⊥ T , E 2x T , E 3x T . From (5.5) we see this choice of framing is geometrically unique up to rigid rotations that act on the normal vectors Under such rotations, v 1 is invariant, while u and v undergo a rigid phase rotation, To proceed we need to write down the tangent vectors of r(x, y) in terms of the parallel framing, The respective projections of r y orthogonal and parallel to r x are given by the scalar variables where, under rigid rotations (5.11) on the normal vectors in the parallel frame, q 1 is invariant while q transforms by a rigid phase rotation, q → e −iφ q.  (x, y). These variables will be seen later (cf. Propositions 7 and 8) to encode both the intrinsic and extrinsic surface geometry of r(x, y) in R 3 .
We will now show that all surfaces r(x, y) with the x coordinate satisfying the non- For a given function u(x, y), the resulting surface r(x, y) and frame E(x, y) are unique up to Euclidean isometries.

VI. SURFACE FLOWS AND 2+1 SOLITON EQUATIONS
The derivation of bi-Hamiltonian soliton equations from non-stretching space curve flows in Theorem 1 and their geometrical correspondence to integrable vector models and dynamical map equations in Theorems 3 and 4 will now be generalized to geometrically analogous surface flows in R 3 . This will mean we consider surfaces that are non-stretching along one coordinate direction yet stretching in all transverse directions. (No assumptions will be place on the topology of the surface.) Such surface flows r(t, x, y) can be written naturally in terms of a parallel frame (5.2)- (5.4) adapted to the non-stretching coordinate lines, for which x will be the arclength coordinate and y will be a transverse coordinate, by an equation of motion Here h 1 and h respectively determine the components of the flow that are tangential and orthogonal to the non-stretching x-coordinate lines on the surface. The relation (6.2) imposes the non-stretching property by which the flow (6.1) preserves the local arclength ds = | r x |dx along these coordinate lines, with u = u 2 + iu 3 given by the components of the parallel frame connection matrix with respect to x from (5.7).
The corresponding evolution of the parallel frame is given by the same matrix equations (2.26)-(2.27) that govern a parallel framing of non-stretching flows of space curves. Equivalently, in the complex variable notation (2.29) for the components of the evolution matrix (2.27), the evolution equations on the frame vectors consist of are given in terms of h and u. These equations have the following Hamiltonian interpretation. H(x, y, u,ū, u x ,ū x , u y ,ū y , u xx ,ū xx , u xy ,ū xy , u yy ,ū yy , . . .)dxdy (6.8) (where C denotes the coordinate domain of (x, y)). In particular, yields a 2+1 Hamiltonian evolution equation corresponding to a surface flow given by (6.1), (6.2), (6.5).
At the bottom of this hierarchy, This is a 2+1 nonlocal bi-Hamiltonian NLS equation which was first derived from Lax pair methods by Zhakarov [29] and Strachan [26].
These 2+1 flow equations have the following integrability properties. and comprising, respectively, all of the 2+1 Hamiltonians (6.14) plus the 1+1 Hamiltonians which is a 2+1 integrable generalization of the SO(3) Heisenberg model. It corresponds to the surface flow This flow equation describes the motion of a sheet of non-stretching filaments in Euclidean space, in analogy with the form of the vortex filament equations (2.8). Some properties of the model (7.1) have been studied recently in [9,30].
Theorem 7 is established as follows. In parts (i) and (ii), the derivation of the equations of motion (7.10) and Hamiltonians (7.12) for S(t, x) involves combining the evolution equation 3) for the frame vector E 1 = S with the hierarchy (6.11) for the variable ̟ by means of the identities (7.28) in addition to holding for vectors f in R 3 , with componentsf = ( E 2 + i E 3 ) · f , such that f · S = 0; here S is the recursion operator (4.24). Similarly, the derivation of the Hamiltonian structures (7.17) in part (iii) for n = 1 relies on applying the previous identities to the bi-Hamiltonian structure (6.13) for the flow equations on u(t, x). The n = 1 case reduces to computing the Hamiltonian derivatives (7.21)- (7.22), which is carried out in appendices B and C. Finally, all of the corresponding results for γ(t, x) are an immediate consequence of the geometric identifications (7.5)-(7.7) in addition to (3.41)-(3.42).

VIII. GEOMETRIC FORMULATION
There is a natural geometric formulation for the surface flows (6.1) corresponding to the 2+1 vector models (7.10) and 2+1 dynamical maps (7.11) in Theorem 7. We begin by writing down the the intrinsic and extrinsic surface geometry of r(x, y) in terms of the variables u, v, q, v 1 , q 1 appearing in the structure equations of the parallel framing for the non-stretching x-coordinate lines. All other aspects of the intrinsic surface geometry can be derived from the line element (8.1). In particular, the 1st fundamental form (i.e. the surface metric tensor) is simply dxdx + 2q 1 dxdy + |q| 2 dydy, from which the Gauss curvature can be directly computed in terms of the x, y coordinates [12].
The extrinsic surface geometry can be determined through the surface normal vector as given by the expression (5.13) for the surface tangent vectors r x and r y in terms of the parallel frame along the x-coordinate lines. This normal vector (8.3) depends on a choice of the transverse coordinate y due to its normalization factor |q|. To proceed, we use the following natural geometric framing [12] that is defined entirely by the non-stretching direction on the surface and the orthogonal direction of the surface unit-normal in R 3 : The Frenet equations of the frame e , * e , e ⊥ directly encode the extrinsic geometry of the surface r(x, y). In matrix notation, the Frenet equations with respect to the x, y coordinates are given by  where α = * e · e x = Re(ue iψ ), β = e ⊥ · e x = Im(ue iψ ), (8.9) γ = e ⊥ · * e x = |q| −1 Im(q x e iψ ) = −ψ x (8.10) and µ = * e · e y = Re(ve iψ ) = q 1 α + |q| x , ρ = e ⊥ · e y = Im(ve iψ ) = q 1 β − |q|ψ x , (8.11) σ = e ⊥ · * e 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), (5.14) together with the structure equations (5.16), (5.17). With respect to the x, y coordinates on the surface, the scalars α and µ are known as the geodesic curvatures; β and ρ as the normal curvatures; γ and σ as the relative torsions [12]. x Re(ūR n−1 (u y )) (8.20) in terms of the recursion operator R = i(D x + uD −1 x Re(uC)). Moreover, these coefficients (8.20) have a geometrical formulation derived from the operator identity combined with e iψ u y = (e iψ u) y − iψ y e iψ u = α y + βψ y + i(β y − αψ y ). (8.22) This leads to the following geometric counterpart of Theorem 7.
Remark: The bottom flow in this hierarchy can be written in the alternative form by means of the relation P −1 (α y + βψ y + i(β y − αψ y )) = ρ − iµ from which we obtain the evolution of the geodesic curvature, normal curvature, and relative torsion of the x-coordinate lines: These developments now lead to the following geometric results.
where H is the mean curvature (8.17) of the surface. The geodesic curvature α, normal curvature β, and relative torsion γ = −ψ x of the x-coordinate lines in each surface flow (8.23) satisfy the integrable system of evolution equations in terms of the recursion operator (8.25) and the pair of geometric operators (8.34), with the geodesic curvature µ and normal curvature ρ of the y-coordinate lines given by (8.27), and where yields (modulo total x, y-derivatives) a set of non-trivial constants of motion C a (2) dxdy = We conclude by pointing out that the surface flows (8.23) in Theorem 8 and the corresponding integrable systems (8.36)-(8.37) in Theorem 9 provide a geometric realization for the 2+1 vector models (7.10) and 2+1 dynamical maps (7.11) in Theorem 7.
This flow is a geometric realization of the 2+1 Heisenberg model (7.1), corresponding to the new integrable 2+1 generalization of the Schrödinger map (7.8).
The corresponding geometric integrable systems on the geodesic curvature α, normal curvature β, and relative torsion γ = −ψ x of the non-stretching x-coordinate lines in these surface flows (8.39) and (8.40) are respectively given by

IX. CONCLUDING REMARKS
There are some directions in which to extend the geometrical correspondence among integrable vector models, Hamiltonian curve and surface flows, and bi-Hamiltonian soliton equations presented in this paper.
First, it would be of interest to generalize this correspondence to integrable models for spin vectors S in Euclidean spaces R N for N ≥ 3. In particular, the Heisenberg model and etc.
which thus establishes parts (i) and (iv) of the theorem. We will first verify the second Hamiltonian for the Schrödinger map equation (4.54) and its 2+1 generalization (7.20). The following preliminaries concerning the tangent space structure of S 2 will be needed. Here u, v, w will be any triple of tangent vectors.
(1) The metric tensor g, complex structure tensor J, and metric-normalized volume form ǫ g on S 2 satisfy the identities (2) In local coordinates on S 2 , the metric-compatible covariant derivative ∇ (i.e. Riemannian connection) and the associated covariant divergence operator div g are given by where Γ denotes the Christoffel symbol [12] determined from g by the properties (3) For an arbitrary variation δγ of the map γ into S 2 , geometrically represented by a tangent vector field, the variation of g| γ and ǫ g | γ (as induced by their evaluation at γ) is given by Now, consider the 1+1 Hamiltonian density (4.52) defined in terms of a vector field ξ(γ) with covariantly-constant divergence, div g ξ(γ) = 1. Through the identity (B1), this density can be written more conveniently as Its variation is given by Integration by parts on the third term in (B9) yields ǫ g (ξ(γ), D x δγ) = D x (ǫ g (ξ(γ), δγ)) − ǫ g (∂ γx ξ(γ), δγ) − tr(Γ γx )ǫ g (ξ(γ), δγ).
By combining the middle terms in (B9) and (B10) via the identity (B2), we get Likewise, combining the first term in (B9) with the third term in (B10), we obtain Hence, modulo total x-derivatives, (B11) and (B12) combine to give which yields the Hamiltonian derivative (4.53).
Hence, modulo total x, y-derivatives, substitution of (B16) into (B15) yields after we use the commutativity identity ∇ x γ y = ∇ y γ x which is a consequence of the first property in (B5).
The above derivation carries over verbatim to also verify the first Hamiltonian (7.13) for the 2+1 Heisenberg model (7.19). To verify the second Hamiltonian (7.14), we will separately consider the two terms in the density H (−1) = H 1 + H 2 given by with v 1x = S x · (S ∧ S y ) = S · (S y ∧ S x ). (C15) The following identity will be useful: holding similarly to (C6).