Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type

The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of non-stretching curves in Riemannian symmetric spaces G/SO(N). These spaces are exhausted by the Lie groups G=SO(N+1),SU(N). The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curves, tied to a zero curvature Maurer-Cartan form on G, and this yields the vector mKdV recursion operators in a geometric O(N-1)-invariant form. The kernel of these recursion operators is shown to yield two hyperbolic vector generalizations of the sine-Gordon equation. The corresponding geometric curve flows in the hierarchies are described in an explicit form, given by wave map equations and mKdV analogs of Schrodinger map equations.


Introduction
There has been much recent interest in the close relation between integrable partial differential equations and the differential geometry of plane and space curves (see [10,11,12,13,20] for an overview and many results). The present paper studies flows of curves in Riemannian manifolds G/SO(N ) for arbitrary N ≥ 2, where G = SO(N + 1), SU (N ). Such symmetric spaces [16] are well-known to exhaust all examples of curved G-invariant geometries that are a natural generalization of Euclidean spaces R N ≃ Euc(N )/SO(N ) modeled by replacing the Euclidean isometry group with a compact semisimple Lie-group G ⊃ SO(N ).
It will be shown that if non-stretching curves are described using a moving parallel frame and an associated frame connection 1-form in G/SO(N ) then the frame structure equations for torsion and curvature encode O(N − 1)-invariant bi-Hamiltonian operators. These operators will be demonstrated to produce a hierarchy of integrable flows of curves in which the frame components of the principal normal along the curve satisfy O(N − 1)-invariant vector soliton equations. The hierarchies for both SO(N + 1)/SO(N ), SU (N )/SO(N ) will be seen to possess a scaling symmetry and accordingly will be organized by the scaling weight of the flows. The 0 flow just consists of a convective (traveling wave) equation, while the +1 flow will be shown to give the two vector generalizations of the mKdV equation known from symmetry-integrability classifications of vector evolution equations in [27]. A recent classification analysis [5] found there are vector hyperbolic equations for which the respective vector mKdV equations are higher symmetries. These two vector hyperbolic equations will be shown to describe a −1 flow in the respective hierarchies for SO(N + 1)/SO(N ) and SU (N )/SO(N ).

S.C. Anco
As further results, the Hamiltonian operators will yield explicit O(N − 1)-invariant recursion operators for higher symmetries and higher conservation laws of the vector mKdV equations and the vector hyperbolic equations. The associated curve flows produced from these equations will describe geometric nonlinear PDEs, in particular given by wave maps and mKdV analogs of Schrödinger maps.
Previous fundamental work on vector generalizations of KdV and mKdV equations as well as their Hamiltonian structures and geometric origin appeared in [6,7,21,22]. In addition, the bi-Hamiltonian structure of both vector mKdV equations was first written down in [30] from a more algebraic point of view, in a multi-component (non-invariant) notation. Special cases of two component KdV-mKdV integrable systems related to vector mKdV equations have been discussed recently in [15,28,25].
2 Curve f lows, parallel frames, and Riemannian symmetric spaces Let γ(t, x) be a flow of a non-stretching curve in some n-dimensional Riemannian manifold (M, g). Write Y = γ t for the evolution vector of the curve and write X = γ x for the tangent vector along the curve normalized by g(X, X) = 1, which is the condition that γ is non-stretching, so thus x represents arclength. In the tangent space T γ M of the two-dimensional surface swept out by γ(t, x) we introduce orthonormal frame vectors e a and connection 1-forms ω ab = ω [ab] related through the Riemannian covariant derivative operator g ∇ in the standard way [17]: (Throughout, a, b = 1, . . . , n denote frame indices which get raised and lowered by the Euclidean metric δ ab = diag(+1, . . . , +1)). Now choose the frame along the curve to be parallel [9], so it is adapted to γ via e a := X (a = 1), (e a ) ⊥ (a = 2, . . . , n) where g(X, (e a ) ⊥ ) = 0, such that the covariant derivative of each of the n−1 normal vectors (e a ) ⊥ in the frame is tangent to γ, holding for some functions v a , while the covariant derivative of the tangent vector X in the frame is normal to γ, Equivalently, along γ the connection 1-forms of the parallel frame are given by the skew matrix where e x a := g(X, e a ) is the row matrix of the frame in the tangent direction. In matrix notation we have with 0, 0 respectively denoting the 1 × (n − 1) zero row-matrix and (n − 1) × (n − 1) zero skew-matrix. (Hereafter, upper/lower frame indices will represent row/column matrices.) This matrix description (3) of a parallel frame has a purely algebraic characterization: e x a is a fixed unit vector in R n preserved by a SO(n − 1) rotation subgroup of the local frame structure group SO(n), while ω x a b belongs to the orthogonal complement of the corresponding rotation subalgebra so(n − 1) in the Lie algebra so(n) of SO(n).
The curve flow has associated to it the pullback of the Cartan structure equations [17] expressing that the covariant derivatives g ∇ x := X g ∇ along the curve and g ∇ t := Y g ∇ along the flow have vanishing torsion and carry curvature determined from the metric g, given by the Riemann tensor R(X, Y ) which is a linear map on T x M depending bilinearly on X, Y . In frame components the torsion and curvature equations look like [17] Here e t a := g(Y, e a ) and ω t a b := Y ω a b = g(e b , g ∇ t e a ) are respectively the frame row-matrix and connection skew-matrix in the flow direction, and R a b (X, As outlined in [2,21], these frame equations (6) and (7) directly encode a bi-Hamiltonian structure based on geometrical variables when the geometry of M is characterized by having its frame curvature matrix R a b (e c , e d ) be constant on M . In this situation the Hamiltonian variable is given by the principal normal v := g ∇ x X = v a (e a ) ⊥ in the tangent direction of γ, while the principal normal in the flow direction ̟ := g ∇ t X = ̟ a (e a ) ⊥ represents a Hamiltonian covector field, and the normal part of the flow vector h ⊥ := Y ⊥ = h a (e a ) ⊥ represents a Hamiltonian vector field 1 . In a parallel frame these variables v a , ̟ a , h a are encoded respectively in the top row of the connection matrices ω x a b , ω t a b , and in the row matrix (e t a ) ⊥ = e t a − h e x a where h := g(Y, X) is the tangential part of the flow vector.
A wide class of Riemannian manifolds (M, g) in which the frame curvature matrix R a b (e c , e d ) is constant on M consists of the symmetric spaces M = G/H for compact semisimple Lie groups G ⊃ H (such that H is invariant under an involutive automorphism of G). In such spaces the Riemannian curvature tensor and the metric tensor are covariantly constant and G-invariant [17], which implies constancy of the curvature matrix R a b (e c , e d ). The metric tensor g on M is given by the Cartan-Killing inner product ·, · on T x G ≃ g restricted to the Lie algebra ⊆ h (corresponding to the eigenspaces of the adjoint action of the involutive automorphism of G that leaves H invariant). A complete classification of symmetric spaces is given in [16]; their geometric properties are summarized in [17]. In these spaces H acts as a gauge group so consequently the bi-Hamiltonian structure encoded in the frame equations will be invariant under the subgroup of H that leaves X fixed 2 . Thus in order to obtain O(N −1)-invariant bi-Hamiltonian operators, as sought here, we need the group O(N −1) to be the isotropy subgroup in H leaving X fixed. Hence we restrict attention to the symmetric spaces M = G/SO(N ) with H = SO(N ) ⊃ O(N − 1). From the classification in [16] all examples of these spaces are exhausted by G = SO(N +1), SU (N ). The example M = SO(N + 1)/SO(N ) ≃ S N is isometric to the N -sphere, which has constant curvature. In this symmetric space, the encoding of bi-Hamiltonian operators in terms of geometric variables has been worked out in [2] using the just intrinsic Riemannian geometry of the N -sphere, following closely the ideas in [20,21]. An extrinsic approach based on Klein geometry [26,5] will be used here, as it applicable to both symmetric spaces SO(N + 1)/SO(N ) and SU (N )/SO(N ).
In a Klein geometry the left-invariant g-valued Maurer-Cartan form on the Lie group G is identified with a zero-curvature connection 1-form ω G called the Cartan connection [26]. Thus where d is the total exterior derivative on the group manifold G. Through the Lie algebra decomposition g = so(N ) ⊕ p with [p, p] ⊂ so(N ) and [so(N ), p] ⊂ p, the Cartan connection determines a Riemannian structure on the quotient space M = G/SO(N ) where G is regarded [26] as a principal SO(N ) bundle over M . Fix any local section of this bundle and pull-back ω G to give a g-valued 1-form g ω at x in M . The effect of changing the local section is to induce a SO(N ) gauge transformation on g ω. Let σ denote an involutive automorphism of g such that so(N ) is the eigenspace σ = +1, p is the eigenspace σ = −1. We consider the corresponding decomposition of g ω: it can be shown that [26] the symmetric part defines a so(N )-valued connection 1-form for the group action of SO(N ) on the tangent space T x M ≃ p, while the antisymmetric part defines a p-valued coframe for the Cartan-Killing inner product ·, · p on T x G ≃ g restricted to T x M ≃ p. This inner product ·, · p provides a Riemannian metric g = e ⊗ e p on M = G/SO(N ), such that the squared norm of any vector X ∈ T x M is |X| 2 g = g(X, X) = X e, X e p .
Moreover there is a G-invariant covariant derivative ∇ associated to this structure whose restriction to the tangent space T γ M for any curve flow γ(t, and These derivatives ∇ x , ∇ t obey the Cartan structure equations (4) and (5), namely they have zero torsion and carry G-invariant curvature where e x := γ x e, e t := γ t e, ω x := γ x ω , ω t := γ t ω .
The G-invariant covariant derivative and curvature on T γ M are thus seen to coincide with the Riemannian ones determined from the metric g. More generally, in this manner [26] the relations (8) and (9) canonically solder a Klein geometry onto a Riemannian symmetric-space geometry.
Geometrically, e x and ω x represent the tangential part of the coframe and the connection 1-form along γ. For a non-stretching curve γ, where x is the arclength, note e x has unit norm in the inner product, e x , e x p = 1. This implies p has a decomposition into tangential and normal subspaces p and p ⊥ with respect to e x such that e x , p ⊥ p = 0, with p = p ⊥ ⊕ p and p ≃ R. Remark 1. A main insight now, generalizing the results in [21,2], is that the Cartan structure equations on the surface swept out by the curve flow γ(t, x) in M = G/SO(N ) will geometrically encode O(N − 1)-invariant bi-Hamiltonian operators if the gauge (rotation) freedom of the group action SO(N ) on e and ω is used to fix them to be a parallel coframe and its associated connection 1-form related by the Riemannian covariant derivative. The groups G = SO(N + 1) and G = SU (N ) will produce a different encoding except when N = 2, since in that case (2) is the same tangent space for M = SO(3)/SO(2) and M = SU (2)/SO(2) due to the Lie-algebra isomorphism so(3) ≃ su(2). This will be seen to account for the existence of the two different vector generalizations of the scalar mKdV hierarchy.
The algebraic characterization of a parallel frame for curves in Riemannian geometry extends naturally to the setting of Klein geometry, via the property that e x is preserved by a SO(N − 1) rotation subgroup of the local frame structure group SO(N ) acting on p ⊂ g, while ω x belongs to the orthogonal complement of the SO(N − 1) rotation Lie subalgebra so(N − 1) contained in the Lie algebra so(N ) of SO(N ). Their geometrical meaning, however, generalizes the Riemannian properties (1) and (2), as follows. Let e a be a frame whose dual coframe is identified with the p-valued coframe e in a fixed orthonormal basis for p ⊂ g. Decompose the coframe into parallel/perpendicular parts with respect to e x in an algebraic sense as defined by the kernel/cokernel of Lie algebra multiplication [e x , · ] g = ad(e x ). Thus we have e = (e C , e C ⊥ ) where the p-valued covectors e C , e C ⊥ satisfy [e x , e C ] g = 0, and e C ⊥ is orthogonal to e C , so [e x , e C ⊥ ] g = 0. Note there is a corresponding algebraic decomposition of the tangent space (10), the derivative ∇ x of a covector perpendicular (respectively parallel) to e x lies parallel (respectively perpendicular) to e x , namely ∇ x e C belongs to p C ⊥ , ∇ x e C ⊥ belongs to p C . In the Riemannian setting, these properties correspond Such a frame will be called SO(N )-parallel and defines a strict generalization of a Riemannian parallel frame whenever p C is larger than p .
Existence of a SO(N )-parallel frame for curve flows in Klein geometries G/SO(N ) is guaranteed by the SO(N ) gauge freedom on e and ω inherited from the local section of G used to pull back the Maurer-Cartan form to G/SO(N ).
parameterized by the N component vector p. The Cartan-Killing inner product on g is given by the trace of the product of an so(N + 1) matrix and a transpose so(N + 1) matrix, multiplied by a normalization factor 1 2 . The norm-squared on the quotient space p thereby reduces to the ordinary dot product of vectors p: Note the Cartan-Killing inner product provides a canonical identification between p ≃ R N and its dual p * ≃ R N . Let γ(t, x) be a flow of a non-stretching curve in M = SO(N +1)/SO(N ) ≃ S N . We introduce a SO(N )-parallel coframe e ∈ T * γ M ⊗ p and its associated connection 1-form ω ∈ T * γ M ⊗ so(N ) along γ by putting 3 and where Note the form of e x indicates the coframe e is adapted to γ, with (1, 0) representing a choice of a constant unit-norm vector in p ≃ R N , so e x , e x p = (1, 0) · (1, 0) = 1. All such choices are equivalent through the SO(N ) rotation gauge freedom on the coframe and connection 1-form.
Consequently, there is a decomposition of SO(N + 1)/SO(N ) matrices into tangential and normal parts relative to e x via a corresponding decomposition of vectors given by p = (p , p ⊥ ) ∈ R N relative to (1, 0). Thus p is identified with p = p C , and p ⊥ with p ⊥ = p C ⊥ .
In the flow direction we put where The components h , h ⊥ correspond to decomposing e t = g(γ t , γ x )e x + (γ t ) ⊥ e ⊥ into tangential and normal parts relative to e x . We now have Hence the curvature equation (12) reduces to while the torsion equation (11) yields Here  (17) and (18) to eliminate in terms of the variables v, h ⊥ , ̟. Then equation (16) gives a flow on v, obtained from equation (19). Here χ = 1 represents the Riemannian scalar curvature of SO(N + 1)/SO(N ) ≃ S N (see [2]). In these equations we read off the operators The results in [21] prove the following properties of H, J .
where R = H • J is a hereditary recursion operator.
On the x-jet space of v(t, x), the variables h ⊥ and ̟ have the respective meaning of a Hamiltonian vector field h ⊥ ∂/∂ v and covector field ̟ d v (see the Appendix of [2]). Thus the recursion operator 4 generates a hierarchy of commuting Hamiltonian vector fields h with

Remark 2.
Using the terminology of [5], h (k) ⊥ will be said to produce a +(k + 1) flow on v. This differs from the terminology in [2] which refers to equation (22) as the +k flow.

The +1 flow given by
which is a vector mKdV equation up to a convective term that can be absorbed by a Galilean transformation x → x − χt, t → t. The +(k + 1) flow gives a vector mKdV equation of higher order 3 + 2k on v.
There is also a 0 flow v t = v x arising from h ⊥ = 0, h = 1, which falls outside the hierarchy generated by R.
All these flows correspond to geometrical motions of the curve γ(t, x), given by subject to the non-stretching condition The equation of motion is obtained from the identifications γ t ↔ e t , ∇ x γ x ↔ D x e x = [ω x , e x ], and so on, using ∇ x ↔ D x + [ω x , ·] = D x . These identifications correspond to T x M ↔ p as defined via the parallel coframe. Note we have and so on. In particular, for the +1 flow, We identify the first term as −∇ g since the Cartan-Killing inner product corresponds to the Riemannian metric g. Hence we have e t ↔ −(∇ 2 x γ x + 3 2 |∇ x γ x | 2 g γ x ). This describes a geometric nonlinear PDE for γ(t, x), which is referred to as the non-stretching mKdV map equation on the symmetric space M = SO(N + 1)/SO(N ) ≃ S N . A different derivation using just the Riemannian geometry of S N was given in [2]. Since in the tangent space T x S N ≃ so(N + 1)/so(N ) the kernel of [e x , · ] is spanned by e x , a parallel frame in the setting of the Klein geometry of SO(N + 1)/SO(N ) is precisely the same as in the Riemannian geometry of S N . The convective term |∇ x γ x | 2 g γ x can be written in an alternative form using the Klein geometry of SO(N + 1)/SO(N ) ≃ S N . Let ad(·) denote the standard adjoint representation acting in the Lie algebra g = p ⊕ so(N ). We first observe Applying ad([ω x , e x ]) again, we obtain Hence, | v| 2 e x ↔ −χ −1 ad(∇ x γ x ) 2 γ x = |∇ x γ x | 2 g γ x , and thus the mKdV map equation is equivalent to Note here that ad Higher flows on v yield higher-order geometric PDEs. The 0 flow on v directly corresponds to which is just a convective (linear traveling wave) map equation.
There is a −1 flow contained in the hierarchy, with the property that h ⊥ is annihilated by the symplectic operator J and hence gets mapped into R( h ⊥ ) = 0 under the recursion operator. Geometrically this flow means simply J ( h ⊥ ) = ̟ = 0 which implies ω t = 0 from equations (14), (15), (20), and hence 0 = [ω t , e x ] = D t e x where D t = D t + [ω t , ·]. The correspondence ∇ t ↔ D t , γ x ↔ e x immediately leads to the equation of motion for the curve γ(t, x). This nonlinear geometric PDE is precisely a wave map equation on the symmetric space SO(N + 1)/SO(N ) ≃ S N . The resulting flow equation on v is where Note this flow equation possesses the conservation law 0 = D Thus a conformal scaling of t can be used to put |γ t | g = 1, and so

reduces the wave map equation to a hyperbolic vector equation
Equivalently, v satisfies a nonlocal evolution equation describing the −1 flow. It also follows from v = h −1 D x h ⊥ combined with the flow equation (28) that h ⊥ obeys the vector SG equation which has been derived previously in [8,19,30] from a different point of view. These equations (30) and (31) possess the mKdV scaling symmetry x → λx, v → λ −1 v, where h ⊥ has scaling weight 0. The hyperbolic vector equation (30) admits the vector mKdV equation (23) as a higher symmetry, which is shown by the symmetry-integrability classification results in [5]. As a consequence of Corollary 1, we see that the recursion operator R = H • J generates a hierarchy of vector mKdV symmetries v (0) and so on, all of which commute with the −1 flow and so on, all of which are conserved densities for the −1 flow. It follows that the hyperbolic vector equations (30) and (31) admit these respective hierarchies of vector mKdV symmetries and conserved densities. Viewed as flows, the entire hierarchy of vector PDEs (35), (32) to (34), etc. possesses the mKdV scaling symmetry x → λx, v → λ −1 v, with t → λ 1+2k t for k = −1, 0, 1, 2, etc. Moreover for k ≥ 0, all these expressions will be local polynomials in the variables v, v x , v xx , . . . as established by general results in [29,24] concerning nonlocal recursion operators.
Theorem 2. In the symmetric space SO(N + 1)/SO(N ) there is a hierarchy of bi-Hamiltonian flows of curves γ(t, x) described by geometric map equations. The 0 flow is a convective (traveling wave) map (26), while the +1 flow is a non-stretching mKdV map (24) and the +2, . . . flows are higher order analogs. The kernel of the recursion operator (21) in the hierarchy yields the −1 flow which is a non-stretching wave map (27).
Recall su(k) is a complex vector space isomorphic to the Lie algebra of k × k skew-hermitian matrices. The real and imaginary parts of these matrices respectively belong to the real vector space so(k) of skew-symmetric matrices and the real vector space s(k) ≃ su(k)/so(k) defined by k × k symmetric trace-free matrices. Hence g = su(N ) has the decomposition g = h + ip where h = so(N ) and p = s(N ). The Cartan-Killing inner product is given by the trace of the product of an su(N ) matrix and a hermitian-transpose su(N ) matrix, multiplied by 1/2. Note any matrix in s(N ) can be diagonalized under the action of the group SO(N ).
Let γ(t, x) be a flow of a non-stretching curve in M = SU (N )/SO(N ) where we identify T x M ≃ p (dropping a factor i for simplicity 5 ). We consider a SO(N )-parallel coframe e ∈ T * γ M ⊗ p and its associated connection 1-form ω ∈ T * γ M ⊗ so(N ) along γ given by 6 up to a normalization factor κ which we will fix shortly, and Since the form of e x is a constant matrix, it indicates that the coframe is adapted to γ provided e x has unit norm in the Cartan-Killing inner product. We have after putting κ 2 = 2N (N − 1) −1 . As a consequence, all matrices in p = s(N ) will have a canonical decomposition into tangential and normal parts relative to e x , Here (p , p ⊥ ) is identified with p C ⊃ p , and p ⊥ with p C ⊥ ⊂ p ⊥ . Note p ⊥ is empty only if N = 2, so consequently for N > 2 the SO(N )-parallel frame (36) and (37) is a strict generalization of a Riemannian parallel frame.
In the flow direction we put and Note the components h , ( h ⊥ , h ⊥ ) correspond to decomposing e t = g(γ t , γ x )e x + (γ t ) ⊥ e ⊥ into tangential and normal parts relative to e x . We thus have Now the curvature equation (12) yields which are unchanged from the case G = SO(N + 1) up to the factor in front of h ⊥ . The torsion equation (11) reduces to which are similar to those in the case G = SO(N + 1), plus Proceeding as before, we use equations (42), (43), (45) to eliminate in terms of the variables v, h ⊥ , ̟. Then equation (41) gives a flow on v, obtained from equation (44) after we combine h v terms. We thus read off the operators where In particular, R = H • J yields a hereditary recursion operator In the terminology of [5], h (k) ⊥ is said to produce the +(k + 1) flow equation (22) on v (cf. Remark 2). Note these flows admit the same mKdV scaling symmetry x → λx, v → λ −1 v as in the case SO(N + 1)/SO(N ). They also have similar recursion relations h (k) The +1 flow is given by Up to the convective term, which can be absorbed by a Galilean transformation, this is a different vector mKdV equation compared to the one arising in the case SO(N + 1)/SO(N ) for N > 2. The +(k + 1) flow yields a higher order version of this equation (49). The hierarchy of flows corresponds to geometrical motions of the curve γ(t, x) obtained in a similar fashion to the ones in the case SO(N + 1)/SO(N ) via identifying γ t ↔ e t , γ x ↔ e x , ∇ x γ x ↔ [ω x , e x ] = D x e x , and so on as before, where ∇ x ↔ D x = D x + [ω x , e x ]. Note here we have and so on. In addition, Thus, for the +1 flow, we obtain (through equation (39)) Then writing these expressions in terms of D x and ad([ω x , e x ]), we get Thus, up to a sign, γ(t, x) satisfies a geometric nonlinear PDE given by the non-stretching mKdV map equation (25) on the symmetric space SU (N )/SO(N ). The higher flows on v determine higher order map equations for γ. The 0 flow as before is v t = v x arising from h ⊥ = 0, h = 1, which corresponds to the convective (traveling wave) map (26).
There is also a −1 flow contained in the hierarchy, with the property that h ⊥ is annihilated by the symplectic operator J and hence lies in the kernel R( h ⊥ ) = 0 of the recursion operator. The geometric meaning of this flow is simply J ( h ⊥ ) = ̟ = 0 implying ω t = 0 from equations (40) and (46) so 0 = [ω t , e x ] = D t e x where D t = D t + [ω t , ·]. Thus, as in the case SO(N + 1)/SO(N ), we see from the correspondence ∇ t ↔ D t , γ x ↔ e x that γ(t, x) satisfies a nonlinear geometric PDE given by the wave map equation (27) on the symmetric space SU (N )/SO(N ).
The −1 flow equation produced on v is again a nonlocal evolution equation with h ⊥ satisfying where it is convenient to introduce the variables These equations (51) to (53) determine the variables h ⊥ , h, h implicitly as nonlocal functions of v (and its x derivatives). To proceed, we will seek an inverse local expression for v in terms of h ⊥ , analogous to the one that arises in the case SO(N + 1)/SO(N ). However, the presence of the additional variable h here leads to a quite different expression for the resulting flow on v. Let for some expressions α(h), β(h). Substitution of v into equation (53) yields where c is a constant of integration (and 1 is the only available constant matrix that is O(N −1)invariant). Then, substitution of h and v into equation (51) gives which also satisfies equation (52). Next, by taking the trace of h from equation (55) and using equation (56), we obtain which enables h to be expressed in terms of h ⊥ and c. To determine c we use the wave map conservation law (29) where, now, |γ t | 2 g = e t , e t p = κ 2 (| h ⊥ | 2 + 1 2 (h 2 + |h| 2 )).
This corresponds to a conservation law admitted by equations (51) to (53), and as before, a conformal scaling of t can now be used to put |γ t | g equal to a constant. A convenient value which simplifies subsequent expressions is |γ t | g = 2, so then (2/κ) 2 = | h ⊥ | 2 + 1 2 |h| 2 + h 2 .
Substitution of equations (55) to (57) into this expression yields associated to the vector SG equation (61), and all of the mKdV symmetries commute with this flow. Hence these hierarchies are admitted symmetries and conserved densities for the hyperbolic vector equation (60). Viewed as flows, the vector PDEs (62) to (64), etc., including the −1 flow (65), is seen to possess the mKdV scaling symmetry x → λx, v → λ −1 v, with t → λ 1+2k t for k = −1, 0, 1, 2, etc.. Moreover for k ≥ 0, all these expressions will be local polynomials in the variables v, v x , v xx , . . . as established by results in [23] applied to the separate Hamiltonian (cosymplectic and symplectic) operators (47) 8 .
Theorem 3. In the symmetric space SU (N )/SO(N ) there is a hierarchy of bi-Hamiltonian flows of curves γ(t, x) described by geometric map equations. The 0 flow is a convective (traveling wave) map (26), while the +1 flow is a non-stretching mKdV map (25) and the +2, . . . flows are higher order analogs. The kernel of the recursion operator (48) in the hierarchy yields the −1 flow which is a non-stretching wave map (27).

Concluding remarks
In the compact Riemannian symmetric spaces G/SO(N ), as exhausted by the Lie groups G = SO(N + 1) and G = SU (N ), there is a hierarchy of integrable bi-Hamiltonian flows of nonstretching curves γ(t, x), where the +1 flow is described by the mKdV map equation (25) and the +2, . . . flows are higher-order analogs, while the wave map equation (27) [5]. Similar results hold for hermitian symmetric spaces G/U (N ). In particular, there is a hierarchy of flows of curves in such spaces yielding scalar-vector generalizations of the mKdV equation and the SG equation. A further generalization of such results for all symmetric spaces G/H will be given elsewhere [3].