integrable systems in three-dimensional riemannian geometry · integrable systems in...

DOI: 10.1007/s00332-001-0472-yJ. Nonlinear Sci. Vol. 12: pp. 143–167 (2002)

© 2002 Springer-Verlag New York Inc.

Integrable Systems in Three-Dimensional RiemannianGeometry

G. Marı Beffa,1 J. A. Sanders,2 and Jing Ping Wang21 Mathematics Department, University of Wisconsin, Madison, Wisconsin 53706, USA2 Vrije Universiteit, Faculty of Sciences, Division of Mathematics and Computer Science,

De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Received May 31, 2001; accepted January 2, 2002Online publication March 11, 2002Communicated by A. Bloch

Summary. In this paper we introduce a new infinite-dimensional pencil of Hamiltonianstructures. These Poisson tensors appear naturally as the ones governing the evolution ofthe curvatures of certain flows of curves in 3-dimensional Riemannian manifolds withconstant curvature. The curves themselves are evolving following arclength-preservinggeometric evolutions for which the variation of the curve is an invariant combinationof the tangent, normal, and binormal vectors. Under very natural conditions, the evolu-tion of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian andcompletely integrable.

1. Introduction

The theory of integrable systems has traditionally made use of geometrical conceptsand procedures. In particular, the majority of completely integrable PDEs, or systemsof PDEs, are connected to the existence of two compatible Hamiltonian structures withrespect to which the systems are Hamiltonian. When that happens we call the system a bi-Hamiltonian system. If one of the compatible Hamiltonian structures is nondegenerate, arecursion operator can be defined, which will generate a family of preserved quantities forthe flow, effectively integrating the system. The field of Poisson geometry, or geometry ofHamiltonian evolutions, is thus a fundamental part in the study of completely integrablesystems.

The study of the relationship between finite-dimensional differential geometry andpartial differential equations, which later came to be known as integrable systems, startedin the nineteenth century. Liouville found and solved the equation describing minimalsurfaces in 3-dimensional Euclidean space [Lio53]. Bianchi solved the general Goursatproblem for the sine-Gordon equation which arises in the theory of pseudospherical

144 G. Mar´ı Beffa, J. A. Sanders, and Jing Ping Wang

surfaces [Bia], [Bia92]. The original work of Darboux [Dar10] is still of interest; see[Zak98] for the modern developments of this line of research

Much later Hasimoto [Has72] found the relation between the equations for curvatureand torsion of vortex filament flow and the nonlinear Schr¨odinger equation which led tomany new developments, cf. [MW83], [LP91], [DS94], [LP96], [YS98], [Cal00]. Theauthor of [MB99] subsequently defined the second Poisson structure for generalized KdV,the Adler–Gel’fand–Diki˘ı bracket [GD78], [Adl79], in terms of an invariant frame alonga parametrized projective curve and its differential invariants. This close relationshipbetween invariant evolutions and the Hamiltonian structures of integrable systems alsoholds for parametrized plane curves under the action ofO(3,1), cf. [MB00b], and, undermore restrictive conditions, for projective reparametrizations of the projective plane, cf.[MB00a].

In this paper we present a quadruplet of compatible Hamiltonian structures that arisesin a natural way from the geometric arclength-preserving evolution of curves in anygiven 3–dimensional Riemannian manifold with constant curvature. We arrived at thesestructures from two different ways. First, one of the authors was asked whether a certainsystem of PDEs, which is currently being studied by Fels and Ivey (see [Ive01]), hasa recursion operator. She found this recursion operator and then proceeded to find theHamiltonian pair, producing in this way two of the Poisson tensors presented here, theones we callE + B andD. These two were discovered independently by Ivey andthey appear in [Ive01]. The other route is described in this paper, where we presentthe geometrical origin of the above-mentioned Hamiltonian pair, as generating the firstHamiltonian structure in a pencil indexed by the curvature of the manifold. We denotethe pencil byB + ~C +D + E . The tensorC in the pencil is compatible with the otherthree, forming this way a Hamiltonian quadruplet. Indeed we show that if a flow ofcurves in a 3–dimensional Riemannian manifold with constant curvature~ follows anarclength-preserving geometric evolution, the evolution of its Riemannian curvatures isalways, under natural conditions, a Hamiltonian flow with respect to the element of thepencil corresponding to the value~.1 The close geometric relationship remains here:The quadruplet can be obtained solely from the geometry of curves on 3-dimensionalRiemannian manifolds with constant curvature.

Having obtained the pencil, we construct hereditary operators by composing the pen-cil with the inverse of a nondegenerate component. We explicitly do this by invertingDandC. We show that, whileB + ~C + D + E andC are the ones used to integrate thebest known versions of the vortex filament equation on constant curvature manifolds,B+ ~C +D+ E andD are used to integrate the system studied by Ivey and its general-izations to manifolds with nonzero constant curvature. Next we describe the effect of theHasimoto transformation on the Hamiltonian pair associated to the vortex filament flow.As is already known, the Hasimoto transform is a Poisson map from the vortex filamentflow to the nonlinear Schr¨odinger equation. We show that the hodographic transforma-tion used by Ivey for the Euclidean case [Ive01] has a role analogous to the Hasimototransformation for the second hierarchy. The hodographic transformation is a Poissonmap which, in the case~ 6= 0, takes the second system in the hierarchy to a system of

1 The study in [YS98] suggests that~ plays the role of a spectral parameter. But that is another story.

Integrable Systems in Three-Dimensional Riemannian Geometry 145

decoupled modified potential KdV equations. The decoupling does not hold in the flatcase, which is degenerate in that sense. When the curvature~ of the manifold is nonzero,as the hodographic transformation is applied to the Hamiltonian pair, the transformedsystem lies in the integrable hierarchy of very simple equations. This relation was notat all clear prior to the application of the transformation. In fact, we can also produceanother integrable equation usingB+ ~C +D+ E andB. However, at this moment wecannot find a proper transformation to simplify it, so we did not treat this case.

Section 2 introduces all concepts of Riemannian geometry needed. Since the paperrelates two somehow separate subjects, we decided to include definitions that some ofthe readers might not be too familiar with, while others might find them very basic.We have done so in the simplest possible way, including only the necessary concepts.Section 3 contains two theorems: Theorem 2 describes the evolution followed by theRiemannian curvatures of a flow of curves satisfying an arclength-preserving geometricevolution. The calculation of this evolution can be carried out in many different ways,as one can see in the Euclidean case in [Cal00] for the vortex filament flow, in [LP91]for general flows, and in the case of spheres in [DS94]. Although the most generalizableway (and also the simplest) would involve the use of Cartan’s definition of connection,we chose to describe it the way we think to be easier for a reader who is not familiarwith Riemannian geometry and its Cartan interpretation. It is a longer procedure butperhaps easier to understand. Theorem 3 shows that the tensor defining the curvatureevolution is a pencil of Hamiltonian structures indexed by the curvature of the manifold.We show that the pencil is formed by four compatible Poisson tensors. The proof ofTheorem 3 needs many specialized definitions and formulae that we have preferred notto include here, since they are only used in the proof. We refer the reader to [Olv93] forthe material needed. In Section 4 we study the two canonical evolution equations, theirhereditary operators, and their integrable hierarchies. We also describe their associatedtransformations, which simplify them. The last section contains comments about theimplications of the results in this paper as well as further open problems.

2. Definitions

2.1. Definitions in Riemannian Geometry

In this subsection we present all concepts about Riemannian manifolds that we need touse in the next section. Definitions and notations are mostly as in [Hic65] and [Pet98].A manifold will be aC∞-manifold.

Definition 1. (i) A connectionon a manifoldM is an operator∇ which assigns to twoC∞ vector fieldsX andY with domainÄ, a thirdC∞ vector field denoted by∇XY withthe same domainÄ, in such a way that the following properties are satisfied:

(1) ∇X(Y + Z) = ∇XY +∇X Z,(2) ∇X+WY = ∇XY +∇WY,(3) ∇ f XY = f∇XY,(4) ∇X( f Y) = X( f )Y + f∇XY,


for any X,W vectors atp ∈ M , Y, Z smooth fields andf a smooth function defined ina neighborhood ofp.

(ii) We say ann–dimensional manifoldM is aRiemannian manifoldif M is endowed witha symmetric and positive definite 2–covariant tensor field〈, 〉. The tensor〈, 〉 is called theRiemannian metricof the manifold, and it allows us to define distances, length, angles,orthogonality, etc., in the natural way. In particular, thelengthof a vectorX is defined as

|X| =√〈X, X〉.

The simplest example of a Riemannian manifold is, of course,Rn with the usual dotproduct.

(iii) A Riemannian connectionon a Riemannian manifoldM is a connection∇ on Msuch that

(5) ∇XY −∇Y X = [X,Y] (the connection has zerotorsion tensor),(6) Z〈X,Y〉 = 〈∇Z X,Y〉 + 〈X,∇ZY〉,

for all fields X,Y, Z with a common domain.

The fundamental theorem of Riemannian manifolds states that on any Riemannian man-ifold there exists a unique Riemannian connection. Riemannian manifolds are thus thenatural generalization of Euclidean spaces and the Riemannian connections the naturalgeneralization of covariant (or directional) differentiation.

(iv) The curvature tensorof a connection∇ is a tensorR that assigns to each pair ofvectorsX,Y at a pointp a linear transformationR(X,Y) of the tangent space top,TpM , into itself. After extendingX,Y, andZ to smooth vectorfields nearp, R(X,Y)Zis defined via the relation

R(X,Y)Z = ∇X∇Y Z −∇Y∇X Z −∇[X,Y] Z. (1)

The value of this expression is independent of the way the vector fields were extended.

(v) TheRiemann–Christoffel curvature tensor(of type(0,4)) is the 4-covariant tensor

K (X,Y, Z,W) = 〈R(Z,W)Y, X〉for any X,Y, Z,W tangent vectors atp.

Apart from being tensors (and thus multilinear with respect toC∞(M) in all theircomponents), curvature tensors are best known for the following properties:

Theorem 1. The following relations are true:

(1) R(X,Y)Z + R(Z, X)Y + R(Y, Z)X = 0 (first Bianchi identity),(2) ∇Z R(X,Y)W +∇X R(Y, Z)W +∇Y R(Z, X)W = 0 (secondBianchi identity),(3) K(X,Y, Z,W) = −K (Y, X, Z,W) = −K (X,Y,W, Z),(4) K(X,Y, Z,W) = K (Z,W, X,Y).


Finally we give the last group of definitions.

Definition 2. (i) Given two independent vectorsX,Y in TpM , the normalized quadraticform,

sec(X,Y) = K(X,Y,X,Y)

〈X,X〉〈Y,Y〉 − 〈X,Y〉2 ,is calledthe sectional curvatureof X,Y. It can easily be checked that sec(X,Y) dependsonly on the planeπ spanned byX andY, and so the sectional curvature is also calledK (π), the Riemannian curvature of the plane sectionπ .

(ii) A Riemannian manifoldM is said to haveconstant Riemannian curvature~ if theRiemannian curvature of all plane sections is the constant~.

(iii) If S is a(0, r ) tensor, one can define thecovariant derivative of the tensoralong thevector fieldX by ensuring that the Leibniz rule holds. That is,∇X(S) is determined bythe relation

∇X(S(Y1, . . . ,Yr )) = (∇X S)(Y1, . . . ,Yr )+ S(∇XY1, . . . ,Yr )+ · · · + S(Y1, . . . ,∇XYr )

to hold for any vectorsY1, . . . ,Yr in TpM .

This proposition can be found in [Pet98].

Proposition 1. The following properties are equivalent:

(1) K(π) = ~ for all 2–planes in TpM.(2) R(X,Y)Z = ~(〈Y, Z〉X − 〈X, Z〉Y) for any X,Y, Z in TpM.

Corollary 1. Assume the manifold M has constant Riemannian curvature. Then

(a) ∇X R= 0 along any direction determined by the vector field X. That is, the Riemanncurvature tensor isparallel.

(b) If Z is orthogonal to X and Y , then R(X,Y)Z = 0.(c) If W is orthogonal to X and Y , then K(W, Z, X,Y) = 0 for any Z.

What follows is the description of a Frénet frame and Frénet formulae for any smoothcurve on a Riemannian manifold under some nondegeneracy conditions that follow fromthe construction.

Let γ: U ⊂ R → M be a smooth curve on a Riemannian manifoldM with Rie-mannian connection∇. From now on we will assume that all vector fields are definedon some common open subset ofU . Let V(x) be the tangent field atx obtained bydifferentiation with respect tox (also called thevelocity vector). We will naturally saythatγ is parametrized by arclengthwhenever|V(x)| = 1 for all x in the domain ofγ.Assume thatγ is nondegenerate, that is,V(x) 6= 0 for all x ∈ U . We can then define thefirst vector in the Frénet frame, theunit tangent vector, as

e1(x) = V(x)

|V(x)| .


Define thegeodesic curvature(or first curvature) ofγ to be the length of the field∇e1e1,that is,k1 = |∇e1e1|.

One immediately sees from property (6) in Definition 1 of the Riemannian connectionthat the vector∇e1e1 must be orthogonal toe1 with respect to the Riemannian metric.In the case for whichk1(x) 6= 0, we can define thefirst normalto γ at x to be the unitvectore2(x) in the direction of∇e1e1(x), so that

∇e1e1 = k1e2.

Also using property (6) we see that

0= 〈∇e1e2,e1〉 + 〈e2,∇e1e1〉 = 〈∇e1e2,e1〉 + k1,

so that

〈∇e1e2+ k1e1,e1〉 = 〈∇e1e2+ k1e1,e2〉 = 0.

We callk2 = |∇e1e2+ k1e1| thetorsionof γ (or second curvature).Wheneverk2 6= 0 we can definethe second normalto γ to be the unit vectore3 in the

direction of∇e1e2+ k1e1, so that

∇e1e2 = k2e3− k1e1.

The process above can be continued to definek3, the third curvature, and wheneverk3 6= 0 we can definee4, the third normal, etc.

Definition 3. The orthonormal vectorsei , i = 1, . . . ,n are called theFrenet vectorsorFrenet frame. Equations

∇e1e1 = k1e2,

∇e1ei = ki ei+1− ki−1ei−1, i = 2, . . . ,n− 1, (2)

∇e1en = −kn−1en−1,

are called theFrenet formulae.

2.2. Definitions in the Theory of Integrable Systems

In this subsection we present the concepts about bi-Hamiltonian integrable systems thatwe need to use in the next section. Definitions and notations are mostly as in [Olv93].Another good introduction is [Dor93].

Let M ⊂ X × U be an open subset of the space of independent and dependentvariablesx = (x1, . . . , xp) andu = (u1, . . . ,uq). The algebra of differential functionsP[u] over M is denoted byA. We define the spaceF of functionalsP = ∫ Pdx as thequotient ofA by the image of the total divergence.

For a linear differential operatorD: Aq → Aq, which we can think of as aq × qmatrix differential operator depending onx, u, and derivatives ofu, we define a bracketonF as follows:

{P,Q} =∫δP ·DδQdx, (3)


whereδP is the variational derivative of functionalP and where by·we denote the usualdot product inRq.

Definition 4. A linear operatorD: Aq → Aq is called Hamiltonian if the bracket (3)satisfies the conditions of skew-symmetry

{P,Q} = −{Q,P}, (4)

and Jacobi identity

{{P,Q},S} + {{S,P},Q} + {{Q,S},P} = 0, (5)

for all functionalsP,Q,S ∈ F . The bracket (3) is calledPoisson bracket.

We say two Hamiltonian operatorsD andE form aHamiltonian pairor arecompatibleif every linear combinationaD + bE , a,b ∈ R is a Hamiltonian operator. In fact, weonly need to check whetherD + E is a Hamiltonian operator (Lemma 7.20 in [Olv93])to prove this. We say that four Hamiltonian operators form aHamiltonian quadruplet,or are compatible, if any two of them are compatible.

An evolution system is a Hamiltonian system if for a Hamiltonian operatorD, thereexists a functionalH, called Hamiltonian, such that

ut = K [u] = DδH, K [u] ∈ Aq.

If for a Hamiltonian pairD andE , there exists corresponding Hamiltonian functionalsH1 andH0 such that

ut = K [u] = DδH1 = EδH0, (6)

we say the evolution system is abi-Hamiltonian system.

Definition 5. A differential operatorD: Aq → Aq is degenerateif there is a nonzerodifferential operatorD: Aq → A such thatD ·D = 0.

In the field of nonlinear evolution equations, one important question to answer iswhether a given equation is integrable, in the sense that it has infinitely many commutingsymmetries.

Definition 6. We sayQ[u] ∈ Aq is asymmetryof ut = K [u] if and only if

[K , Q] = DQ[K ] − DK [Q] = 0,

whereDQ[K ] is the Frechet derivative ofQ in the direction ofK . If

DQ[K ] + D?K [Q] = 0,

whereD?K is the formal adjoint ofDK , thenQ[u] is acosymmetryof the equation.

For system (6), all variational derivatives of its Hamiltonian functionals are cosym-metries.


Definition 7. A linear differential operatorR: Aq → Aq is a recursion operatorofut = K [u] if it maps a symmetry to a new symmetry.

It follows thatR is a recursion operator ofut = K [u] if and only if

Rt = [DK ,R],

whereRt = ∂R∂t + DR[K ], assumingD−1

x Dx = 1. For more details, see [SW01a],[SW01b].

Related to the concept of the recursion operator is that of thehereditary operator. AnoperatorR is said to be hereditary if its Nijenhuis tensor vanishes, i.e.,

[RX,RY] −R[RX,Y] −R[X,RY] +R2[X,Y] = 0, for all X,Y ∈ domR.

Many recursion operators are hereditary, but one should notice that the definition ofhereditarydoes not need any specific equation; it is a geometric property of the operatordefining a structure on the space. Given a Hamiltonian pair(D, E), one constructs ahereditary operator by takingR = ED−1 if D is nondegenerate(cf. Theorem 7.24 in[Olv93] or Theorem 3.12 in [Dor93]).

The following is an important property: IfX is a symmetry of the hereditary operatorR, that is,DXR = [DK ,R], then for anyk, l ,

[Rk X,Rl X] = 0.

If there existsH1 ∈ F such thatR?δH0 = δH1, then for anyn ∈ N , there existsHn ∈ Fsuch thatR?nδH0 = δHn. This explains how the infinitely many conserved densities (orHamiltonians) arise, implying the integrability of the Hamiltonian system.

3. Hamiltonian Quadruplet

In this section we assume that we are working on a 3-dimensional Riemannian man-ifold M with constant curvature~. We remark that there are obvious generalizationsto n–dimensional Riemannian manifolds, but the exact connection with integrable sys-tems needs further study, which we plan to undertake. There are also generalizations toother homogeneous spaces. Many of these cases are still open. The following theoremgeneralizes results if [LP91] to the case of nonzero curvature.

Theorem 2. Let M be a 3–dimensional Riemannian manifold with constant curvature~, and letγ(x, t) be a family of curves on M satisfying a geometric evolution system ofequations of the form

γt = h1e1+ h2e2+ h3e3, (7)

where{e1,e2,e3} is the Frenet frame ofγ, and where h1, h2, h3 are arbitrary smoothfunctions of the curvatures k1, k2 and their derivatives with respect to x. Since we are in3–dimensional space, from now on we use the notationκ, τ for k1, k2.

Assume that x is the arclength parameter and that evolution (7) is arclength preserv-ing.


Then, the curvaturesκ, τ satisfy the evolution(κ

τ

)t

= P

(h3

h1

), (8)

where, if we denote by Dx the total differentiation operator with respect to x,

P =(

−τDx − Dxτ D2x

1κ

Dx − τ 2

κDx + Dxκ

Dx1κ

D2x − Dx

τ 2

κ+ κDx Dx

(τκ2 Dx + Dx

τκ2

)Dx + τDx + Dxτ

)

+ ~(

0 1κ

Dx

Dx1κ

0

). (9)

Proof. A short comment on the calculations to follow: Let us denote byT = γt ande1 =γx, assumingx to be arclength. These vectors are defined as the push-forward vectors ofthe vectors∂

∂t ,∂∂x , tangent to the domain ofγ, throughγ. That is, ifγ: U ⊂ R2 → M ,

thenγt = γ∗ ∂∂t acting on functions asγt ( f ) = ∂∂t f (γ(t, x)); likewise for x. Thus, by

applyingT or e1 to functions defined alongγ, we are indeed taking their derivativeswith respect tot or x, respectively. If (7) is arclength preserving, these two vectors willcommute since∂

∂t and ∂∂x commute andγ∗ preserves Lie brackets. The condition needed

to guarantee that (7) is arclength preserving will be given below.In order to find the evolution of the curvaturesκ andτ , we will also find the evolution

of the two first members of the frame,e1 ande2.It follows from property (5) of a Riemannian connection and the fact thatt andx

differentiation commute, ifγ is a solution of ( 7), that

∇Te1 = ∇e1T = ∇e1(h1e1+ h2e2+ h3e3). (10)

Using property (4) of∇ and Frenet formulae ( 2), evolution (10) can be rewritten as

∇Te1 = (h′1− κh2)e1+ (h′2+ h1κ − τh3)e2+ (h′3+ h2τ)e3, (11)

where we denote by′ the total differentiation of the functions with respect tox. In orderfor the flow to be arclength preserving, it suffices to have∇Te1 to be orthogonal toe1.This leads to

h2 = h′1κ; (12)

see [Ive01].The evolution ofκ can be found from here. On the one hand, we have

2κκt = (κ2)t = 2〈∇T (∇e1e1),∇e1e1〉 = 2κ〈∇T∇e1e1,e2〉. (13)

Meanwhile, from the definition of the curvature tensor

∇T∇e1e1 = ∇e1∇Te1+ R(T,e1)e1, (14)

whereR is the curvature tensor of the manifold. Therefore

κt = 〈∇e1∇Te1,e2〉 + K (e2,e1, T,e1), (15)


where K is the Riemann-Christoffel curvature tensor ofM . On the other hand, byapplying∇e1 to both sides of ( 11), one obtains

∇e1∇Te1 = ∇2e1

T = [(h′1− κh2)′ − κ(h′2+ h1κ − τh3)]e1 (16)

+ [(h′2+ h1κ − τh3)′ + κ(h′1− κh2)− τ(h′3+ h2τ)]e2

+ [(h′3+ h2τ)′ + τ(h′2+ h1κ − τh3)]e3,

with the simple use of the Fr´enet formulae. From these two relations we obtain theevolution ofκ as given by

κt = (h′2+ κh1− τh3)′ + κ(h′1− κh2)− τ(h′3+ τh2)+ K (e2,e1, T,e1). (17)

Applying the tensorial properties ofK and Theorem 1, we obtain

κt = (h′2+ κh1− τh3)′ + κ(h′1− κh2)− τ(h′3+ τh2)

+ h2K (e2,e1,e2,e1)+ h3K (e2,e1,e3,e1).(18)

We can now find the evolution ofe2 from the Frenet relationship

e2 = 1

κ∇e1e1. (19)

Indeed, if we apply∇T to (19), and we use that the result should be orthogonal toe2

(since it is a unit vector), this leads to

∇Te2 = 1

κ∇T∇e1e1− 1

κ〈∇T∇e1e1,e2〉e2. (20)

Substituting (14) into it, we obtain

∇Te2 = 1

κ∇2

e1T + 1

κR(T,e1)e1− 1

κ〈∇2

e1T,e2〉e2− 1

κK (e2,e1, T,e1)e2. (21)

We are now in a position to find the evolution ofτ . As we did in (13) forκ, it is verysimple to see that

τt = 〈∇T (∇e1e2+ κe1),e3〉. (22)

But

∇T∇e1e2 = ∇e1∇Te2+ R(T,e1)e2,

so that, applying∇e1 to (21) we obtain, after some short calculations,

〈∇e1∇Te2,e3〉 =(

1

κ〈∇2

e1T,e3〉

)′+(

1

κK (e3,e1, T,e1)

)′, (23)

and from here we obtain

τt = [ 1κ(h′3+ h2τ)

′ + τκ(h′2+ h1κ − τh3)]′ + κ(h′3+ h2τ)

+ [ h2κ

K (e3,e1,e2,e1)+ h3κ

sec(e1,e3)]′

+ h2K (e3,e2,e2,e1)+ h3K (e3,e2,e3,e1).

(24)


If we finally impose arclength preserving condition (12) to evolutions (17) and (24), weobtain that the evolution ofκ andτ can be written as(

κ

τ

)t

= P

(h3

h1

),

where, if Dx is the total derivative with respect tox, and we denoteKi jkr = K (ei ,ej ,

ek,er ),

P =(

−τDx − Dxτ D2x

1κ

Dx − τ 2

κDx + Dxκ

Dx1κ

D2x − Dx

τ 2

κ+ κDx Dx

(τκ2 Dx + Dx

τκ2

)Dx + τDx + Dxτ

)

+(

K21311κsec(e1,e2)Dx

K3231+ Dx1κsec(e1,e3)

1κK3221Dx + Dx

K3121κ2 Dx

). (25)

If the manifoldM has constant curvature~, Proposition 1 and Corollary 1 provide thevalues

K (e3,e1,e2,e1) = K (e3,e2,e1,e2) = K (e1,e3,e2,e3) = 0,

and if i 6= j ,

sec(ei,ej) = ~,for any i, j = 1, . . . ,3. We simply need to substitute the values in ( 25), to obtain theresult of this theorem.

The general scheme to obtain the evolution equations in then-dimensional arclength-preserving case runs as follows. Let us definek0 = 0. Then we use the following formulaeto inductively compute all derivatives with respect tot .

∇Tei =

= ∇e1T if i = 1,(∇e1∇Tei−1− ki−1,tei + ki−2,tei−2,

+ki−2∇Tei−2+ R(T,e1)ei−1)/ki−1 if i = 2, . . . ,n,

and

kit = 〈∇e1∇Tei + ki−1∇Tei−1,ei+1〉 + K (ei+1,ei , T,e1).

We now turn back to our 3-dimensional problem.

Theorem 3. The skew-symmetric operators

B =(−τDx − Dxτ − τ 2

κDx

−Dxτ 2

κ0

), (26)

C =(

0 1κ

Dx

Dx1κ

0

), (27)

D =(

0 Dxκ

κDx τDx + Dxτ

), (28)

E =(

0 D2x

1κ

Dx

Dx1κ

D2x Dx(

τκ2 Dx + Dx

τκ2 )Dx

), (29)


form a quadruplet of compatible Hamiltonian operators (where P= B+ ~C +D+ E ,cf. (9)).

Proof. We prove this by checking the conditions of Theorem 7.8 and Corollary 7.21 in[Olv93]. Here we give only the details to show thatD andE are compatible and leavethe remaining, identical computations to the reader.

First we check that the operatorsD andE are indeed Hamiltonian operators. Theassociated bivector ofD is by definition

2D = 12

∫(θ∧Dθ)dx

= 12

∫(κϑ ∧ ζ1+ κ1ϑ ∧ ζ + κζ ∧ ϑ1+ 2τζ ∧ ζ1)dx

=∫(κζ ∧ ϑ1+ τζ ∧ ζ1)dx,

whereθ = (ϑ, ζ ) andϑi = ∂ i ϑ∂xi , etc. Here∧ means that one needs to take the ordinary

inner product between the vectorsθ andDθ . The elements of these vectors are thenmultiplied using the ordinary wedge product. We need to check the vanishing of theSchouten bracket [D,D] which is equivalent to the Jacobi identity for the Lie bracketdefined byD.

[D,D] = pr vDθ (2D)

=∫((κ1ζ + κζ1) ∧ ζ ∧ ϑ1+ (κϑ1+ 2τζ1+ τ1ζ ) ∧ ζ ∧ ζ1)dx

=∫(κζ1 ∧ ζ ∧ ϑ1+ κϑ1 ∧ ζ ∧ ζ1)dx

= 0.

The proof thatE is Hamiltonian is a rather laborious calculation. Its associated bivector is

2E = 12

∫ (θ∧Eθ)dx

= 12

∫ (ϑ ∧ (D2

x

ζ1

κ)+ ζ ∧ (Dx

ϑ2

κ)+ ζ ∧ (Dx

τζ2

κ2)+ ζ ∧ (D2

x

τζ1

κ2)

)dx

=∫ (

1

κϑ2 ∧ ζ1+ τ

κ2ζ2 ∧ ζ1

)dx.

The vanishing ofpr vEθ (2E) can be proved by the fact that

pr vEθ (κ) = D2x

1

κζ1 ≡ −2κ1

κ2ζ2+ 1

κζ3 modζ1,

pr vEθ (τ ) = Dx1

κϑ2+ Dx

(τκζ2+ Dx

τ

κ2ζ1

)≡ − κ1

κ2ϑ2+ 1

κϑ3+ 2τ

κ2ζ3 modζ1 ∧ ζ2,


and by integration by parts:

[E, E ] = pr vEθ (2E)

=∫− 1

κ2pr vEθ (κ)ϑ2 ∧ ζ1− 2τ

κ3pr vEθ (κ)ζ2 ∧ ζ1+ 1

κ2pr vEθ (τ )ζ2 ∧ ζ1 dx

=∫− 1

κ2(−2κ1

κ2ζ2+ 1

κζ3) ∧ ϑ2 ∧ ζ1− 2τ

κ4ζ3 ∧ ζ2 ∧ ζ1

+ 1

κ2(− κ1

κ2ϑ2+ 1

κϑ3+ 2τ

κ2ζ3) ∧ ζ2 ∧ ζ1 dx

=∫

3κ1

κ4ζ2 ∧ ϑ2 ∧ ζ1− 1

κ3ζ3 ∧ ϑ2 ∧ ζ1+ 1

κ3ϑ3 ∧ ζ2 ∧ ζ1 dx

=∫

Dx

(1

κ3ϑ2 ∧ ζ2 ∧ ζ1

)dx = 0.

Now we prove thatD andE form a Hamiltonian pair by checking

[D, E ] + [E,D] = pr vDθ (2E)+ pr vEθ (2D) = 0.

We compute

[D, E ] + [E,D] = pr vDθ (2E)+ pr vEθ (2D)

=∫ (

1

κ2pr vDθ (κ) ∧ ζ1 ∧ ϑ2+ 2τ

κ3pr vDθ (κ) ∧ ζ1 ∧ ζ2− 1

κ2pr vDθ (τ ) ∧ ζ1 ∧ ζ2

+ pr vEθ (κ) ∧ ζ ∧ ϑ1+ pr vEθ (τ ) ∧ ζ ∧ ζ1

)dx

=∫ (

κ1

κ2ζ ∧ ζ1 ∧ ϑ2+ 2τκ1

κ3ζ ∧ ζ1 ∧ ζ2− 1

κ2(κϑ1+ τ1ζ ) ∧ ζ1 ∧ ζ2

+(

D2x

1

κζ1

)∧ ζ ∧ ϑ1+

(Dx

1

κϑ2

)∧ ζ ∧ ζ1+

(Dx

τ

κ2ζ2

)∧ ζ ∧ ζ1

+(

D2x

τ

κ2ζ1

)∧ ζ ∧ ζ1

)dx

=∫ (

κ1

κ2ζ ∧ ζ1 ∧ ϑ2+ 2τκ1

κ3ζ ∧ ζ1 ∧ ζ2− 1

κ2(κϑ1+ τ1ζ ) ∧ ζ1 ∧ ζ2

+1

κζ1 ∧ (ζ2 ∧ ϑ1+ ζ ∧ ϑ3)− 1

κϑ2 ∧ ζ ∧ ζ2− τ

κ2ζ ∧ ζ1 ∧ ζ3

)dx

=∫ (

2τκ1

κ3ζ ∧ ζ1 ∧ ζ2− τ

κ2ζ ∧ ζ1 ∧ ζ3− τ1

κ2ζ ∧ ζ1 ∧ ζ2

+ κ1

κ2ζ ∧ ζ1 ∧ ϑ2+ 1

κζ1 ∧ ζ ∧ ϑ3− 1

κζ ∧ ζ2 ∧ ϑ2

)dx

= −∫

Dx

(τ

κ2ζ ∧ ζ1 ∧ ζ2+ 1

κζ ∧ ζ1 ∧ ϑ2

)dx = 0.

Thus the result follows.


4. Integrable Evolutions

4.1. Two Integrable Canonical Evolution Equations

Having a Hamiltonian quadruplet, with three of its members,B, C, andD, being nonde-generate, allows us to produce two hereditary operators. We insist again on the fact thatthese are tensors linked to the intrinsic geometry of Riemannian curves and not to anyintegrable system in particular. On the other hand, they are indeed recursion operatorsfor two canonical integrable systems.

The Vortex Filament FlowThe Hamiltonian pairP = B + ~C +D + E andC gives us the hereditary operator:

R1 = PC−1

=(

D2x − τ 2+ κ2+ ~ −2κτ

2D2xτκ− Dx(

τ1κ− 2κ1τ

κ2 )+ 2κτ D2x + 2Dx

κ1κ+ κ2

κ− τ 2+ κ2+ ~

)+(−2τκ1− κτ1κ3κ− κ1κ2

κ2 − 2ττ1+ κκ1

)D−1

x (0 1)

+(κ1

τ1

)D−1

x (κ 0).

From its expression, one can identify equations{κt1 = −2τκ1− κτ1,

τt1 = κ3κ− κ1κ2

κ2 − 2ττ1+ κκ1,(30)

and {κt3 = κ3+ 3

2κ2κ1− 3κ1τ

2− 3κττ1+ ~κ1,

τt3 = Dx(τ2+ 32κ

2τ − τ 3+ 3κ2τκ+ 3κ1τ1

κ+ ~τ), (31)

the latter beingR1(κ1

τ1), as symmetries of the operatorR1. One can also identify their

cosymmetries to be(κ,0) and (0,1), deriving from conservation laws with densitiesκ2

2 andτ , respectively. These (commuting) equations have the operatorR1 as recursionoperator ([SW01a]), which generates a hierarchy of symmetries, cosymmetries, andconservation laws for the equations.

Equations (30) and (31) are the evolutions of curvature and torsion associated to thebest-known versions of the vortex filament equations. Vortex filament equations are thenonlinear evolution of curves describing the time development of a very thin vortex tube.The ones associated to our integrable systems are

γt1 = κe3; (32)

γt3 =1

2κ2e1+ κ ′e2+ κτe3. (33)

In the Euclidean case, Hasimoto found the transformation from ( 32) to the nonlinearSchrodinger equation, which enabled him to study its geometry and integrability, cf.


[Has72]. Some generalizations have been extensively studied in [LP91], [DS94], and[YS98]. The curve evolution (33) and its relation to (32) was also studied in [LP91].

Evolution equation (30) is easily seen to be a bi-Hamiltonian system since

ut1 = CδH1 = (B + ~C +D + E)δH0,

whereu = (κ, τ ),H0 =∫κ2

2 dx, andH1 =∫ (

18κ

4− 12κ

21 − 1

2τ2κ2)

dx. In fact, noticethatH0 is in the kernel ofC. This fact allows us to drop~C from the Hamiltonian pair ofthis particular equation and to consider the simpler recursion operator(B+D+ E)C−1,corresponding to the flat case, as valid for the general case. This simplification does nothold for equation ( 31). However, it is also bi-Hamiltonian, as shown in [LP91], since

ut3 = CδH3 = (B + ~C +D + E)δH2,

withH2 =∫

12κ

2τdx andH3 =∫(κκ2τ − 1

2κ21τ − 1

2τ3κ2+ 3

8κ4τ)dx+ ~H2.

Therefore, evolutions (30) and (31) are completely integrable systems. Geometricevolutions (32) and (33) would also be integrable in the sense that their associatedκ, τ

evolutions are, and given thatκ(t, x), τ (t, x) determineγ (t, x) up to the action of thegroup.

After minor calculations, one can easily see that the operatorD is not a Hamiltonianoperator for either of these two equations.

A Second Integrable SystemSince the operatorD is also nondegenerate, the different choice of Hamiltonian pairJ = B + ~C + E andD (we will see shortly the reason why we dropD in this pair)gives us the hereditary operator:

R2 = JD−1

=(

1κ2 D2

x − 5κ1κ3 Dx − 4κ2

κ3 + 12κ21

κ4 + 3τ 2

κ2 − 2τκ

− 2τ1κ3 Dx + 2τ 3

κ3 − 3τ2κ3 + 9κ1τ1

κ41κ2 D2

x − 3κ1κ3 Dx − κ2

κ3 + 3κ21

κ4 − τ 2

κ2

)

+ ~( 1κ2 0−2 τ

κ31κ2

)

−(κ3κ3 − 9κ1κ2

κ4 + 12κ3

1κ5 − 3ττ1

κ2 + 3τ2κ1κ3 + ~ κ1

κ3

τ3κ3 − 6κ1τ2

κ4 − 3κ2τ1κ4 − 3τ

2τ1κ3 + 3κ1τ

3

κ4 + 12κ2

1τ1

κ5 + ~(τκ3

)1

)D−1

x (1 0)

−(

τ12ττ1κ− τ 2κ1

κ2 + ~ κ1κ2

)D−1

x

(− τκ2

1

κ

).

Again, from its expression we can identify the cosymmetries inR2, that is,(1, 0) and(− τ

κ2 ,1κ), deriving from conservation laws, with densitiesκ and τ

κ, respectively.

The equations {κt1 = τ1,

τt1 = 2ττ1κ− τ 2κ1

κ2 + ~ κ1κ2 ,

(34)


and κt3 = κ3κ3 − 9κ1κ2

κ4 + 12κ3

1κ5 − 3ττ1

κ2 + 3τ2κ1κ3 + ~ κ1

κ3 ,

τt3 = τ3κ3 − 6κ1τ2

κ4 − 3κ2τ1κ4 − 3τ

2τ1κ3 + 3κ1τ

3

κ4 + 12κ2

1τ1

κ5 + ~( τκ3 )1,(35)

are symmetries of the operator, and so they haveR2 as recursion operator.Equation (35) is a generalization of the equations studied by Ivey to the case of

manifolds with constant curvature.Again, evolution equation ( 34) is easily seen to be a bi-Hamiltonian system. This is

true since

ut1 = DδH1 = J δH0,

whereu = (κ, τ ),H0 = −∫κdx, andH1 =

∫( τ

2

2κ +~ 12κ )dx. We also have that equation

(35) is bi-Hamiltonian:

ut3 = DδH3 = J δH2,

with H2 = −∫τκdx andH3 =

∫ ( τκ21

κ5 − κ1τ1κ4 − τ 3

2κ3 − ~ τ2κ3

)dx. However, operatorC

is not a Hamiltonian operator for either of these two equations. This implies thatC is notin the hierarchy generated byD andE . Therefore,the quadrupletB, C,D, E is a trueHamiltonian quadruplet.

Notice that bothH0 andH2 lie on the kernel of the operatorD. This explains why fromthe beginning we droppedD from the Hamiltonian pair, considering(B + ~C + E, C)rather than the expected, but more complicated,(B + ~C + D + E, C). Thus, (34) and(35) still have geometric evolutions of curves associated to them. Namely,

γt1 = −e3

is associated with (34), and the one associated with (35) is

γt3 = −1

κe1+ κ1

κ3e2+ τ

κ2e3.

They would also be integrable in the same sense that was mentioned above for theprevious set of integrable equations.

4.2. Some Transformations

In this section we describe the transformations that simplify our integrable hierarchies,the well–known Hasimoto transformation, which takes the vortex filament flow to thenonlinear Schr¨odinger equation, and the hodographic transformation taking the secondhierarchy in the nonflat case to a decoupled potential KdV system. We first describebriefly how a change of variable will affect the Poisson brackets. Suppose we havecoordinates(x,u) and a Poisson bracket in these coordinates given by

{ f, g} =∫δ f

δu·D δg

δudx.

We map(y, v) by an invertible map8 to (x,u) coordinates. The effect of this change ofvariables on the variational derivatives off andg is known to be given by

(δvδu

)∗ δ8∗ fδv

and


( δvδu )∗ δ8∗gδv

, respectively, where( δvδu )∗ denotes the operator conjugate toδv

δu and8∗ f =f (8(y, v)). From here we obtain the transformation of our Poisson bracket under themap8

8∗{ f, g} =∫ [(

δv

δu

)∗δ8∗ f

δv

]·[D(δv

δu

)∗δ8∗gδv

] ∣∣∣∣δxδy∣∣∣∣ dy.

Let8∗D = δvδuD(

δvδu )∗| δxδy |. Then we define

{ f , g}′ =∫δ f

δv·8∗D δg

δvdy (36)

to be the new bracket, with associated tensor8∗D. Since we have

8∗{ f, g} = {8∗ f,8∗g}′,the skew-symmetry and Jacobi identity follow automatically.

The Hasimoto TransformationWe describe how the first integrable group of equations changes under the Hasimototransformation:

φ = κei∫τdx.

Notice that

φt = (κt + i κ∫τt dx)ei

∫τdx,

φ1 = (κ1+ i κτ)ei∫τdx,

φ2 = (κ2+ 2i κ1τ + i κτ1− κτ 2)ei∫τdx,

φ3 = (κ3+ 3i κ2τ + 3i κ1τ1+ i κτ2− 3κττ1− 3κ1τ2− i κτ 3)ei

∫τdx.

We can now rewrite (30) as

φt2 = iφ2+ i

2|φ|2φ, (37)

which is the nonlinear Schr¨odinger equation, and ( 31) as

φt3 = φ3+ 3

2|φ|2φ1,

which is the modified KdV equation.To show how the transformation affects the Hamiltonian pair, we write it as

u+ i v = κei∫τdx,

which gives the matrix

Q =(

cos(∫τdx) −κ sin(

∫τdx)D−1

x

sin(∫τdx) κ cos(

∫τdx)D−1

x

)


describing the action of the transformation on the vector fields. The Hamiltonian operator,sayC, transforms asC 7→ QCQ∗, according to (36), sincex = y so thatδx

δy = 1. Wefind that

QCQ∗ =(

0 −1

1 0

),

and

QDQ∗ =(−vD−1

x uDx + DxuD−1x v −DxuD−1

x u− vD−1x vDx

uD−1x uDx + DxvD−1

x v uD−1x vDx − DxvD−1

x u

),

Q(B + E)Q∗ =(

0 −D2x

D2x 0

).

The new recursion operator, whichR1 transforms to, is

R1 = ~I

+(

D2x+u2+v2+u1D−1

x u−vD−1x v1 vD−1

x u1+u1D−1x v

uD−1x v1+v1D−1

x u D2x+u2+v2−uD−1

x u1+v1D−1x v

)

= ~I−( −vD−1

x u −Dx − vD−1x v

Dx+uD−1x u uD−1

x v

)2

= ~I−R2nls,

where I denotes the identity matrix. It is interesting to notice thatRnls is a recursionoperator of ( 37), as found by Magri [Mag78]. In fact, if we look at the recursion operatorprior to the Hasimoto transformation, we now realize that

R1 = ~I −(

−τ −DxκD−1x

Dx1κ

Dx + κ −DxτD−1x

)2

.

Furthermore, if we let

F =(

Dxτκ

Dx

Dxτκ−Dx

),

and

G =(

0 00 Dx

1κ

Dx1κ

Dx

),

thenF andG are compatible Hamiltonian operators, such that

((F − G)C−1)2 = ~ I −R1.

Indeed, we can now see thatB+D+ E = −(F − G)C−1(F − G) and hence, evolution(30) is bi-Hamiltonian with respect to the simpler Hamiltonian pair{G −F, C}. Its newassociated Hamiltonian is given byH0 = 1

2

∫κ2τdx. This is exactly the Hamiltonian

functional corresponding to (31). Thusin the flat casethe second evolution (31) is nowin the hierarchy of (30) with respect to the new pair, as is already known (see [LP91]).


Proposition 2. {B, C,F,G} form a Hamiltonian quadruplet.

Proof. The proof of this proposition is a straightforward calculation of the kind per-formed in Theorem 3.

Notice that the classical filament flow equation (30) is identically induced for bothflat and nonflat cases, sinceH0 is in the kernel ofC. We consider the Hamiltonian pair(E+D, C), and indeed(G−F, C), as integrating it. The second evolution in the hierarchywouldnot be (31) for the nonflat case, but rather its flat analogue.

The Hodographic TransformationWe describe first the transformation, used in [Ive01], that affects the second integrablegroup of equations. We will see later the effect on the associated Hamiltonian pair.

The first step in this transformation is to defineκ = p1 andτ = q1, so that bothsystems (34) and (35) become{

pt1 = q1,

qt1 = q21

p1− ~ 1

p1,

(38)

and pt3 = p3

p31− 3

p22

p41− 3

2q2

1

p21− ~ 1

2p21,

qt3 = q3

p31− 3 p2q2

p41− q3

1

p31+ ~ q1

p31.

(39)

We now define thehodographtransformation (cf. [Olv93]) by

u = x, v = q, y = p.

Later on we refer to the composition of taking the potential and the hodograph transfor-mation simply as the hodographic transformation. If the old independent variables are(x, t), we call the new(y, t

′) and we find that

1 = ∂x

∂x= ∂u

∂y

∂p

∂x+ ∂u

∂t ′∂t

∂x= ∂u

∂y

∂p

∂x= u1 p1,

0 = ∂x

∂t= ∂u

∂y

∂p

∂t+ ∂u

∂t ′∂t

∂t= ∂u

∂y

∂p

∂t+ ∂u

∂t ′= u1 pt + ut ′ ,

q1 = ∂q

∂x= ∂v

∂y

∂p

∂x+ ∂v

∂t ′∂t

∂x= ∂v

∂y

∂p

∂x= v1 p1,

qt = ∂q

∂t= ∂v

∂y

∂p

∂t+ ∂v

∂t ′∂t

∂t= v1 pt + vt ′ .

This implies

u1 = 1

p1, u2 = ∂u1

∂x

∂x

∂y= − p2

p21

u1 = − p2

p31

,

u3 = ∂uyy

∂x

∂x

∂y= −u1

∂

∂x

p2

p31

= − p3

p41

+ 3p2

2

p51

,


ut ′ = −u1 pt = − pt

p1, v1 = q1

p1,

vt ′ = qt − v1 pt = qt − q1

p1pt .

We can now rewrite the equations as{ut′1= −v1,

vt′1= −~u1,

(40)

and {ut′3= u3+ 3

2u1v21 + 1

2~u31,

vt′3= v3+ 1

2v31 + 3

2~v1u21.

(41)

We show later that the second equation is in the hierarchy of the first, also in the nonflatcase!

Let us look at equation (41) a little bit closer. Assume~ 6= 0 and letu = u andv = 1√

~v. Then {

ut′3= u3+ 3

2~u1v21 + 1

2~u31

vt′3= v3+ 1

2~v31 + 3

2~u21v1

.

Let X = u+ v andY = u− v. Then{Xt′3= X3+ 1

2~X31

Yt′3= Y3+ 1

2~Y31

, (42)

which is a decoupled set of potential modified KdV equations. This simplification onlyoccurs in the nonflat case, while the flat case cannot be decoupled. In this sense, the flatcase is a singular case in the family.

Notice that if we takeτ = 0, the 2-dimensional case, system (35) reduces to

κt3 =κ3

κ3− 9

κ1κ2

κ4+ 12

κ31

κ5+ ~ κ1

κ3, (43)

the generalization of the evolution found in [Ive01] to nonflat Riemannian manifolds withconstant curvature. Using the hodographic transformation, this equation would become

ut′3= u3+ 1

2~u3

1. (44)

Next we describe the transformation as applied to the Hamiltonian pairs. The matrices

Q =(

D−1x 0

0 D−1x

),

and

R=(− 1

p10

− q1

p11

),


describe the action of the transformations on the vector fields(κt

τt

),

(pt

qt

),

respectively, the Hamiltonian operator, sayD, transforms as

D 7→ QDQ∗ 7→ RQDQ∗R∗.

We remark thatDx = 1u1

Dy, as can easily be checked from the definitions of the secondtransformation. We find that

B = RQBQ∗R∗∣∣∣∣δxδy

∣∣∣∣ =(

u1D−1y v1+ v1D−1

y u1 v1D−1y v1

v1D−1y v1 0

),

C = RQCQ∗R∗∣∣∣∣δxδy

∣∣∣∣ =(

0 u1D−1y u1

u1D−1y u1 v1D−1

y u1+ u1D−1y v1

),

D = RQDQ∗R∗∣∣∣∣δxδy

∣∣∣∣ =(

0 D−1y

D−1y 0

),

and

E = RQEQ∗R∗∣∣∣∣δxδy

∣∣∣∣ =(

0 Dy

Dy 0

).

The hodographic transformation is a Poisson transformation between the quadru-plet {B, C,D, E} and the new quadruplet{B, C, D, E}. The following corollary is thusobvious.

Corollary 2. The operatorsB, C, D, E form a Hamiltonian quadruplet. Equation (41)is bi-Hamiltonian with respect to the Hamiltonian pair(B+~C+ E, D), with associatedHamiltonian functionals

H2 = −∫

u1v1 dy andH3 =∫ (

u2v2− 1

2v3

1u1− 1

2~v1u3

1

)dy,

respectively.

Note: Strictly speaking, the definition of Poisson brackets given in Section 2.2 does notinclude these tensors, since their entries are not defined in terms of differential operators.Nevertheless, they are if we allow nonlocal terms and the formalization of this kind ofPoisson geometry.

Our final theorem states that the hodographic transformation, followed by the decou-pling map, is in fact a Poisson map between the space endowed with the Poisson bracketdefined byE + B + ~C and the space endowed with a decoupled pair of the knownpotential modified KdV Hamiltonian structure. We are not entering here into the detailsof how these spaces should actually be defined, to avoid further complications.


Theorem 4. Consider the decoupling transformationσ(y,u, v) = (y, X,Y), where Xand Y are defined as in (42). Denote by H the hodographic transformation. Then, if~ 6= 0, H followed byσ is a Poisson map takingE + B + ~C to a decoupled pair ofmodified potential KdV Hamiltonian structures

B + ~C + E = 2√~

(X1D−1

y X1+ ~Dy 0

0 −Y1D−1y Y1− ~Dy

). (45)

Proof. It is a straightforward calculation to show that

σ ∗H∗(B + ~C + E) = 2√~

(X1D−1

y X1+ ~Dy 0

0 −Y1D−1y Y1− ~Dy

),

by conjugation, as usual (see (36)). The tensorX1D−1y X1+ ~Dy is well-known to be a

Hamiltonian structure for the potential KdV. Potential modified KdV equation is obtainedwith the particular choice of HamiltonianH = −

√~

4

∫X2

1dy.

The new recursion operator, whichR2 transforms to, is

R2 =(

u1D−1y v1+ v1D−1

y u1 v1D−1y v1+ Dy

v1D−1y v1+ Dy 0

)(0 Dy

Dy 0

)

+ ~(

0 u1D−1y u1

u1D−1y u1 v1D−1

y u1+ u1D−1y v1

)(0 Dy

Dy 0

)

=(

Dy2+ v2

1 + ~u21 2u1v1

2~u1v1 Dy2+ v2

1 + ~u21

)

−N

(u1

v1

)D−1

y (v2 u2)−(

u1

v1

)D−1

y (v2 u2)N,

where

N =(

0 1

~ 0

).

Applying R2 to (u1

v1

),

we obtain (u3+ 3

2u1v21 + 1

2~u31

v3+ 12v

31 + 3

2~u21v1

),


so now this third-order equation is in the image ofR2. This was not the case for thecorresponding equation (35). The present coordinates give us a clearer idea of the in-terrelation of the two integrable systems (34) and (35) as finally belonging to the samehierarchy.

We notice that apparently we can multiply a symmetry withN and obtain a newsymmetry. This is becauseN and R2 commute. Thus we can considerN as anotherrecursion operator. It is easy to check that it is hereditary. The complete hierarchy isgenerated by applying powers ofR2 to the trivial symmetry(

u1

v1

),

and then adding the image of these underN to this collection.

5. Conclusions

In this paper we associate a quadruplet of compatible Hamiltonian structures to the in-trinsic geometry of curves in 3-dimensional Riemannian manifolds. From Theorems 2and 3 we can conclude the following interesting fact: If a flow of curves evolves followingan arclength-preserving evolution of the form (7) with(h3, h1) being the gradient of acertain functionalHwith respect toκ andτ , then the associated flow of the curvatures isHamiltonian with respect to the Hamiltonian structure ( 9) presented here, with Hamilto-nian functionalH. The condition of being a gradient amounts to the Fr´echet derivative of(h3, h1) being self-adjoint, and it is a rather natural condition one needs to impose to havea Hamiltonian. Furthermore, even if no Hamiltonian functional is associated to(h3, h1),the flow of the curvature evolution (8) will still lie on the Poisson leaves correspondingto the Poisson structure defined by (9), so that any possible Casimir element would beconstant along the flow. This would be true forany arclength-preserving evolution ofcurvatures induced by geometric evolutions of Riemannian curves.

If in addition to (h3, h1) representing the gradient of a Hamiltonian functional, theassociated(κ, τ ) evolution is also Hamiltonian with respect to either (28) or (29), thenthe PDE evolution would be completely integrable. We presented several examplesof such evolutions, namely the well-known vortex filament equations and the systemof equations announced in [Ive01]. We unified the study of the vortex filament flowequations on Riemannian manifolds with constant curvature and we studied the geometryof the second group of integrable systems in detail, showing that, in the nonflat case, it isPoisson equivalent to a system of decoupled potential KdV equations. To our knowledge,a complete classification of integrable systems associated to this Hamiltonian quadruplethas not yet been found and it is an interesting problem in itself. Ivey sets out to classifyintegrable arclength-preserving geometric evolution equations for the planar Euclideancase in [Ive01].

The Poisson geometry of infinite-dimensional Poisson brackets is largely unexplored,except for some special examples. The implications of the geometrical relationshippresented here for the Poisson geometry of the associated brackets is a problem whichhas not been studied and that could have very important consequences. The presence of


the projective group is essential in the understanding of the Poisson leaves of the Adler–Gel’fand–Dikiı bracket, and so one could expect a similar role in the case of the Euclideangroup as related to these Riemannian manifolds. The geometrical reasons that allow theappearance of these tensors are not at all understood—not only which geometrical factsproduce a skew-symmetric tensor but, more interestingly, which geometrical propertiesallow one to obtain the Jacobi identity. The resolution of this question would be ofthe highest interest, and it might link these Hamiltonian evolutions to evolutions of theDrinfel’d and Sokolov type (cf. [DS84]). The study of other homogeneous spaces is stillopen. Some cases are currently under consideration by the authors.

Acknowledgment

J. P. Wang gratefully acknowledges support from Netherlands Organization for ScientificResearch (NWO) for this research. The authors would like to thank the referees for theircareful reading of the manuscript.

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