Integrable Flows for Starlike Curves in Centroaffine Space

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and prove that the restricted hierarchy of flows on curves that project to conics in ${\mathbb{RP}}^2$ induces the Kaup-Kuperschmidt hierarchy at the curvature level.


Integrable evolutions of space curves
Much of the work on integrable curve evolution equations has been guided by the fundamental role played by the differential invariants of the curve (e.g., curvature and torsion in the Euclidean setting) in helping identify the curve evolution as an integrable one. Perhaps the most important example in the case of space curves is that of the Localized Induction Equation (LIE) describing the evolution of a curve with position vector γ (x, t) in R 3 , and Euclidean arclength parameter x. The complete integrability of equation (1.1) was uncovered by the realization, due to Hasimoto [9], that the function ψ = κ exp(i τ dx), of the curvature κ and torsion τ of γ , is a solution of the cubic focusing nonlinear Schrödinger (NLS) equation iψ t + ψ xx + 1 2 |ψ| 2 ψ = 0, one of the two best-known integrable nonlinear wave equations (the other being the KdV equation).
In this paper, we also use as a guiding principle the observation that many (but not all) integrable curve evolutions have the property of local preservation of arclength, i.e., the associated This paper is a contribution to the Special Issue "Symmetries of Differential Equations: Frames, Invariants and Applications". The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html vector fields satisfy a non-stretching condition. For example, the LIE vector field W = γ x × γ xx satisfies the condition δ W γ x = 0, where δ W denotes the variation in the direction of W . Thus the local arclength parameter x is independent of t, and the compatibility conditions γ xt = γ tx , γ xxt = γ txx , γ xxxt = γ txxx (more commonly written as compatibility conditions of the Frenet equations and the evolution equations for the Frenet frame) turn out to be equivalent to the Lax pair of the NLS equation for ψ.
Indeed, many integrable curve evolutions in various geometries have been found by looking for non-stretching vector fields that produce compatible equations for the moving frame of the evolving curve; in the case of space curves, the geometries explored include Euclidean [11], spherical [6], Minkowski [21], affine and centroaffine [4]. (Moreover, integrable curve evolutions without preservation of arclength have been found in projective [17], conformal [16] and other parabolic geometries.) The approach in these investigations involves finding suitable choices for the coefficients of the non-stretching vector fields (relative to a Frenet-type frame) and often assuming special relations among the differential invariants; thus it can be challenging to identify integrable hierarchies.
Another approach to investigating the relation between a non-stretching curve evolution and the integrable PDE system satisfied by the differential invariants is to seek a natural Hamiltonian setting for the curve flow. The LIE was shown by Marsden and Weinstein [18] to be a Hamiltonian flow on a suitable phase space endowed with a symplectic form of hydrodynamic origin (see also [1,2]). In a fundamental paper [14] Langer and Perline used this framework to explore in depth the correspondence between the LIE and NLS equations and, along the way, derived a geometric recursion operator at the curve level that made it easy to obtain the integrable hierarchies of both curve and curvature flows, as well as meaningful reductions thereof [12,13].
In this article we study integrable evolution equations for closed curves in centroaffine R 3 beginning, as in [14], with a natural pre-symplectic form on an appropriate infinite-dimensional phase space. The Hamiltonian setting allows us to construct integrable hierarchies of curve flows and the associated families of integrable evolution equations for the centroaffine differential invariants (which turn out to be equivalent to the Boussinesq hierarchies). The motivation for addressing the centroaffine case comes from an interesting article by Pinkall [26], who derived a Hamiltonian evolution equation on the space of closed nondegenerate curves in the centroaffine plane. The simple definition of the symplectic form in the planar case (related to the SL(2)invariant area form) suggests that an analogous description may be possible in the 3-dimensional case, where a parallel could be drawn with the more familiar Euclidean case treated by [14].
Before describing the organization of the paper, we briefly discuss Pinkall's original setting and some results of ours for the planar case.

Pinkall's f low in R 2
Centroaffine differential geometry in R n refers to the study of submanifolds and their properties that are invariant under the action of SL(n), not including translations 1 . For example, a parametrized curve γ : for all x ∈ I, and this property is clearly invariant under the action of SL(n). Thus, for these curves the integral γ, γ , . . . , γ (n−1) 2/n(n−1) dx (1.2) is SL(n)-invariant, and represents the centroaffine arclength, where for the sake of convenience we use the notation |X 1 , . . . , X n | := det(X 1 , . . . , X n ) for n-tuples of vectors X i ∈ R n . (The fractional power in (1.2) is necessary to make the integral invariant under reparametrization.) In the case where n = 2, Pinkall [26] defined a geometrically natural flow for nondegenerate curves in R 2 , which he referred to as star-shaped curves, as follows. Suppose that γ is parametrized by centroaffine arclength s, so that | γ , γ | = 1 identically. It follows that γ ss = −p(s) γ where p(s) is defined as the centroaffine curvature. Along a closed curve γ , one defines the skew-symmetric form where X and Y are vector fields along γ . This pairing is nondegenerate on the space of vector fields that locally preserve arclength. Then the symplectic dual with respect to (1.3) of the functional γ p(s) ds is the vector field Pinkall's flow γ t = 1 2 p s γ − p γ s induces an evolution equation for curvature that coincides with the KdV equation, up to rescaling. In an earlier paper [3], we showed how to use solutions of the (scalar) Lax pair for KdV to generate solutions of Pinkall's flow. In particular, we showed that varying the spectral parameter in the Lax pair for a fixed KdV potential q corresponds to constructing a solution to the flow with curvature given by a Galileian KdV symmetry applied to q. We also derived conditions under which periodic KdV solutions corresponded to smoothly closed loops (for appropriate values of the spectral parameter) and illustrated this using finitegap KdV solutions.

Organization of the paper
In Section 2 we introduce basic notions concerning the differential geometry of nondegenerate curves in centroaffine R 3 , including centroaffine arclength, differential invariants, and nonstretching curve variations. This section also contains a discussion of the relation between nondegenerate curves and parametrized maps into RP 2 . In Section 3 we generalize Pinkall's setting to R 3 by introducing a pre-symplectic form on the space of closed unparametrized starlike curves; we also compute Hamiltonian vector fields associated with the total length and total curvature functionals. Flow by these vector fields induces evolution equations for the differential invariants; we discuss these equations in Section 4, including their bi-Hamiltonian formulation, Lax representation, and the connection with the Boussinesq equation. In Section 5 we show that the Poisson operators introduced in Section 4 give rise to the Boussinesq recursion operator, generating a (double) hierarchy of commuting evolution equations for the differential invariants. In Theorem 5.4, we relate the Hamiltonian structure for starlike curves and the Poisson structure for the differential invariants, and obtain a double hierarchy of centroaffine geometric evolution equations. We conclude Section 5, and the paper, by considering which of these flows preserve the property that γ corresponds to a conic under the usual projectivization map π : R 3 → RP 2 . We show that the sub-hierarchy of conicity-preserving curve evolutions induces the Kaup-Kuperschmidt hierarchy at the curvature level.
2 Centroaf f ine curve f lows in R 3

Centroaf f ine invariants
Let γ : I → R 3 be nondegenerate. We parameterize γ by centroaffine arclength, so that | γ , γ , γ | = 1. (2.1) We assume for the rest of this subsection that x is an arclength parameter. It follows by differentiating (2.1) with respect to x that for some functions p 0 (x) and p 1 (x). As explained below, these constitute a complete set of differential invariants for nondegenerate curves.
They define invariants κ and τ which correspond to −p 1 and p 0 respectively. Labeling p 0 as torsion is appropriate, since nondegenerate curves that lie in a plane in R 3 (not containing the origin) are exactly those for which p 0 is identically zero.
Remark 2.2. Some insight into the meaning of the centroaffine curve invariants can be gained by considering the relationship between γ and the corresponding parametrized curve Υ = π • γ in RP 2 , where π : R 3 → RP 2 is projectivization. The nondegeneracy condition on γ corresponds to Υ being regular and free of inflection points. Conversely, any such parametrized curve Υ : R → RP 2 has a unique lift to γ : R → R 3 which is centroaffine arclength-parametrized; we refer to γ as the canonical lift of Υ. When written in terms of Υ instead of γ , the invariants p 0 and p 1 are (up to sign) the well-known Wilczynski invariants [30]. Since these invariants define a differential equation whose solution determines the curve uniquely up to the action of the group SL(3), any other differential invariant must be functionally dependent on p 0 , p 1 and their x-derivatives. According to Ovsienko and Tabachnikov [25], the cubic differential (p 0 − 1 2 p 1 )(dx) 3 has the interesting property that it is invariant under reparametrizations of Υ. Curves in RP 2 for which this differential vanishes identically are conics. For curves for which the coefficient p 0 − 1 2 p 1 is nowhere vanishing, one can define the projective arclength differential (p 0 − 1 2 p 1 ) 1/3 dx. Those parametrized curves in RP 2 for which p 0 − 1 2 p 1 = C (a nonzero constant) are parametrized proportional to projective arclength, and we use the same terminology for their canonical lifts into R 3 . (Note that, in this case, the projective arclength differential is C 1/3 times the centroaffine arclength differential dx.) Along a nondegenerate curve, an analogue of the Frenet frame is provided by vectors γ , γ , γ . In fact, if we combine them as columns in an SL(3)-valued matrix W = ( γ , γ , γ ), then the analogue of the Frenet equations is However, for later use it will be convenient to define a different SL(3)-valued frame F (x) = ( γ , γ , γ − p 1 γ ) which satisfies the Frenet-type equation where k 1 = p 1 , k 2 = p 0 − p 1 . Of course, k 1 , k 2 also constitute a complete set of differential invariants, and we will come to use these in place of the Wilczynski invariants from Section 4 onwards.
In particular, the variation X preserves the centroaffine arclength differential if and only if i.e., We refer to vector fields of this form as non-stretching, since not only do such variations preserve the overall arclength of, say, a closed loop, but also no small portion of the curve is stretched or compressed.
3 Hamiltonian curve f lows

Symplectic structure on starlike loops
Generalizing Pinkall's setting [26] for planar star-shaped loops to the three-dimensional case, we introduce the infinite-dimensional space , γ is a closed starlike curve parametrized by centroaffine arclength; then a vector field X = a γ + b γ + c γ is in the tangent space T γ M if and only if X is of the form (2.5), where the coefficients a and c are 2π-periodic functions of x and satisfy the "zero mean" condition γ a + 2 3 p 1 c dx = 0, ensuring that the coefficient of γ in (2.5) is also periodic.
On V , define the skew-symmetric form Note that ω is automatically closed (that is, dω = 0) since the integrand in (3.1) is a volume form on R 3 [1,2].
Assuming that γ ∈ M and X, Y ∈ T γ M , then ω γ (X, Y ) = 0 for all Y if and only if a = − 2 3 p 1 c and c is constant. Thus, the restriction of ω to M is a degenerate closed 2-form (a pre-symplectic form), with kernel given by the subspace RZ 0 + RZ 1 of constant-coefficient linear combinations of the vector fields Z 0 = γ and Z 1 = γ − 2 3 p 1 γ (corresponding to c = 0 and c = 1 respectively). Note that this degeneracy is the result of restricting the 2-form (3.1) to the space of closed curves satisfying the arclength constraint. Degeneracy coming from constraints is common when defining symplectic structures on loop spaces [19] and phase spaces of nonlinear evolution equations [7].
Remark 3.1. The generators Z 0 and Z 1 of the kernel of ω will turn out to be the seeds of the double hierarchy of curve flows discussed in Section 5. A similar situation is encountered for the LIE hierarchy [14], where the seed γ spans the kernel of the natural pre-symplectic form on loops in Euclidean R 3 .
One could attempt to remove the degeneracy by constructing a quotient of M with respect to group actions generated by flowing by Z 0 and Z 1 . Flow by Z 0 generates an action of the additive group R, simply by translation in x. The resulting quotient space M = M /R can be identified with the space of unparametrized starlike loops. Because of translation invariance, ω descends to a give a well-defined closed 2-form on M , with one-dimensional kernel RZ 1 . On the other hand, we do not know of a natural geometric interpretation for the quotient of M by flow under Z 1 , on which ω would become non-degenerate.
However, ω can still be used to define a link between vector fields and functionals, and we will see that Z 1 is linked in this way to the arclength functional.

Examples
Recall that the correspondence between vector fields X H and (differentials of) Hamiltonians H ∈ C ∞ (M ) on a manifold M with symplectic form ω is defined by the relation

3)
X H being the Hamiltonian vector field corresponding to H. However, when ω is degenerate the correspondence is no longer an isomorphism: for those functionals H for which there is a Hamiltonian vector field, X H is only defined up to addition of elements in the kernel of ω. We will use (3.3) to compute Hamiltonian vector fields for a few interesting functionals; to do so, we will initially work in the ambient space V , and use the arclength-preserving condition (2.4) to rewrite the differential in a form suitable for applying (3.3).
We first consider the arclength functional on the space V . Given an arbitrary vector field X = a γ + b γ + c γ (not necessarily arclength preserving), the variation of the determinant in (3.4) along X is given by Assume now that γ ∈ M , so that | γ , γ , γ | = 1 and We now seek a vector field Using the non-stretching condition (2.4), we obtain the following Hamiltonian vector field (which is unique only up to adding a constant times Z 0 ). In Section 4.4 we will see that the associated curve flow γ t = X L leads to the Boussinesq equation for the curvatures k 1 , k 2 . Next, we introduce the total curvature functional From γ = p 0 γ + p 1 γ and (2.1), it follows that p 1 = | γ , γ , γ |. Then the variation of p 1 along an arbitrary vector field X = a γ + b γ + c γ is given by Then, up to perfect derivatives, Assuming X is an arclength-preserving vector field, we set b = −a− 1 3 (c +2p 1 c) and compute Integrating by parts, we arrive at Suppose that X P =ã γ +b γ +c γ is also arclength-preserving. Setting the right-hand side of (3.7) equal to ω γ (X, Thus, a Hamiltonian vector field corresponding to (3.5) is (This is only unique up to adding a linear combination of Z 0 and Z 1 .) Again, we will see that the associated curve flow γ t = X P is also directly related, at the level of the curvatures, to one of the flows in the Boussinesq hierarchy.
4 Integrable centroaf f ine curve f lows In this section we will examine the evolution of centroaffine curvatures induced by the curve flows defined in Section 3.2. We begin by computing the evolution of invariants under more general curve flows.

Evolution of invariants
First, we consider how the centroaffine invariants of a starlike curve evolve under a general non-stretching evolution equation . (Thus, we assume from now on that γ (x, t) is parametrized by arclength x at each time.) Of course, in order for (4.1) to represent a geometric evolution equation, r 1 and r 2 should be functions of the invariants p 0 , p 1 and their arclength derivatives.
From now on, we will take k 1 = p 1 and k 2 = p 0 − p 1 as fundamental invariants; one reason for doing this is that the evolution equations for these invariants induced by (4.1) take the form where P is the skew-adjoint matrix differential operator D stands for the derivative with respect to x and [·, ·] denotes the commutator on pairs of operators 2 . This operator P, which arises naturally when using k 1 , k 2 instead of p 0 , p 1 , will play a significant role in the integrable structure of the flows we study.

Two integrable f lows
The vector field X L induces a non-stretching evolution equation (This will be the first non-trivial curve evolution in the hierarchy discussed in Section 5, where the right-hand side is labeled as Z 1 .) By setting r 1 = 0, r 2 = 1 in (4.4), we obtain the corresponding curvature evolution (4.7) This PDE system for curvatures is Hamiltonian, since it can be written in the form where E denotes the vector-valued Euler operator on scalar functions f of k 1 , k 2 and their higher x-derivatives k 2 . (One can check that the Poisson bracket defined using the Hamiltonian operator P on the appropriate function space -see Section 5.2 below -satisfies the usual requirements of skew-symmetry and the Jacobi identity.) Moreover, (4.7) can also be written in Hamiltonian form as for a different Hamiltonian operator and density (4.10) (The notation ρ 3 is explained below.) Since the curvature evolution can be written in Hamiltonian form in two ways (4.7) and (4.9), the integrals k 2 dx and ρ 3 dx are conserved by the flow (for appropriate boundary conditions).

Remark 4.2.
In fact, the curvature evolution here is a bi-Hamiltonian system, because P and Q are a Hamiltonian pair, i.e., their linear combinations form a pencil of Hamiltonian operators, and a pencil of compatible Poisson structures. This assertion can be verified mechanically (see, e.g., Section 7.1 in [22] for details), but it also follows from the fact that, at least in the periodic case, the Poisson structures are reductions of a well-known compatible pencil of Poisson brackets on the space of loops in sl (3). (Indeed, when γ is periodic -or more generally has monodromythe matrix K in (2.3) provides a lift into this loop space.) The proof of the reduction of these brackets can be found in [8], where P is linked to the Adler-Gel'fand-Dikii bracket for sl (3) and Q is associated to its companion. The brackets were later linked to curve evolutions and differential invariants in [15], where more details are available.
The (negative of the) Hamiltonian vector field X P of Section 3.2 induces the non-stretching evolution where (The right-hand side of (4.11) is labeled as Z 2 in the hierarchy discussed in Section 5.) We similarly obtain the curvature evolution equations induced by this flow by setting r 1 = k 2 , r 2 = k 1 in (4.4). We remark that the resulting system is also bi-Hamiltonian, since it can be written as Thus, ρ 2 dx and ρ 4 dx are conserved integrals for (4.11). (Because X P corresponds symplectically to the Hamiltonian k 1 dx, it is automatic that this integral is also conserved.) Remark 4.3. The arclength normalization (2.1) is preserved by the simultaneous rescaling x → λx, γ → λ −1 γ . Under this rescaling, k 1 and k 2 scale by multiples λ 2 and λ 3 respectively. Thus, we may assign scaling weights 2 and 3 respectively to these curvatures, and each xderivative taken increases weight by one. It will turn out (see Section 5.1 below) that the conserved densities for evolution equations (4.6) and (4.11) are all of homogeneous weight, with one density for each positive weight not congruent to 1 modulo 3. We will number the densities in order of increasing weight, letting ρ 0 = k 1 , ρ 1 = k 2 and so on; thus, the density in (4.9) is denoted by ρ 3 , since its weight falls between those of ρ 2 and ρ 4 .
The curve flows (4.6) and (4.11) turn out to share the same conservation laws; for example, k 1 dx is conserved by (4.6) because (4.7) implies that Similarly, (4.11) conserves k 2 dx because (4.13) implies that In Section 5 we will show that these flows share an infinite sequence of conservation laws.

Lax representation
In this subsection we use geometric considerations to derive Lax pairs for curvature evolution equations induced by (4.6) and (4.11).
In [3] we found that the components of the solution γ(x, t) of Pinkall's flow satisfied the scalar Lax pair for the KdV equation. In the same spirit, we seek a system of the form satisfied by each component of γ , where L and M are differential operators in x with coefficients involving k 1 , k 2 . Using (2.2), we see that every component of γ satisfies the scalar ODE y = (k 1 y) + k 2 y, and so we will let L := D 3 − Dk 1 − k 2 and seek operators M 1 for (4.6) and M 2 for (4.11).
In the case of (4.6), the components of γ also satisfy y t = y − 2 3 k 1 y, so we choose One can then verify that (4.7) implies that In the case of (4.11), the components of γ satisfy y t = k 1 y + k 2 y + r 0 y, with r 0 as given by (4.12). So, we might set M 2 = k 1 D 2 + k 2 D + r 0 . However, (4.14) would also be satisfied if we modify M 2 by adding N L, where N is an arbitrary differential operator. In fact, the system (4.13) actually implies that L t = [M 2 , L] for Writing these systems in Lax form (4.14) enables us to interpolate a spectral parameter into the linear equations satisfied by the components. Thus, consider solutions of the compatible system For fixed λ ∈ R, let y 1 , y 2 , y 3 be linearly independent solutions of (4.16), with Wronskian W . Then W is constant in x and t, and γ = W −1/3 (y 1 , y 2 , y 3 ) T is arclength-parametrized at each time t, with centroaffine invariants k 1 andk 2 = k 2 + λ. Furthermore, γ satisfies the evolution equation Proof . If we let y = (y 1 , y 2 , y 3 ) T and form the matrix F = (y, y , y ), then F satisfies differential equations of the form where both right-hand side matrices have trace zero. For example, when j = 1 one can directly calculate, by differentiating y t = M 1 y, that Thus, the Wronskian W is constant in x and t. Because γ = (k 1 γ ) + (k 2 + λ) γ , the centroaffine invariants of γ are k 1 andk 2 . It is straightforward to compute γ t in the j = 1 case, using y t = M 1 y. In the j = 2 case, we compute y t = M 2 y = k 1 y + (k 2 − λ)y + r 0 y − 3DLy = k 1 y + (k 2 − 4λ)y + r 0 y.

Connection with Boussinesq equations
In [4] Chou and Qu note that, under the centroaffine curve flow (4.6), the curvatures k 1 , k 2 satisfy a two-component system of evolution equations that is equivalent to the Boussinesq equation. This suggests that the other integrable flow (4.11) under discussion may be related to the Boussinesq hierarchy.
Dickson et al. [5] write the (first) Boussinesq equation as a system (q 0 ) t + 1 6 q 1 + 2 3 q 1 q 1 = 0, (q 1 ) t − 2q 0 = 0. (4.17) They embed this in a hierarchy of integrable equations, each of which is written in Lax form as where P m is a differential operator of order m ≡ 0 mod 3, with coefficients depending on q 0 , q 1 and their x-derivatives. Note that P m must be chosen so that [P m , L] has order one. For example, while P 1 = D yields the trivial evolution (q 1 ) t = q 1 , (q 0 ) t = q 0 , setting P 2 = D 2 + 2 3 q 1 gives the Boussinesq equation (4.17).
Given the resemblance between (4.15) and (4.18), it is tempting to find substitutions to connect the Boussinesq equation with (4.7). In fact, we can make L and L coincide by setting With this substitution, M 1 coincides with P 2 , so it follows that (4.7) and (4.17) are equivalent.
In [5] it is shown how the coefficients of the operators P m can be obtained solving a recursive system of differential equations, and thus these depend on a number of constants of integration. For example, the expression for P 4 is where f 0 = 0, g 0 = 1, f 1 = 1 3 q 1 + c 1 , g 1 = 1 3 q 0 + d 1 , and k 4,0 , k 4,1 , c 1 , d 1 are arbitrary constants. For convenience, we will set all these arbitrary constant to zero, so that P 4 = D 4 + 4 3 q 1 D 2 + 4 3 (q 1 + q 0 )D + 5 9 q 1 + 2 3 q 0 + 2 9 q 2 1 .
Again, if we use the substitutions (4.19), we find that the operator M 2 coincides with −3P 4 . Thus, (4.13) is equivalent to the second nontrivial flow in the Boussinesq hierarchy, provided we also rescale time by t → −3t.

Hierarchies
In [22] the Boussinesq hierarchy is discussed as an example of a bi-Hamiltonian system, in which two sequences of commuting flows (and conservation laws) are generated by applying recursion operators. Thus, given the equivalences established in Section 4.4, it is not surprising that the Poisson operators defined in Section 4 can be combined to give a recursion operator that generates a double hierarchy of commuting evolution equations for k 1 , k 2 . In fact, we will show that our recursion operator is equivalent to the Boussinesq recursion operator as given in Example 7.28 of [22]. The new information we add is that each of these evolution equations is induced by a centroaffine geometric evolution equation for curves, which is itself Hamiltonian relative to the pre-symplectic structure defined in Section 3.1 (see Theorem 5.4 below).

Recursion operators
We define a sequence of evolution equations for k 1 , k 2 with initial data given by (Note that F 1 is the right-hand side of (4.7), while for j = 0 (5.1) gives a simple transport equation for k 1 , k 2 , corresponding to flow in the direction of the tangent vector γ .) In order to assert that the F j defined by (5.2) are local functions of k 1 , k 2 and their derivatives -i.e., in calculating each F j , the operator D −1 is only applied to exact x-derivatives of local functions -we cite well-known results on the Boussinesq hierarchy. For example, the version of the first Boussinesq equation used by Olver [22] is where τ is the time variable. If one considers linear transformations on the variables, it is necessary to use some imaginary coefficients to make our version (4.7) of the first Boussinesq equation for k 1 , k 2 equivalent to (5.3): Proposition 5.1. Under the above change of variables, the recursion operator PQ −1 is equivalent to the Boussinesq recursion operator in [22].
Proof . The transformation between k 1 , k 2 and u, v can be written as is an evolution equation for u, v, the right-hand side of the corresponding evolution for k 1 , k 2 is G • F . Thus, our recursion operator PQ −1 for flows on the k 1 , k 2 variables corresponds to a recursion operator on flows in the u, v variables. In fact, when one calculates (5.5) and substitutes for k 1 , k 2 in terms of u, v using (5.4), the result is exactly −i times the Boussinesq recursion operator given in [22].
Since in [27] (see Section 5.4 in that paper) it is proven that the Boussinesq recursion operator from [22] always produces local flows when applied to the 'seed' evolution equations (i.e., the tangent flow and first Boussinesq), it follows that the same is true for our recursion operator.
Remark 5.2. Once one checks that the evolution equations (5.1) for j = 0 and j = 1 commute, it is automatic from the bi-Hamiltonian structure that all evolution equations in the sequence (5.1) commute in pairs (see, e.g., Theorem 7.24 in [22]).
It is easy to check that the 'seeds' F 0 , F 1 for the recursion are related to the initial conserved densities by Non-stretching vector field Conserved density (The coefficient of γ in some vector fields is omitted for reasons of space, but can be determined from the non-stretching condition.) • The γ and γ coefficients of Z j match the components of Eρ j .
(The second set of equations was derived in Section 4.2.) While PQ −1 is the recursion operator for commuting flows, it is evident from (5.6) that Q −1 P should be the recursion operator for conservation law characteristics (i.e., the result of applying the Euler operator E to a density).
In fact, we may define an infinite sequence of conserved densities by The first few densities calculated using this recursion appear in Table 1.
We now use these densities to define a sequence of flows for centroaffine curves, and relate each of them to a curvature evolution equation in the sequence (5.1). Namely, if f is any local function of k 1 , k 2 and their derivatives, we define X f := (Ef ) 1 γ + (Ef ) 2 γ + r 0 γ , (5.8) where the subscripts indicate the components given by (4.8) and r 0 is determined by the nonstretching condition. Then for the sequence of densities defined recursively by (5.7) we define the vector fields and the corresponding sequence of curve flows γ t = Z j . Proof . From (5.6) and the recursion relations, it follows by induction that Then the result follows immediately from (4.4).

Hamiltonian structure at the curve level
We now consider the question of how the Hamiltonian operator P is related to the Hamiltonian structure defined at the curve level in Section 3.1. Recall from Section 3.2 that X is a Hamiltonian vector field associated to the functional H if for any non-stretching vector field Y . i.e., ρ is next after ρ in the sequence of densities generated by the recursion operator Q −1 P.
Then −X ρ (as defined by (5.8)) is Hamiltonian for H.
Proof . Based on the definition (3.1) of ω, we need to show that dH[Y ] equals However, using (2.4) to eliminate a andã, we obtain where the last equation follows by integration by parts. Thus, c · PEρ dx, using (5.11) in the last step. Then, because P is skew-adjoint, On the other hand, using the properties of the Euler operator we have where δ Y denotes the first variation in the direction of Y . Now using (4.4) we have This concludes the proof.
The following corollaries are immediate consequences of the theorem. where G(k 1 , k 2 ) = g dx and H(k 1 , k 2 ) = h dx, and g, h are functions of periodic k 1 , k 2 and their derivatives. Then dH[X g ] = {H, G}.
Corollary 5.6. The vector fields Z j defined by (5.9) are Hamiltonian for j ≥ 2.
Corollary 5.7. A closed curve γ is critical for the functional H( γ ) = γ ρ j dx with respect to non-stretching variations if and only if γ is stationary for a constant-coefficient linear combination Z j+2 + c 0 Z 0 + c 1 Z 1 .
Proof . By Theorem 5.4, −Z j+2 is Hamiltonian for H. Thus, a curve γ is H-critical if and only if ω(Y, Z j+2 ) = 0 for all Y ∈ T γ M . This condition is satisfied if and only if Z j+2 is in the kernel of ω γ , i.e., along γ it is equal to a constant-coefficient linear combination of vectors Z 0 and Z 1 . Equivalently, Z j+2 + c 0 Z 0 + c 1 Z 1 = 0 along γ for some constants c 0 , c 1 , expressing the property that γ is stationary for a linear combination of these vector fields.

Projective properties
As stated in Remark 2.2, a centroaffine curve γ projects to give a conic in RP 2 if and only if the Wilczynski invariants satisfy p 0 − 1 2 p 1 = 0. (The corresponding condition in terms of k 1 , k 2 is k 2 + 1 2 k 1 = 0.) In this subsection we will investigate flows in the hierarchy having the property that, if γ projects to a conic at time zero, then it continues to have a conical projection at subsequent times. We will show later that the equation of the conic in homogeneous coordinates is fixed in time. We will also discuss flows that preserve a parametrization that is proportional to projective arclength; in that case, the corresponding condition in terms of curvatures is that k 2 + 1 2 k 1 is a nonzero constant along the curve. These investigations are much easier if, instead of k 2 , we use an invariant that vanishes precisely when the condition we are investigating holds. Accordingly, we fix a constant C, and define an alternative pair of invariants Thus, the curve is a conic if v = 0 when C = 0, and the curve has a constant-speed parametrization (relative to projective arclength) if v = 0 when C = 0. We will convert the evolution equations in the hierarchy (at the level of the invariants) to these variables. Suppose that a curve flow causes the invariants to evolve by where F is a vector-valued function of k 1 , k 2 and their derivatives. Then the corresponding evolution equation for the alternative invariants is Similarly, if R is the recursion operator generating the hierarchy of evolution equations for k 1 , k 2 , then the recursion operator for the corresponding flows on u, v differs by a gauge transformation: where (It is understood that, in R on the right-hand side, k 1 , k 2 are substituted for in terms of u, v.) Specifically, using R = PQ −1 as defined by (4.5), (4.10), we computẽ and N is the scalar differential operator 1 6 D 4 − 5 6 uD 2 − 5 4 u D + 2 3 u 2 − 3 4 u . One can check that the vectorsF 0 ,F 1 are the time derivatives of u, v, corresponding to the 'seeds' Z 0 and Z 1 for the hierarchy of curve flows.
By applyingR toF 0 ,F 1 , one can generate the right-hand sides of the evolution equations in the hierarchy in terms of u and v. LettingF j denote these vectors, we compute (for example) that Here, when applying D −1 to differential polynomials in u, v, the constant of integration is taken to be zero. Notice in particular that if v ≡ 0, then the bottom component ofF 3 − 3CF 1 vanishes. Thus, the flow Z 3 − 3CZ 1 preserves the condition that v is identically zero. In fact, we can calculate two infinite sequences of evolution equations for u, v that preserve this condition; the right-hand sides of these are While it is routine to verify that any individual curvature evolution equation in these sequences preserves v ≡ 0, it is easier to observe that the members of these sequences satisfy the recursion relation G k+2 = (R − 3C) 2 G k . Then the fact that they all preserve v ≡ 0 is a consequence of the following: Proof . We assume that G k = (D 1 , D 2 ) T for local functions 1 , 2 of u, v and their derivatives. (This form for G k is necessary if we are able to apply operatorR to it and produce local functions.) Within the ring of polynomials in u, v and their derivatives, let V denote the ideal generated by v, v , v , etc. By hypothesis, D 2 ∈ V, and the same is true for 2 .
We compute (5.14) Thus, the bottom component of (R − 3C) 2 G k is given by (Here,F j1 andF j2 denote the top and bottom entries in the vectorF j .) The second factor in the top line is the top entry of (R − 3C)G k . This polynomial lies in V, and the same is true if we apply N or D −1 to it. On the other hand, becauseF 02 = v ∈ V, the coefficient in front on the second line also vanishes when v ≡ 0.
In the special case when C = 0, we see that the following curve flows (as defined by (5.9)) preserve conicity: However, when C = 0, some care needs to be taken in matching the evolution equations for u, v that preserve v ≡ 0 with the corresponding linear combinations of the curve flows Z j .
In the proof of Proposition 5.8, we used the fact that if an exact derivative D lies in V, then by choosing the constant of integration equal to zero, the antiderivative also lies in V. However, if we express D in terms of k 1 , k 2 instead of u, v, and then take an antiderivative, a particular constant of integration must be chosen in order to belong in V. Thus, when we compute the kth evolution equation for u, v by using the recursion operatorR (which involves applying D −1 ), then convert this to an evolution equation for k 1 , k 2 , and finally try to match it with a curve flow in the hierarchy (5.10), we get a linear combination of Z k with lower-order flows of the same parity. For example, if we substitute (5.12) intoF 2 , and then apply the operator G −1 , we get Thus,F 2 is induced by the curve flow Z 2 −CZ 0 ; similarly,F 3 is induced by Z 3 −2CZ 1 , and so on.
Similarly, when we apply the recursion operator (R − 3C) 2 to generate higher-order flows that preserve v ≡ 0, these constants of integration accumulate and change the relatively nice pattern of the coefficients exhibited by (5.13). Here is what we get when we compute the first few curve evolutions corresponding to the evolutions G k : u, v evolution centroaffine curve flow

Conical evolutions and the Kaup-Kuperschmidt hierarchy
In this section we examine special properties of the conicity-preserving flows (5.15) (hence, from now on we are assuming C = 0). These properties will enable us to connect our hierarchy of centroaffine curve flows in R 3 with the Kaup-Kuperschmidt hierarchy and with curve flows in centroaffine R 2 . We begin with the observation that, when restricted to conical curves, the coefficient of γ vanishes for as many of the vector fields in (5.15) as one cares to check. In other words, this coefficient belongs to V, the ideal within the ring of differential polynomials in k 1 , k 2 generated by k 2 + 1 2 k 1 and its derivatives. In fact, this is true in general, as shown in the following: Proposition 5.9. If j ≡ 0 or j ≡ −1 modulo 4, then the bottom component of Eρ j belongs in V; hence, the γ coefficient of Z j vanishes on conical curves.
Proof . The statement can be verified directly for j = 0 and j = 3. For higher values, we use the recursion relation between the characteristics, which implies that Eρ j+4 = (Q −1 P) 2 Eρ j . From Proposition 5.8 we know that the bottom component ofF j = GPEρ j lies in V. By inserting powers of G and G −1 into the recursion relation, we get As in the proof of Proposition 5.8, we can assume thatF j = (D 1 , D 2 ) T where 2 ∈ V. Taking C = 0 in equation (5.14), we see that the top entry of which clearly is in V. (Note that v = k 2 + 1 2 k 1 here.) Noting the form of Q, we see that the bottom entry of Eρ j+4 is D −1 applied to (5.16), so it must also belong to V.
Next, we make a connection with curve flows in centroaffine R 2 to show that, for the flows in (5.15), the cone that the curve lies on is preserved by the time evolution.
Proposition 5.10. If γ (x, t) evolves by any of the vector fields in (5.15), and γ (x, 0) lies on a cone through the origin in R 3 , then γ (x, t) lies on the same cone at later times.
Proof . Using the action of SL(3) we can, without loss of generality, assume that the equation of the cone is y 1 y 3 − (y 2 ) 2 = 0. We fix a map V from R 2 onto this cone: Of course, when we projectivize on each end this gives the Veronese embedding of RP 1 as a quadric in RP 2 . The scale factor of 2 −1/3 is chosen so that if X(x) is a parametrized curve in R 2 satisfying the centroaffine normalization |X, X | = 1, then Γ = V • X satisfies the normalization |Γ, Γ , Γ | = 1. Moreover, if p(x) is the curvature of X, then the invariants of Γ are k 1 = −4p and k 2 = 2p . Finally, if X evolves by the non-stretching flow X t = rX − 1 2 r X, (5.17) then Γ(x, t) = V • X(x, t) satisfies Γ t = rΓ − r Γ. By Proposition 5.9 all the flows in (5.15), when restricted to conical curves, are of this form, for some choice of differential polynomial r in k 1 . For any initial data γ (x, 0), we can define a curve X(x, 0) in R 2 such that γ (x, 0) = V • X(x, 0) and make X(x, t) evolve by (5.17). Because Γ(x, t) = V • X(x, t) satisfies the same initial value problem, then γ (x, t) = Γ(x, t) at all times, and γ (x, t) lies on the cone defined by y 1 y 3 − (y 2 ) 2 = 0 at all times.
In [4] Chou and Qu discovered a non-stretching flow for curves in centroaffine R 3 which preserves the conicity condition k 2 + 1 2 k 1 = 0 and which causes the curvature k 1 to evolve by the Kaup-Kuperschmidt equation: u t = u − 5uu − 25 2 u u + 5u 2 u (see Case 3 in Section 3 of their paper, taking λ = 0). In fact, up to a multiplicative factor of 1/3, Chou and Qu's flow is the same as the restriction of Z 3 to conical curves.
This agrees with the known recursion operator for symmetries of the Kaup-Kuperschmidt hierarchy. (See, e.g., Example 2.20 in [29], where the operator differs by changing u to −u and rescaling time by a factor of 1/3.) Using this, one can check that the curvature flows induced by (5.15) for conical curves coincide with the commuting flows of the Kaup-Kuperschmidt hierarchy.
Remark 5.11. The curve flow discovered by Chou and Qu is in fact also defined for centroaffine curves parametrized proportional to projective arclength (i.e., those for which k 2 + 1 2 k 1 is a constant), and nevertheless still induces Kaup-Kuperschmidt evolution for k 1 . Recently, Musso [20] has extended this to a hierarchy of flows for arclength-parametrized curves in RP 2 which induces the Kaup-Kuperschmidt hierarchy for curvature evolution. We suspect that these flows coincide with the restrictions of the flows studied in Section 5.3 (for C = 0) to the centroaffine lifts of such curves in RP 2 .
Remark 5.12. Schwartz and Tabachnikov [28] showed that certain maps defined on the space of convex polygons preserve the subset of polygons that are inscribed (or circumscribed) on a conic: that is, if the vertices of the polygon (or those of its projective dual) lie on a conic, then the same is true for its image under the map. The building blocks for these maps are elementary maps T r that associate to a given polygon another polygon obtained from the intersections of diagonals joining each vertex to the vertex located r positions to the left or right of it. In fact, the maps preserving conicity are particular combinations of T r for certain values of r.
The map corresponding to r = 2 is called the pentagram map and it is known to be a discretization (in both time and space) of the Boussinesq equation [24]. It is natural to wonder if the maps in [28] are somehow associated to flows in the Boussinesq hierarchy. We are currently investigating this.