Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208...

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Дата:	2008
Автор:	Beffa, G.M.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2008
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems / G.M. Beffa // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 51 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1490502025-02-10T00:04:48Z Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems Beffa, G.M. In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). 2008 Article Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems / G.M. Beffa // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 51 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K25; 53A55 https://nasplib.isofts.kiev.ua/handle/123456789/149050 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection.
format	Article
author	Beffa, G.M.
spellingShingle	Beffa, G.M. Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Beffa, G.M.
author_sort	Beffa, G.M.
title	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
title_short	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
title_full	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
title_fullStr	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
title_full_unstemmed	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
title_sort	geometric realizations of bi-hamiltonian completely integrable systems
publisher	Інститут математики НАН України
publishDate	2008
url	https://nasplib.isofts.kiev.ua/handle/123456789/149050
citation_txt	Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems / G.M. Beffa // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 51 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT beffagm geometricrealizationsofbihamiltoniancompletelyintegrablesystems
first_indexed	2025-12-01T23:49:58Z
last_indexed	2025-12-01T23:49:58Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 034, 23 pages Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems? Gloria MARÍ BEFFA Department of Mathematics, University of Wisconsin, Madison, WI 53705, USA E-mail: maribeff@math.wisc.edu URL: http://www.math.wisc.edu/∼maribeff/ Received November 14, 2007, in final form March 13, 2008; Published online March 27, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/034/ Abstract. In this paper we present an overview of the connection between completely in- tegrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161–213; 55 (1999), 127–208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection. Key words: invariant evolutions of curves; Hermitian symmetric spaces; Poisson brackets; differential invariants; projective differential invariants; equations of KdV type; completely integrable PDEs; moving frames; geometric realizations 2000 Mathematics Subject Classification: 37K25; 53A55 1 Introduction Example 1. One of the simplest examples of a geometric realization of a completely integrable system is that of the nonlinear Schrödinger equation (NLS) realized by the self-induction Vortex Filament flow (VF). The VF flow is a flow in the Euclidean space SO(3) n R3/SO(3) (see [3]). In [21] Hasimoto showed that, if u(x, t) is a flow solution of the VF equation ut = κB, where κ is the Euclidean curvature of the curve u(·, x) ∈ R3, x is the arc-length and B is the binormal, then the evolution of the curvature and torsion of u is equivalent to the NLS equation via the Hasimoto transformation Φ = κei ∫ τ . The Hasimoto transformation (κ, τ) → (ν, η), with Φ = ν + iη, is, in fact, induced by a change from classical Euclidean moving frame to the natural moving frame (ν and η are the natural curvatures, see [28]). Thus, VF is a Euclidean geometric realization of NLS if we use natural moving frames. Equivalently, NLS is the invariantization of VF. The relation to the Euclidean geometry of the flow goes further; consider the evolution ut = hT + h′ κ N + gB, (1) where {T,N, B} is the classical Euclidean moving frame and h and g are arbitrary smooth functions of the curvature, torsion and their derivatives. Equation (1) is the general form of ?This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at http://www.emis.de/journals/SIGMA/symmetry2007.html mailto:maribeff@math.wisc.edu http://www.math.wisc.edu/~maribeff/ http://www.emis.de/journals/SIGMA/2008/034/ http://www.emis.de/journals/SIGMA/symmetry2007.html 2 G. Maŕı Beffa an arc-length preserving evolution of space curves, invariant under the action of the Euclidean group (i.e., E(n) takes solutions to solutions). Its invariantization can be written as( κ τ ) t = P ( g h ) , (2) where P defines a Poisson bracket generated by the second Hamiltonian structure for NLS via the Hasimoto transformation, i.e. the Hasimoto transformation is a Poisson map (see [28, 38, 39]). (For more information on infinite dimensional Poisson brackets see [40], and for more information on P see [38].) Clearly, P can be generated using a classical Euclidean moving frame and the invariants κ and τ . The NLS equation is a bi-Hamiltonian system, i.e., Hamiltonian with respect to two compatible Hamiltonian structures. One of the structures is invertible and a recursion operator can be constructed to generate integrals of the system (see [30] or [40]). The first Hamiltonian structure for NLS is invertible and can also be proved to be generated by the geometry of the flow, although it is of a different character as we will see below. Example 2. A second example is that of the Korteweg–de Vries (KdV) equation. If a 1-pa- rameter family of functions u(t, ·) ∈ R evolves following the Schwarzian KdV equation ut = u′S(u) = u′′′ − 3 2 (u′′)2 u′ , where S(u) = u′′′ u′ − 3 2 ( u′′ u′ )2 is the Schwarzian derivative of u, then k = S(u) itself evolves following the KdV equation kt = k′′′ + 3kk′, one of the best known completely integrable nonlinear PDEs. The KdV equation is Hamiltonian with respect to two compatible Hamiltonian structures, namely D = d dx and D3+2kD+k′ (called respectively first and second KdV Hamiltonian structures). As before, if we consider the general curve evolution given by ut = u′h, (3) where h is any smooth function depending on S(u) and its derivatives with respect to the parameter x, then k = S(u) evolves following the evolution kt = (D3 + 2kD + k′)h. (4) This time S(u) is the generating differential invariant associated to the action of PSL(2) on RP1. That is, any projective differential invariant of curves u(x) is a function of S(u) and its deriva- tives. Equation (3) is the most general form for evolutions of reparametrizations of RP1 (or parametrized “curves”) invariant under the action of PSL(2). Evolution (4) can be viewed as the invariantization of (3). Equivalently, the family of evolutions in (3) provides RP1 geomet- ric realizations for the Hamiltonian evolutions defined by (4). Thus, one can obtain geometric realizations in RP1 not only for KdV, but also for any system which is Hamiltonian with respect to the second KdV Hamiltonian structure. For example, the Sawada–Koterra equation kt = (D3 + 2kD + k′) ( 2k′′ + 1 2 k2 ) = 2k(5) + 5kk′′′ + 5k′k′′ + 5 2 k′k2 is bi-Hamiltonian with respect to the same Hamiltonian structures as KdV is. Its Hamiltonian functional (4) is h(k) = ∫ (1 6k3 − (k′)2)dx. Therefore, the Sawada–Koterra equation has ut = u′ ( 2S(u)′′ + 1 2 S(u)2 ) Geometric Realizations of Bi-Hamiltonian Integrable Systems 3 as RP1 realization. (Incidentally, Sawada–Koterra has a second realization as an equi-affine flow, see [41].) The manifold RP1 is an example of a parabolic homogeneous space, i.e., a manifold of the form G/P with G semisimple and P a parabolic subgroup. From (3) and (4) we can see that the second Hamiltonian structure for KdV can be generated with the sole knowledge of u′ (a classical projective moving frame along u) and S(u), its projective differential invariant. The first KdV structure is also similarly generated, although, again, it is of a different nature. These two simple examples illustrate the close relationship between the classical geometry of curves and bi-Hamiltonian completely integrable PDEs. In the last years many examples of geometric realizations for most known completely integrable systems have been appearing in the literature. Some are linked to the geometric invariants of the flow (see for example [1, 2, 11, 13, 17, 25, 26, 28, 29, 31, 36, 38, 39, 46, 47, 48, 49, 51]). This list is, by no means, exhaustive as this paper is not meant to be an exhaustive review of the subject. Perhaps the simplest way to understand the close relationship between differential invariants and integrable systems is through the AKNS representation on one hand and group-based moving frames on the other. If G is a Lie group, a G-AKNS representation of a nonlinear PDE kt = F (k, kx, kxx, . . . ) (5) is a linear system of equations ϕx = A(t, x, λ)ϕ, ϕt = B(x, t, λ)ϕ, ϕ(t, x, λ) ∈ G, A(x, t, λ), B(x, t, λ) ∈ g such that the compatibility condition for the existence of a solution, At = Bx + [B,A], is independent of λ and equivalent to the nonlinear PDE (5). Such a representation is a basis for generating solutions and integrating the system. Indeed, most integrable systems have an AKNS representation. Geometrically, this is thought of as having a 2-parameter flat connection defined by − d dx + A and − d dt + B along the flow (see [22]). The bridge to differential invariants and differential geometry a-la-Cartan appears when one realizes that this 2-parameter connection is a reduction of the Maurer–Cartan connection of G along the flow ϕ and the AKNS system could be interpreted as the Serret–Frenet equations and the t-evolution of a group-based (right) moving frame (ϕ) along a flow u : R2 → G/H in a certain homogeneous space. This is explained in the next section. In this paper we describe how the background geometry of affine and some symmetric mani- folds generates Hamiltonian structures and geometric realizations for some completely integrable systems. Our affine manifolds will be homogeneous manifolds of the form G n Rn/G with G semisimple. They include Euclidean, Minkowski, affine, equi-affine and symplectic geometry among others. We will also discuss related geometries, like the centro-affine or geometry of star- shaped curves, for which the action of the group is linear instead of affine. On the other hand, our symmetric manifolds are locally equivalent to a homogeneous manifold of the form G/H where g, the Lie algebra associated to G, has a gradation of the form g−1 ⊕ g0 ⊕ g1 and where g0 ⊕ g1 = h is the Lie algebra of H. These includes projective geometry (G = PSL(n + 1)), the Grassmannian (G = SL(p + q)), the conformally flat Möbius sphere (G = O(n + 1, 1)), the La- grangian Grassmannian (G = Sp(2n)), the manifold of reduced pure spinors (G = O(n, n)) and more. We will see how the Cartan geometry of curves in these manifolds induces a Hamiltonian structure on the space of differential invariants. In the last part of the paper we look closely at the case of symmetric spaces. We define differential invariants of projective type as those genera- ted by the action of the group on second order frames. We then describe how, in most cases, the 4 G. Maŕı Beffa reduced Hamiltonian structure can be further reduced or restricted to the space of curves with vanishing non-projective differential invariants. On this manifold the Hamiltonian structure has a (geometrically defined) compatible Poisson companion. They define a bi-Hamiltonian pencil for some integrable equations of KdV-type, and they provide geometric realizations for them. We finally state a conjecture by M. Eastwood on what the presence of these flows might say about the geometry of curves in symmetric spaces. As group based moving frames are relatively new, our next section will describe them in detail. We will also describe their role in AKNS representations. 2 Moving frames The classical concept of moving frame was developed by Élie Cartan [8, 9]. A classical moving frame along a curve in a manifold M is a curve in the frame bundle of the manifold over the curve, invariant under the action of the transformation group under consideration. This method is a very powerful tool, but its explicit application relied on intuitive choices that were not clear on a general setting. Some ideas in Cartan’s work and later work of Griffiths [19], Green [20] and others laid the foundation for the concept of a group-based moving frame, that is, an equivariant map between the jet space of curves in the manifold and the group of transformations. Recent work by Fels and Olver [14, 15] finally gave the precise definition of the group-based moving frame and extended its application beyond its original geometric picture to an astonishingly large group of applications. In this section we will describe Fels and Olver’s moving frame and its relation to the classical moving frame. We will also introduce some definitions that are useful to the study of Poisson brackets and bi-Hamiltonian nonlinear PDEs. From now on we will assume M = G/H with G acting on M via left multiplication on representatives of a class. We will also assume that curves in M are parametrized and, therefore, the group G does not act on the parameter. Definition 1. Let Jk(R,M) the space of k-jets of curves, that is, the set of equivalence classes of curves in M up to kth order of contact. If we denote by u(x) a curve in M and by ur the r derivative of u with respect to the parameter x, ur = dru dxr , the jet space has local coordinates that can be represented by u(k) = (x, u, u1, u2, . . . , uk). The group G acts naturally on parametrized curves, therefore it acts naturally on the jet space via the formula g · u(k) = (x, g · u, (g · u)1, (g · u)2, . . . ), where by (g ·u)k we mean the formula obtained when one differentiates g ·u and then writes the result in terms of g, u, u1, etc. This is usually called the prolonged action of G on Jk(R,M). Definition 2. A function I : Jk(R,M) → R is called a kth order differential invariant if it is invariant with respect to the prolonged action of G. Definition 3. A map ρ : Jk(R,M) → G is called a left (resp. right) moving frame if it is equivariant with respect to the prolonged action of G on Jk(R,M) and the left (resp. right) action of G on itself. Geometric Realizations of Bi-Hamiltonian Integrable Systems 5 If a group acts (locally) effectively on subsets, then for k large enough the prolonged action is locally free on regular jets. This guarantees the existence of a moving frame on a neighborhood of a regular jet (for example, on a neighborhood of a generic curve, see [14, 15]). The group-based moving frame already appears in a familiar method for calculating the curvature of a curve u(s) in the Euclidean plane. In this method one uses a translation to take u(s) to the origin, and a rotation to make one of the axes tangent to the curve. The curvature can classically be found as the coefficient of the second order term in the expansion of the curve around u(s). The crucial observation made by Fels and Olver is that the element of the group carrying out the translation and rotation depends on u and its derivatives and so it defines a map from the jet space to the group. This map is a right moving frame, and it carries all the geometric information of the curve. In fact, Fels and Olver developed a similar normalization process to find right moving frames (see [14, 15] and our next theorem). Theorem 1 ([14, 15]). Let · denote the prolonged action of the group on u(k) and assume we have normalization equations of the form g · u(k) = ck, where ck are constants (they are called normalization constants). Assume we have enough normalization equations so as to determine g as a function of u, u1, . . . . Then g = ρ is a right invariant moving frame. The direct relation between classical moving frames and group-based moving frames is stated in the following theorem. Theorem 2 ([32]). Let Φg : G/H → G/H be defined by multiplication by g. That is Φg([x]) = [gx]. Let ρ be a group-based left moving frame with ρ · o = u where o = [H] ∈ G/H. Iden- tify dΦρ(o) with an element of GL(n), where n is the dimension of M . Then, the matrix dΦρ(o) contains in its columns a classical moving frame. This theorem illustrates how classical moving frames are described only by the action of the group-based moving frame on first order frames, while the action on higher order frames is left out. Accordingly, those invariants determined by the action on higher order frames will be not be found with the use of a classical moving frame. Next is the equivalent to the classical Serret–Frenet equations. This concept if fundamental in our Poisson geometry study. Definition 4. Consider Kdx to be the horizontal component of the pullback of the left (resp. right)-invariant Maurer–Cartan form of the group G via a group-based left (resp. right) moving frame ρ. That is K = ρ−1ρx ∈ g (resp. K = ρxρ−1) (K is the coefficient matrix of the first order differential equation satisfied by ρ). We call K the left (resp. right) Serret–Frenet equations for the moving frame ρ. Notice that, if ρ is a left moving frame, then ρ−1 is a right moving frame and their Serret– Frenet equations are the negative of each other. A complete set of generating differential in- variants can always be found among the coefficients of group-based Serret–Frenet equations, a crucial difference with the classical picture. The following theorem is a direct consequence of the results in [14, 15]. A more general result can be found in [23]. Theorem 3. Let ρ be a (left or right) moving frame along a curve u. Then, the coefficients of the (left or right) Serret–Frenet equations for ρ contain a basis for the space of differential invariants of the curve. That is, any other differential invariant for the curve is a function of the entries of K and their derivatives with respect to x. 6 G. Maŕı Beffa Example 3. Assume G = PSL(2) so that M = RP1. The action of G on RP1 is given by fractional transformations. Assume ρ = ( a b c d ) ∈ G is a (right) moving frame satisfying the normalization equations ρ · u = au + b cu + d = 0, ρ · u1 = au1 cu + d − (au + b)cu1 (cu + d)2 = 1, ρ · u2 = au2 cu + d − 2 acu2 1 (cu + d)2 + au + b (cu + d)3 ( cu2(cu + d) + 2 c2u2 1 (cu + d)3 ) = 2λ. Then it is straightforward to check that ρ is completely determined to be ρ = ( 1 0 1 2 u2 u1 − λ 1 )( u −1/2 1 0 0 u 1/2 1 )( 1 −u 0 1 ) . A moving frame satisfying this normalization will have the following right Serret–Frenet equation ρx = ( −λ −1 1 2S(u) + λ2 λ ) ρ. (6) This equation is gauge equivalent to the λ = 0 equation via the constant gauge g = ( 1 0 λ 1 ) . (7) This gauge g will take the second normalization constant to zero. Furthermore, if u is a solution of (3), it is known (see [31]) that the t-evolution induced on ρ is given by ρt = ( −1 2hx − λh −h 1 2hxx + λhx + λ2h + 1 2S(u)h 1 2hx + λh ) ρ. (8) We can now see the link between the AKNS representation of KdV and the evolution of a right moving frame. Assume a completely integrable system (5) has a geometric realization which is invariant under the action of the geometric group G ut = f(λ, u, u1, u2, . . . ). (9) Then, under regularity assumptions of the flow, the invariantization of (9) is the integrable system (5). A right moving frame along u will be a solution of its Serret–Frenet equation ρx = K(t, x, λ)ϕ and the time evolution will induce a time evolution on ρ of the form ρt = N(t, x, λ)ϕ. Furthermore, since (9) is invariant under the group, both K and N will depend on the differential invariants of the flow. These equations are defined by the horizontal component of the pullback of the Maurer–Cartan form of the group, ω = dgg−1 by the moving frame ρ, that is Kdx+Ndt. Geometric Realizations of Bi-Hamiltonian Integrable Systems 7 If we now evaluate the structure equation for the Maurer–Cartan form, i.e., dω + 1 2 [ω, ω] = 0, along ρx and ρt, we get Kt = Nx + [N,K] which is exactly the invariantization of the flow (9); therefore it is independent of λ. Hence, a λ-dependent geometric realization of an integrable system provides an AKNS representation of the system. See [7] for more information. Example 4. The AKNS representation for KdV is very well known. It is given by the system ϕx = ( −λ −1 −q λ ) ϕ, ϕt = ( −1 2qx − λq + 2λ3 −q + 2λ2 1 2qxx + λqx + q(−q + 2λ2) 1 2qx + λq − 2λ3 ) ϕ. Comparing it to (6) and (8) we see that q = 1 2S(u) + λ2 and hence u will depend on λ. Furthermore, h = q − 2λ2 provides λ-dependent RP1 geometric realizations for KdV, namely ut = ux ( −1 2 S(u)− 3λ3 ) . A complete description of this example can be found in [7]. (Notice that the KdV equation they represent is different, but equivalent, to our introductory example. This is merely due to a different choice of invariant.) In this paper we will not focus on the study of solutions (see [7] instead) but rather on the interaction between geometry and integrable systems. Hence we will largely ignore the spectral parameter λ and its role. 3 Hamiltonian structures generated by group-based moving frames Consider the group of loops LG = C∞(S1, G) and its Lie algebra Lg = C∞(S1, g). Assume g is semisimple. One can define two natural Poisson brackets on Lg∗ (see [45] for more information), namely, if H,F : Lg∗ → R are two functionals defined on Lg∗ and if L ∈ Lg∗, we define {H,F}1(L) = ∫ S1 〈( δH δL (L) ) x + ad∗ ( δH δL (L) ) (L), δF δL (L) 〉 dx, (10) where 〈 , 〉 is the natural coupling between g∗ and g, and where δH δL (L) is the variational derivative of H at L identified, as usual, with an element of Lg. One also has a compatible family of second brackets, namely {H,F}2(L) = ∫ S1 〈 ad∗ ( δH δL (L) ) (L0), δF δL (L) 〉 dx, (11) where L0 ∈ g∗ is any constant element. Since g is semisimple we can identify g with its dual g∗ and we will do so from now on. From now on we will also assume that our curves on homogeneous manifolds have a group monodromy, i.e., there exists m ∈ G such that u(t + T ) = m · u(t), 8 G. Maŕı Beffa where T is the period. Under these assumptions, the Serret–Frenet equations will be periodic. One could, instead, assume that u is asymptotic at ±∞, so that the invariants will vanish at infinity. We would then work with the analogous of (10) and (11). The question we would like to investigate next is whether or not these two brackets can be reduced to the space of differential invariants, or the space of differential invariants associated to special types of flows. We will describe affine and symmetric cases separately. 3.1 Affine manifolds Assume M = (G n Rn)/G is an affine manifold, G semisimple. In this case a moving frame can be represented as ρ = ( 1 0 ρu ρG ) (12) acting on Rn as ρ · u = ρGu + ρu. A left invariant moving frame with ρ · o = u will hold ρu = u and, in view of Theorem 2, ρG will have in its columns a classical moving frame. In this case K = ρ−1ρx is given by K = ( 0 0 ρ−1 G (ρu)x ρ−1 G (ρG)x ) . In [32] it was shown that ρ−1 G (ρu)x contains all first order differential invariants. It was also explained how one could make this term constant by choosing a special parametrization if necessary. Let’s call that constant ρ−1 G (ρu)x = Λ. Our main tool to find Poisson brackets is via reduction, and as a previous step, we need to write the space of differential invariants as a quotient in Lg∗. The proof of the following Theorem can be found in [32]. Theorem 4 ([32]). Let N ⊂ G be the isotropy subgroup of Λ. Assume that we choose moving frames as above and let K be the space of Serret–Frenet equations determined by these moving frames for curves in a neighborhood of a generic curve u. Then, there exists an open set of Lg∗, let’s call it U , such that U/LN ∼= K, where LN acts on Lg∗ using the gauge (or Kac–Moody) transformation a∗(n)(L) = n−1nx + n−1Ln. (13) In view of this theorem, our next theorem comes as no surprise. Theorem 5 ([32]). The Hamiltonian structure (10) reduces to U/LN ∼= K to define a Poisson bracket in the space of differential invariants of curves. Example 5. If we choose G = SO(3) and M the Euclidean space, for appropriate choice of normalization constants our left moving frame is given by ρ = ( 1 0 u T N B ) , where {T,N, B} is the classical Euclidean Serret–Frenet frame. Its Serret–Frenet equations will look like K = ρ−1ρx =  0 0 0 0 (u1 · u1) 0 −κ 0 0 κ 0 −τ 0 0 τ 0  . Geometric Realizations of Bi-Hamiltonian Integrable Systems 9 In this case, if we choose to parametrize our curve by arc-length, Λ = e1 where, as usual, we denote by ek the standard basis of Rn. The matrix K can be clearly identified with its o(3) block and hence K can be considered as a subspace of Lo(3)∗. The isotropy subgroup N is given by matrices of the form ( 1 0 0 Θ ) with Θ ∈ SO(2). Using this information we can find the reduced bracket algebraically. For this we take any functional h : K → R. Let’s call H an extension of h to Lo(3)∗, constant on the gauge leaves of LN . Its variational derivative at K needs to look like δH δL (K) =  0 δh δκ α − δh δκ 0 δh δτ −α − δh δτ 0  for some α to be determined. Since H is constant on the gauge leaves of LN〈 n−1nx + n−1Kn, δH δL (K) 〉 = 0, for any n ∈ LN . This is equivalent to( δH δL (K) ) x + [ K, δH δL (K) ] ∈ Lno, where n is the Lie algebra of N and no is its annihilator. From here0 ( δh δκ)x − ατ αx − κ δh δτ + τ δh δκ ∗ 0 ( δh δτ )x + κα ∗ ∗ 0  = 0 ∗ ∗ ∗ 0 0 ∗ 0 0  , where ∗ indicates entries that are not, at least for now, relevant. Hence α = − 1 κ( δh δτ )x. The bracket is thus given by {f, h}R 1 (K) = ∫ S1 tr   0 ∗ ∗ −( δh δκ)x − τ κ( δh δτ )x 0 0( 1 κ ( δh δτ ) x ) x + κ δh δτ − τ δh δκ 0 0   0 δf δκ − 1 κ( δf δτ )x − δf δκ 0 δf δτ 1 κ( δf δτ )x − δf δτ 0   dx = −2 ∫ S1 ( δf δκ δf δτ ) R ( δh δκ δh δτ ) dx, where R is R = ( D τ κD D τ κ −D −D 1 κD 1 κD ) . The second Hamiltonian structure (11) can also be reduced to K with the general choice L0 = 0 a b −a 0 c −b −c 0 . The reduced bracket is found when applying (11) to the variational derivatives of extensions that, as before, are constant on the LN leaves. Thus, it is straightforward to check that the second reduced bracket is given by {f, h}R 2 (K) = 2 ∫ S1 ( δf δκ δf δτ ) (aA+ bB + cC) ( δh δκ δh δτ ) dx, 10 G. Maŕı Beffa where A = ( 0 0 0 1 κD −D 1 κ ) , B = ( 0 1 −1 0 ) , C = ( 0 1 κD D 1 κ 0 ) . These are all Hamiltonian structures and they appeared in [38]. In fact, the structure P shown in the introduction can be written as P = −RC−1R and, hence, P is in the Hamiltonian hierarchy generated by R and C. A study of integrable systems associated to these brackets, and their geometric realizations, was done in [38]. See also [24]. Our first reduced bracket is directly related to geometric realizations. In fact, by choosing a special parameter x we can obtain geometric realizations of systems that are Hamiltonian with respect to the reduced bracket. We explain this next. Let ρ be given as in (12) and assume u is a solution of the invariant equation ut = ρGr, (14) where r = (ri) is a differential invariant vector, that is, ri are all functions of the entries of K and its derivatives. If the parameter has been fixed so as to guarantee that ρ−1 G ρu = Λ is constant, then r has to be modified to guarantee that the evolutions (14) preserve the parameter. (In the running example ρG = (T,N, B) and ρGr = r1T + r2N + r3B with r2 = r′ 1 κ once the arc-length is chosen as parameter.) Theorem 6 ([32]). If there exists a Hamiltonian h : K → R and a local extension H constant on the leaves of N such that δH δL (K)Λ = rx + Kr, (15) then the invariantization of evolution (14) is Hamiltonian with respect to the reduced bracket { , }R 1 and its associated Hamiltonian is h. If (after choosing a special parameter if necessary) relation (15) can be solved for r given a certain Hamiltonian h, then the Theorem guarantees a geometric realization for the reduced Hamiltonian system. Such is the case for the VF flow, Sawada–Koterra [32, 41], modified KdV and others [38]. One can find many geometric realizations of integrable systems in affine manifolds (see, for example, [2, 11, 24, 25, 26, 28, 46, 47, 49, 51]). Many of these are realizations of modified KdV equations (or its generalizations), sine-Gordon and Schrödinger flows. These systems have geometric realizations also in non-affine manifolds (see [1, 25, 26, 47, 48, 49]). Nevertheless, a common feature to the generation of these realizations is the existence of a classical moving frame that resembles the classical natural moving frame, that is, the derivatives of the non- tangential vectors of the classical frame all have a tangential direction. Thus, it seem to be the case that the existence of geometric realizations for these systems is linked to the existence of a natural frame. This close relationship between geometry and the type of integrable system is perhaps clearer in our next study, that of symmetric manifolds. Before moving on, we have one final comment in this line of thought. There are other manifolds whose geometry is given by a linear (rather than affine) action of the group. We can still follow a similar approach, reduce the brackets and study Hamiltonian structures on the space of differential invariants. For example, in the case of centro-affine geometry one considers the linear action of SL(n) on Rn and the associated geometry is that of star-shaped curves. If we assume that curves are parametrized by the centro-affine arc-length, we can reduce both brackets and obtain a pencil of Poisson brackets. This pencil coincides with the Geometric Realizations of Bi-Hamiltonian Integrable Systems 11 bi-Hamiltonian structure of KdV. Indeed, a geometric realization for KdV was found by Pinkall in [44]. This realization is the one guaranteed by the reduction, as explained in [7]. The interesting aspect of the centro-affine case is the following: there is a natural identification of a star-shaped curve with a projective curve. The identification is given by the intersection of the curve with the lines going through the origin. If the star-shaped curve is nondegenerate (that is, det(γ, γx, . . . , γ(n−1)) 6= 0. For example, in the planar case the curve is never in the radial direction), the identification is well-defined. Furthermore, if we parametrized star-shaped curves with centroaffine arc-length (that is, if det(γ, γx, . . . , γ(n−1)) = 1), the identification is 1-to-1 and the geometries are Poisson-equivalent, the Poisson isomorphism given by the identification. In fact, Pinkall’s geometric realization is the star-shaped version of the Schwarzian KdV under this relation (see [7] for more details). The existence of a geometric realization for KdV seems to imply the existence of a background projective geometry. 3.2 Symmetric manifolds Assume that M is a symmetric manifold which is locally equivalent to G/H with: (a) G semisim- ple; (b) its (Cartan) connection given by the Maurer–Cartan form (i.e. the manifold is flat); (c) the Lie algebra g has a gradation of length 1, i.e. g = g−1 ⊕ g0 ⊕ g1 (16) with h = g0 ⊕ g1, where h is the Lie algebra of H. If M is a symmetric manifold of this type, G splits locally as G−1 ·G0 ·G1 with H given by G0 ·G1. The subgroup G0 is called the isotropic subgroup of G and it is the component of G that acts linearly on G/H (for more information see [4] or [42]). That means G0 is the component of the group acting on first order frames. According to Theorem 2, the ρ0 factor of a left moving frame ρ will determine a classical moving frame (see also [31]). As in the previous case, the basis for the definition of a Poisson bracket on the space of invariants is to express that space as a quotient in Lg∗. This is the result in the following Theorem. For a complete description and proofs see [31]. Notice that, if ρ is a (right) moving frame along a curve in a symmetric manifold with ρ · u = o, then ρ · u1 is always constant. In general ρ · u1 is described by first order invariants, but curves in symmetric manifolds do not have non-constant first order differential invariants (invariants are third order or higher), and hence ρ · u1 must be constant. Theorem 7 ([31]). Let M = G/H be a symmetric manifold as above. Assume that for every curve in a neighborhood of a generic curve u in M we choose a left moving frame ρ with ρ ·o = u and ρ−1 · u1 = Λ̂ constant. Assume that we choose a section of G/H so we can locally identify the manifold with G−1 and its tangent with g−1. Let Λ ∈ g−1 represent Λ̂ and let K be the manifold of Serret–Frenet equations for ρ along curves in a neighborhood of u. Clearly K ⊂ Lg∗. Denote by 〈Λ〉 the linear subspace of C∞(S1, g∗) given by 〈Λ〉 = {αΛ, α(x) > 0}. Then the space K can be described as a quotient U/N , where U is an open set of 〈Λ〉⊕Lg0⊕Lg1 and where N = N0·LG1 ⊂ LG0·LG1 acts on U via the Kac–Moody action (13). The subgroup N0 is the isotropy subgroup of 〈Λ〉 in LG0. As before, after writing K as a quotient, one gets a reduction theorem. Theorem 8 ([31]). The Poisson bracket (10) can be reduced to K and there exists a well-defined Poisson bracket { , }R 1 defined on a generating set of independent differential invariants. Example 6. As we saw before, the (left) Serret–Frenet equations for the RP1 case are given by K = ( 0 1 k 0 ) 12 G. Maŕı Beffa where k = −1 2S(u). The splitting of the Lie algebra sl(2) is given by( 0 β 0 0 ) + ( α 0 0 −α ) + ( 0 0 γ 0 ) ∈ g−1 + g0 + g1. In this case Λ = ( 0 1 0 0 ) ∈ g−1. The isotropic subgroup of 〈Λ〉 in G0 is G0 itself, and so N = LG0 · LG1 or subspace of lower triangular matrices. To reduce the bracket (10) we need to have a functional h : K → R and to find an extension H : M→ R such that( δH δL (K) ) x + [ K, δH δL (K) ] ∈ n0 (17) for any K ∈ K, where n = Lg0 ⊕ Lg1 and n0 is its annihilator (which we can identify with Lg1 = Lg∗−1). Also, if H is an extension of h, its variational derivative at K will be given by δH δL (K) = ( a δh δk (k) b −a ) , where a and b are to be determined. On the other hand condition (17) translates into( ax + b− k δh δk (k) ( δh δk (k) ) x − 2a bx + 2ka −ax − b + k δh δk (k) ) = ( 0 0 ∗ 0 ) . From here a = 1 2 ( δh δk (k) ) x and b = k δh δk (k) − 1 2 ( δh δk (k) ) xx . We are now ready to find the reduced bracket. If f, h : K → R are two functionals and F and H are extensions vanishing on the N -leaves, the reduced bracket is given by {f, h}R 1 (K) = ∫ S1 tr ( δF δL (K) {( δH δL (K) ) x + [ K, δH δL (K) ]}) dx = ∫ S1 tr (( ∗ δf δk (k) ∗ ∗ )( 0 0 k ( δh δk (k) ) x + (k δh δk (k))x − 1 2 ( δh δk (k) ) xxx 0 )) dx = ∫ S1 δf δk (k) ( −1 2 D3 + kD + Dk ) δh δk (k)dx. The difference in coefficients as compared to the introductory example is due to the fact that k = −1 2S(u) and not S(u). As it happens, the companion bracket also reduces for L0 = Λ∗ = ( 0 0 1 0 ) . Indeed, it is given by {f, h}R 2 (K) = ∫ S1 tr { δF δL (K) [( 0 0 1 0 ) , δH δL (K) ]} = ∫ S1 tr  1 2 ( δf δk (k) ) x δf δk (k) ∗ −1 2 ( δf δk (k) ) x ( − δh δk (k) 0( δh δk (k) ) x δh δk (k) ) dx = 2 ∫ S1 δf δk (k)D δh δk (k)dx. It is not true in general that (11) is also reducible to K. In fact, one finds that for M = RPn and G = PSL(n + 1) the second bracket (11) is also reducible to K when L0 = Λ∗ ∈ g∗. The Geometric Realizations of Bi-Hamiltonian Integrable Systems 13 resulting two brackets are the first and second Hamiltonian structure for Adler–Gel’fand–Dikii flows. But if M is the so-called Lagrangian Grassmannian, G = Sp(4), the second bracket is never reducible to K [34]. One interesting comment on the connection to AKNS representations: as before, the reduc- tion of the bracket (10) is directly linked to geometric realizations. But the reduction of (11) indicates the existence of an AKNS representation and an integrable system. In the KdV ex- ample we described how the Serret–Frenet equations (6) were gauge equivalent to λ = 0 using the gauge (7). If we gauge the x-evolution of the KdV AKNS representation in Example 4 by that same element we get that the matrix A changes into Aλ = ( 0 1 q − λ2 0 ) = A− λ2L0. Therefore, up to a constant gauge, L0 indicates the position of the spectral parameter in the KdV AKNS representation in Example 4. In fact, it goes further. One can prove that the coefficient h (as in (3)) of the λ-dependent realizations for KdV determined by this AKNS representation (that is, h = q − 2λ2) is given by the variational derivative of the Hamiltonian functional used to write KdV as Hamiltonian system in the pencil { , }R 1 − λ2{ , }R 2 . In the same fashion, the NLS, when written in terms of κ and τ as in (2), has a second Hamiltonian structure obtained when reducing (11) with the choice L0 = 0 0 0 0 0 1 0 −1 0 . One can see that this system has an AKNS representation with x evolution given by ρx =  0 κ 0 −κ 0 τ − λ 0 −τ + λ 0  ρ and where the t component is determined by the evolution induced on the right moving frame ρ by a u evolution whose invariant coefficients h and g as jn (1) are given by the variational derivative of the Hamiltonian used to write this Euclidean representation of NLS as Hamiltonian with respect to the pencil { , }R 1 − λ{ , }R 2 . This must be a known fact on integrable systems, but we could not find it in the literature. For a complete description of this relation see [7]. Back to symmetric spaces. Recall that g = g−1 ⊕ g0 ⊕ g1 with h = g0 ⊕ g1, so that we can identify TxM with g−1. Recall also that dΦρ(o) = (T1 . . . Tn) is a classical moving frame. Theorem 9 ([31]). Assume ρ is as in the statement of Theorem 7. Assume u(t, x) is a solution of the evolution ut = dΦρ(o)r = r1T1 + · · ·+ rnTn, (18) where r = (ri) is a vector of differential invariants. Then, if there exists a Hamiltonian functional h : K → R and an extension of h, H : U → R, constant of the leaves of N and such that[ δH δL (L) ] −1 = r (the subindex −1 indicates the component in g−1), then the evolution induced on the generating system of differential invariants defined by K is Hamiltonian with associated Hamiltonian h. Let us call the generating differential invariants k. Given a Hamiltonian system kt = ξh(k) = P(k) δh δk , its associated geometric evolution (18) will be its geometric realization in M . Therefore they always exist, the previous theorem guarantees their existence. Indeed, one only needs to extend the functional h preserving the leaves. As explained in [31] and described in our running examples, one can find δH δL (K) along K explicitly using a simple algebraic process. Then, the 14 G. Maŕı Beffa coefficients ri of the realization are given by ( δH δL (K) ) −1 , as identified with the tangent to the manifold using our section of G/H. Some examples are given in [31, 33, 34, 35]. As we have previously pointed out, the second bracket does never reduce to the space of invariants when G = Sp(4), the Lagrangian Grassmannian. Still, one can select a submanifold of invariants and study reductions of the brackets on the submanifold where the other invariants vanish. This is equivalent to studying Hamiltonian evolutions of special types of flows on Sp(4)/H. In particular, the author defined differential invariants of projective type in [33]. She then showed how both brackets (10) and (11) (for some choice of L0) could be reduced on flows of curves with vanishing non-projective differential invariants. The reductions produced Hamiltonian structures and geometric realizations for integrable systems of KdV-type. This, and its implication for the geometry of G/H is described next. 4 Completely integrable systems of KdV type associated to differential invariants of projective type There are some differential invariants of curves in symmetric spaces that one might call of pro- jective type. They are generated by the action of the group on second frames (hence they cannot be found using a classical moving frame) and most of them closely resemble the Schwarzian derivative. In terms of the gradation g = g−1 ⊕ g0 ⊕ g1, if we choose an appropriate moving frame, these invariants will appear in K1, where K = ρ−1ρx = K−1 + K0 + K1 is the graded splitting of the Serret–Frenet equation associated to the moving frame ρ. Of course, the simplest example of differential invariants of projective-type are projective differential invariants. As it was shown in [31], a moving frame can be chosen so that all differential invariants appear in the g1 component of the Serret–Frenet equations, while all entries outside g1 are constant. More examples of differential invariants of projective type appear in [33, 34, 35] and [37]. In this series of papers the author showed how in many symmetric spaces one can find geometric realizations inducing evolutions of KdV type on the differential invariants of projective type. Indeed, if G/H is a symmetric space as above, it is known [27] that g is the direct sum of the following simple Lie algebras: 1. g = sl(p + q) with p, q ∈ Z+. If q = 1 then G/H ≡ RPn. In general G/H is the Grassmannian. 2. g = sp(2n), the manifold G/H is called the Lagrangian Grassmannian and it can be identified with the manifold of Lagrangian planes in R2n. 3. g = o(n, n), the manifold G/H is called the manifold of reduced pure spinors. 4. g = o(p + 1, q + 1) with p, q ∈ Z+. If q = 0 the manifold G/H is isomorphic to the Möbius sphere, the local model for flat conformal manifolds. 5. Two exceptional cases, g = E6 and g = E7. We will next describe the situation for each one of the 1–4 cases above. The Grassmannian case (1) (other than q = 1) and the exceptional cases (5) have not yet been studied. 4.1 Projective case Let G = PSL(n + 1). If g ∈ G then, locally g = g−1g0g1 = ( I u 0 1 )( Θ 0 0 (detΘ)−1 )( I 0 vT 1 ) Geometric Realizations of Bi-Hamiltonian Integrable Systems 15 with u, v ∈ Rn and Θ ∈ GL(n). If we define H by the choice u = 0, then G/H ∼= RPn, the n-projective space. This factorization corresponds to the splitting given by the gradation (16). A section for G/H can be taken to be the g−1 factor. As with any other homogeneous space, the action of G on this section is completely determined by the relation g ( I u 0 1 ) = ( I g · u 0 1 ) h for some h ∈ H. The corresponding splitting of the Lie algebra is given by V = V−1 + V0 + V1 = ( 0 a 0 0 ) + ( A 0 0 −trA ) + ( 0 0 bT 0 ) ∈ g−1 ⊕ g0 ⊕ g1. We can identify the first term above with the tangent to the manifold. The following theorem describes the type of moving frame and invariant manifold we will choose. It is essential to choose a simple enough representation for K so that one can readily recognize the result of the reduction. Naturally, any choice of moving frame will produce a choice for K and a Hamiltonian structure. But showing the equivalence of Poisson structures is a non- trivial problem, and hence its recognition is an important part of the problem. As we said before, all differential invariants are of projective-type. Theorem 10 (reformulation of Wilczynski [50]). There exists a left moving frame ρ along nondegenerate curves in RPn such that its Serret–Frenet equations are defined by matrices of the form K =  0 1 0 . . . 0 0 0 1 . . . 0 ... ... . . . . . . ... 0 0 . . . 0 1 k1 k2 . . . kn 0  , where ki, i = 1, . . . , n are, in general, a generating combination of the Wilczynski projective invariants and their derivatives. For the precise relation between these invariants and Wylczynski’s invariants, see [7]. The following result was originally obtained by Drinfel’d and Sokolov in [12]. Their description of the quotient is not the same as ours, but [31] showed that our reduction and theirs are equivalent. Theorem 11 ([12]). Assume K is represented by matrices of the form above. Then, the reduc- tion of (10) to K is given by the Adler–Gel’fand–Dikii (AGD) bracket or second Hamiltonian structure for generalized KdV. The bracket (11) reduces for the choice L0 = e∗n and its reduction is the first Hamiltonian structure for generalized KdV equations. Finally, the following theorem identifies geometric realizations for any flow Hamiltonian with respect to the reduction of (10); in particular, it provides geometric realizations for generalized KdV equations or AGD flows. The case n = 2 was originally proved in [18]. Theorem 12 ([36]). Assume u : J ⊂ R2 → PSL(n + 1)/H is a solution of ut = h1T1 + h2T2 + · · ·+ hnTn, where Ti form a projective classical moving frame. 16 G. Maŕı Beffa Then, k = (ki) satisfies an equation of the form kt = Ph, where k = (k1, . . . , kn)T , h = (h1, . . . , hn)T and where P is the Poisson tensor defining the Adler–Gel’fand–Dikii Hamiltonian structure. In particular, we obtain a projective geo- metric realization for a generalized KdV system of equations. In our next section we look at some cases for which not all differential invariants of curves are of projective type. 4.2 The Lagrangian Grassmannian and the manifold of reduced pure Spinors These two examples are different, but their differential invariants of projective type behave similarly and so we will present them in a joint section. Lagrangian Grassmannian. Let G = Sp(2n). If g ∈ G then, locally g = g−1g0g1 = ( I u 0 I )( Θ 0 0 Θ−T )( I 0 S I ) with u and S symmetric n× n matrices and Θ ∈ GL(n). Again, this factorization corresponds to the splitting given by the gradation (16). The subgroup H is locally defined by the choice u = 0 and a local section of the quotient can be represented by g−1. As usual, the action of the group is determined by the relation g ( I u 0 I ) = ( I g · u 0 I ) h for some h ∈ H. The corresponding splitting of the algebra is given by V = V−1 + V0 + V1 = ( 0 S1 0 0 ) + ( A 0 0 −AT ) + ( 0 0 0 S2 ) , where S1 and S2 are symmetric matrices and A ∈ gl(n). The manifold G/H is usually called the Lagrangian Grassmanian in R2n and it is identified with the manifold of Lagrangian planes in R2n. The following theorem describes a representation of the manifold K for curves of Lagrangian planes in R2n under the above action of Sp(2n). Theorem 13 ([34]). There exists a left moving frame ρ along a generic curve of Lagrangian planes such that its Serret–Frenet equations are given by K = ρ−1ρx = ( K0 I K1 K0 ) , where K0 is skew-symmetric and contains all differential invariants of order 4, and where K1 = −1 2Sd. The matrix Sd is diagonal and contains in its diagonal the eigenvalues of the Lagrangian Schwarzian derivative (Ovsienko [43]) S(u) = u −1/2 1 ( u3 − 3 2 u2u −1 1 u2 ) (u−1/2 1 )T . The entries of K0 and K1 are functionally independent differential invariants for curves of Lagrangian planes in R2n under the action of Sp(2n). They generate all other differential in- variants. The n differential invariants that appear in K1 are the invariants of projective type. Geometric Realizations of Bi-Hamiltonian Integrable Systems 17 Now we describe some of the geometric flows that preserve the value K0 = 0. Therefore, geometric flows as below will affect only invariants of projective type, if proper initial conditions are chosen. Theorem 14 ([34]). Assume u : J ⊂ R2 → Sp(2n)/H is a flow solution of ut = ΘT u 1/2 1 hu 1/2 1 Θ, (19) where Θ(x, t) ∈ O(n) is the matrix diagonalizing S(u) (i.e., ΘS(u)ΘT = Sd) and where h is a symmetric matrix of differential invariants. Assume h is diagonal. Then the flow preserves K0 = 0. Finally, our next theorem gives integrable PDEs with geometric realizations as geometric flows of Lagrangian planes. Theorem 15 ([34]). Let K1 be the submanifold of K given by K0 = 0. Then, the reduced bracket on K restricts to K1 to induce a decoupled system of n second Hamiltonian structures for KdV. Bracket (11) also reduces to K1 (even though it does not in general reduce to K for any value of L0). The reduction for the choice L0 = ( 0 0 I 0 ) = ( 0 I 0 0 )∗ ∈ g1 is a decoupled system of n first KdV Hamitonian structures. Furthermore, assume u(t, x) is a flow solution of (19) with h = Sd. Then (19) becomes the Lagrangian Schwarzian KdV evolution ut = u3 − 3 2 u2u −1 1 u2. If we choose initial conditions for which K0 = 0, then the differential invariants Sd of the flow satisfy the equation (Sd)t = ( D3 + SdD + (Sd)x ) h, where D is the diagonal matrix with d dx down its diagonal. Accordingly, if we choose h = Sd, then Sd is the solution of a decoupled system of n KdV equations (Sd)t = (Sd)xxx + 3Sd(Sd)x. Reduced pure Spinors. A parallel description can be given for a different case, that of G = O(n, n). In this case, if g ∈ G, locally g = g−1g0g1 = ( I − u −u u I + u ) 1 2 ( Θ−1 + ΘT ΘT −Θ−1 ΘT −Θ−1 Θ−1 + ΘT )( I − Z Z −Z I + Z ) , where u and S are now skew-symmetric matrices and where Θ ∈ GL(n, R). The corresponding gradation of the algebra is given by V = V−1 + V0 + V1 ∈ g−1 ⊕ g0 ⊕ g1 with V−1 = V−1(y) = ( −y −y y y ) , V0 = V0(C) = ( A B B A ) , V1 = V1(z) = ( −z z −z z ) , y and z skew symmetric and C = A + B given by the symmetric (B) and skew-symmertric (A) components of C. Assume now that G = O(2m, 2m). This case has been worked out in [33]. The homogeneous space is locally equivalent to the manifold of reduced pure spinors in the sense of [4]. The odd dimensional spinor case is worked out in [37]. Although somehow similar it is more cumbersome to describe, so we refer the reader to [37]. 18 G. Maŕı Beffa Theorem 16 ([33]). Let u be a generic curve in O(2m, 2m)/H. There exists a left moving frame ρ such that the left Serret–Frenet equations associated to ρ are defined by K = V−1(J) + V0(R) + 1 8 V1(D), where J = ( 0 Im −Im o ) and R is of the form R = ( R1 R2 R3 −RT 1 ) ∈ Sp(2m) with R2 and R3 symmetric, R1 ∈ gl(m). The matrix R contains in the entries off the diagonals of Ri, i = 1, 2, 3, a generating set of independent fourth order differential invariants. The diagonals of Ri, i = 1, 2, 3 contain a set of 3m independent and generating differential invariants of order 5 for m > 3 and of order 5 and higher if m ≤ 3. The matrix D is the skew-symmetric diagonalization of the Spinor Schwarzian derivative. The Spinor Schwarzian derivative [33] is described as follows: if u is a generic curve repre- sented by skew-symmetric matrices, u1 is non degenerate and can be brought to a normal form using µ ∈ gl(2m). The matrix µ is determined up to an element of the symplectic group Sp(2m) (see [33]) and µu1µ T = J. We define the Spinor Schwarzian derivative to be S(u) = µ ( u3 − 3 2 u2u −1 1 u2 ) µT , again, unique up to the action of Sp(2m). One can then prove [33] that, for a generic curve, the Schwarzian derivative can be diagonalize using an element of the symplectic group. That is, there exists θ ∈ Sp(2m) such that θS(u)θT = D = ( 0 d −d 0 ) with d diagonal. The matrix D is the one appearing in the Serret–Frenet equations and it contains in its entries m differential invariants of projective type. The Spinor case seems to be different from others and the behavior of the Poisson brack- ets (10) and (11) is not completely understood yet. Still we do know how the invariants of projective type behave under the analogous of the KdV Schwarzian evolution. That is described in the following theorem. The Spinor KdV Schwarzian evolution is defined by the equation ut = u3 − 3 2 u2u −1 1 u2. Theorem 17 ([33]). Let ρ is a moving frame for which normalization equations of fourth order are defined by constants c4. Assume that, as the fourth order invariants vanish, [R, R̂] =̂̂ R + block diagonals, where R̂ and ̂̂R are any matrices whose only non-zero entries are in the same position as the nonzero normalized entries in R. Assume also that [R, [R, R̂]]d = 0 for R̂ as above, where d indicates the diagonals in the main four blocks. Then, if we choose initial conditions with vanishing fourth order invariants, these remain zero under the KdV Schwarzian flow, Dt and (Rd)t decouple, and D evolves as DtJ = DxxxJ + 3DDx, following a decoupled system of KdV equations. Geometric Realizations of Bi-Hamiltonian Integrable Systems 19 Although the hypothesis of this theorem seem to be very restrictive, they aren’t. In fact, they are easily achieved when we construct moving frames in a dimension larger that 4, although they are more restrictive for the lower dimensions. The need for these conditions was explained in [33]. This is the first case where the manifold of vanishing non-projective invariants is not pre- served. Not only the vanishing of fifth order invariants is not preserved, but the system blows up when we approach the submanifold of vanishing fifth order invariants. This situation (and the analogous one for the odd dimensional case) is not well understood. It is possible that choosing normalization equations for which c4 involve derivatives of the third order differential invariants will produce a better behaved moving frame. Or perhaps the fifth order differential invariants are also of projective type and the Hamiltonian behavior is more complicated that appears to be. The main problem understanding this case is the choice of a moving frame that simplifies the study of the reduced evolution. In the spinorial case such a choice is highly not trivial and we do not know whether there is no such choice or it is just very involved. It is also not known whether or not (10) and (11) reduce to K1. For more information see [33] and [37]. The Lagrangian and Spinorial examples describe a certain type of behavior (evolving as a decoupled system of KdVs) of invariants of projective type that is certainly different from our first example, that of RPn. Still, there is a third behavior that is different from these two. These evolutions of KdV type appear in conformal manifolds. In the conformal case we have only two differential invariants of projective type, and their Hamiltonian behavior is that of a complexly coupled system of KdVs. That is what we describe in the next subsection. 4.3 Conformal case In this section we study the case of G = O(p+1, q +1) acting on Rp+q as described in [42]. The case q = 0 was originally studied in [35]. Using the gradation appearing in [27] we can locally factor an element of the group as g = g1g0g−1 (this is a factorization for a right moving frame, not a left one, so it is slightly different from the previous examples), with gi ∈ Gi where g−1(Y ) =  1− 1 2 \|\|Y \|\| 2 −Y T 1 −1 2 \|\|Y \|\| 2 Y T 2 Y1 Ip Y1 0 1 2 \|\|Y \|\| 2 Y T 1 1 + 1 2 \|\|Y \|\| 2 −Y T 2 Y2 0 Y2 Iq , g0(a, b, Θ) =  a 0 b 0 0 Θ11 0 Θ12 b 0 a 0 0 Θ21 0 Θ22 , g1(Z) =  1− 1 2 \|\|Z\|\| 2 ZT 1 1 2 \|\|Z\|\| 2 ZT 2 −Z1 Ip Z1 0 −1 2 \|\|Z\|\| 2 ZT 1 1 + 1 2 \|\|Z\|\| 2 ZT 2 Z2 0 −Z2 Iq  . The splitting Z = ( Z1 Z2 ) and Y = ( Y1 Y2 ) is into p and q components, \|\|X\|\|2 is given by the flat metric of signature (p, q) and also Θ = ( Θ11 Θ12 Θ21 Θ22 ) ∈ O(p, q), a2 − b2 = 1. Without loosing generality we will assume that the flat metric is given by \|\|X\|\|2 = XT JX, where J = ( Ip 0 0 −Iq ) . The corresponding splitting in the algebra is given by V−1(y) =  0 −yT 1 0 yT 2 y1 0 y1 0 0 yT 1 0 −yT 2 y2 0 y2 0  , V0(α, A) =  0 0 α 0 0 A11 0 A12 α 0 0 0 0 A21 0 A22  , 20 G. Maŕı Beffa V1(z) =  0 zT 1 0 zT 2 −z1 0 z1 0 0 zT 1 0 zT 2 z2 0 −z2 0  , where y = ( y1 y2 ) , z = ( z1 z2 ) are the p and q components and where A = (Aij) ∈ o(p, q). The algebra structure can be described as [V0(α, A), V1(z)] = V1(JAJz + αz), [V0(α, A), V−1(y)] = V−1(Ay − αy), [V1(z), V−1(y)] = 2V0 ( zT y, JzyT J − yzT ) , [V0(α, A), V0(β, B)] = V0(0, [A,B]). With this factorization one chooses H to be defined by Y = 0 and uses G−1 as a local section of G/H (as such Y = −u, not u). As before, the following theorem describes convenient choices of moving frames. Theorem 18 ([33]). There exists a left moving frame ρ such that the Serret–Frenet equation for ρ is given by ρ−1ρx = K = K1 +K0 +K−1 where K−1 = V−1(e1), K1 = V1(k1e1 + k2e2) and K0 = V0(0, K̂0) with K̂0 = ( A0 B0 BT 0 0 ) , and A0 =  0 0 0 0 . . . 0 0 0 −k3 −k4 . . . −kp 0 k3 0 0 . . . 0 ... ... ... ... ... 0 0 kp 0 . . . 0 0  , B0 =  0 0 . . . 0 −kp+1 −kp+2 . . . −kp+q 0 0 . . . 0 ... ... ... ... 0 0 . . . 0  . In this case there are only two generating differential invariants of projective type, namely k1 and k2. Again, the behavior of the Poisson brackets (10) and (11) with respect to this subman- ifold is spotless. Theorem 19 ([33]). Let K1 be the submanifold of K given by K0 = 0. Then, the reduction of (10) to K restricts to K1 to induce the second Hamiltonian structure for a complexly coupled KdV system. Bracket (11) also reduced to K1 to produce the first Hamiltonian structure for this system. And, again, the geometric realization for complexly coupled KdV is found. Theorem 20 ([33]). Assume u : J ⊂ R2 → O(p + 1, q + 1)/H is a solution of ut = h1T + h2N, where T and N are conformal tangent and normal (see [33]) and h1, h2 are any two functions of k1, k2 and their derivatives. Then the flow has a limit as K0 → 0. As K0 → 0, the evolution of k1 and k2 becomes( k1 k2 ) t = ( −1 2D3 + k1D + Dk1 k2D + Dk2 k2D + Dk2 1 2D3 − k1D −Dk1 )( h1 h2 ) . If we choose h1 = k1 and h2 = k2, then the evolution is a complexly coupled system of KdV equations. Geometric Realizations of Bi-Hamiltonian Integrable Systems 21 5 Discussion The aim of this paper is to review some of the known evidence linking the character of differential invariant of curves in homogeneous spaces and the geometric realizations of integrable systems in those manifolds. In particular, we have described how projective geometry and geometric realizations of KdV-type evolutions seem to be very closely related. A similar case can per- haps be made for Schrödinger flows, mKdV and sine-Gordon flows as linked to Riemannian geometry. As we said before, several authors [1, 25, 26, 28, 29, 38, 46, 48, 49], have described geometric realizations of these evolutions on manifolds that have what amounts to be a classical natural moving frame, i.e., a frame whose derivatives of non tangential vectors have a tangen- tial direction. This frame appears in Riemannian manifolds and is generated by the action of the group in first order frames. Thus, one could call the invariants they generate invariants of Riemannian-type. In fact, the most interesting question is how the geometry of the manifold itself generates these geometric realizations. And, further, if a manifold hosts a geometric realization of an integrable system of a certain type, does that fact have any implications for the geometry of the manifold? In the case of projective geometry, the following conjecture due to M. Eastwood points us in this direction. Conjecture. In this type of symmetric spaces there exists a natural projective structure along curves that generates Hamiltonian structures of KdV type along some flows. In the conformal case G = O(p + 1, q + 1) the two invariants of projective type are directly connected to invariant differential operators that appear in the work of Bailey and Eastwood (see [5, 6]). The authors defined conformal circles as solutions of a differential equation. The equation defines the curves together with a preferred parametrization. The parametrizations endow conformal circles with a projective structure (theirs is an explicit proof of Cartan’s ob- servation that a curve in a conformal manifold inherits a natural projective structure, see [10]). We now know that the vanishing of the differential equation in [5] implies the vanishing of both differential invariants of projective type found in [33]. Therefore, the complexly coupled sys- tem of KdV equations could be generated by the projective structure on conformal curves that Cartan originally described. Natural projective structures on curves have only been described for the cases O(p+1, q+1) [5] and SL(p + q) [6], but they do perhaps exist for \|r\|-graded parabolic manifolds. Thus, resolving this conjecture and its generalizations would help to understand the more general situation of parabolic manifolds. In [12] Drinfel’d and Sokolov described many evolutions of KdV-type linked to parabolic gradations of the Lie algebra g. It would be interesting to learn if parabolic manifolds (flag manifolds) can be used to generate geometric realizations for these systems. References [1] Anco S., Hamiltonian flows of curves in G/SO(N) and vector soliton equations of mKdV and sine-Gordon type, SIGMA 2 (2006), 044, 18 pages, nlin.SI/0512046. 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[50] Wilczynski E.J., Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig, 1906. [51] Yasui Y., Sasaki N., Differential geometry of the vortex filament equation, J. Geom. Phys. 28 (1998), 195–207, hep-th/9611073. http://arxiv.org/abs/math.AP/0301212 http://arxiv.org/abs/math.DG/0108154 http://arxiv.org/abs/math.DG/9901086 http://arxiv.org/abs/math.DG/9805074 http://arxiv.org/abs/hep-th/9611073 1 Introduction 2 Moving frames 3 Hamiltonian structures generated by group-based moving frames 3.1 Affine manifolds 3.2 Symmetric manifolds 4 Completely integrable systems of KdV type associated to differential invariants of projective type 4.1 Projective case 4.2 The Lagrangian Grassmannian and the manifold of reduced pure Spinors 4.3 Conformal case 5 Discussion References

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

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