Kronecker webs, Nijenhuis operators, and nonlinear PDEs
aa r X i v : . [ m a t h . DG ] M a y Kronecker webs, Nijenhuis operators, and nonlinear PDEs
Andriy PanasyukFaculty of Mathematics and Computer ScienceUniversity of Warmia and Mazuryul. Soneczna 54, 10-710 Olsztyn, [email protected]
In memory of Stanis law Zakrzewski (1951–1998)
Abstract
The aim of this paper is two-fold. First, a survey of the theory of Kronecker webs and theirrelations with bihamiltonian structures and PDEs is presented. Second, a partial solution to theproblem of bisymplectic realization of a bihamiltonian structure is given. Both the goals are achievedby means of the notion of a partial Nijenhuis operator, which is studied in detail.
Contents Introduction
A seminal paper of F. Magri [Mag78] gave rise to a notion of a bihamiltonian structure , i.e. a pair ofcompatible Poisson structures η , η (here compatibility means that η λ = η + λη is a Poisson structurefor any λ ), which proved to be a very effective tool in the study of integrable systems and has beendeveloped by many authors. F. J. Turiel [Tur89] and I. M. Gelfand and I. Zakharevich [GZ89], [GZ93]initiated the investigation of the local structure of pairs of compatible Poisson brackets. It turns outthat there are two classes of bihamiltonian structures of principally different nature (on the level of localgeometry as well as in applications to integrable systems). The bihamiltonian structures of first classcalled Jordan (cf. Section 2) or bisymplectic consist of pairs η , η such that in the pencil { η λ } almostall members are nondegenerate Poisson structures, i.e. inverse to symplectic forms. On the contrary, inthe pencils corresponding to the second class of Kronecker bihamiltonian structures all the members aredegenerate of the same rank.It is worth mentioning that for both Jordan and Kronecker cases the classical Darboux theorem fails:in general there is no local coordinate system in which η , η simultaneously have constant coefficients.In order to understand local behaviour of Kronecker bihamiltonian structures Gelfand and Zakharevich[GZ91] proposed a procedure which reduces the geometry of pairs of compatible Poisson brackets tothe geometry of webs . Recall that a classical web is a finite number of foliations in general positionon a smooth manifold and that the main question in the theory of webs is to describe obstructions tosimultaneous local straightening of these foliations, i.e. transforming them by a local diffeomorphismto foliations of parallel plains. The reduction mentioned consists in a passage to a local base B of thelagrangian foliation L = T λ S λ , where S λ is the symplectic foliation of η λ (the Kroneckerity of the pair η , η guarantees that indeed the distribution T λ T S λ has constant rank and, moreover, the leaves of L are lagrangian in S λ ). The induced by {S λ } one-parameter family of foliations {S ′ λ } on B is calleda Kronecker web . This notion naturally generalizes the notion of a classical web and the problem ofexisting of the “Darboux coordinates” for η , η can be treated in spirit of the web theory as the problemof simultaneous straightening of the foliations S ′ λ . Moreover, Gelfand and Zakharevich conjectured thatthe Kronecker web is a complete local invariant of a Kronecker bihamiltonian structure, that is, onecan reconstruct η , η from {S ′ λ } up to a local diffeomorphism. This conjecture was proved by Turiel[Tur99b], [Tur00].Later another side of Kronecker webs appeared in the literature: their relation with nonlinear PDEs.First discovery was made by Zakharevich [Zak00] who found a relation of Veronese webs in 3 dimensions(special class of Kronecker webs) with a nonlinear second order PDE called the dispersionless Hirotaequation (originally called by Zakharevich a nonlinear wave equation). The last was studied in [Zak00]from the point of view of twistor theory. Then M. Dunajski and W. Kry´nski [DK14] found relations ofthe Hirota equation with the so-called hyper-CR Einstein–Weyl structures previously described by the hyper-CR nonlinear PDE . Finally, B. Kruglikov and A. Panasyuk [KP17] built several series of contactlynonequivalent PDEs whose solutions are in a 1–1-correspondence with Veronese webs and which includethe Hirota equation as a particular case.This paper is intended as a survey of the above mentioned and related results. However, there are2wo aspects which hopefully allow to treat it partially as an independent research article. First, weintroduce in full generality the notion of a patial Nijenhuis operator , which was outlined in [KP17], anduse it as a convenient tool for defining and working with Kronecker webs, in particular in their relationswith bihamiltonian structures and nonlinear PDEs. Second, we apply the machinery of partial Nijenhuisoperators to the problem of local bisymplectic realizations of Kronecker bihamiltonian structures (seeSection 9) generalizing results by F. Petalidou [Pet00]. Also Theorem 13.1 is new.Let us overview the content of the paper. In Section 1 we discuss relations between vanishing of theNijenhuis torsion of linear operators and compatibility of Lie brackets. We present some examples thatmotivate Definition 1.5 of a partial Nijenhuis operator (PNO) N : h → g , where g is a Lie algebra and h ⊂ g is a Lie subalgebra.Section 2 is devoted to the so-called Jordan–Kronecker decomposition theorem, a classical purelylinear algebraic result on the normal form of a pair of linear operators. This result is a base of classificationof bihamiltonian structures (cf. Jordan and Kronecker cases discussed above) and is also permanentlyused in the context of the pair of operators N, I : h → g , where I : h → g is the canonical inclusion.In Section 3 we study algebraic PNOs: we observe some important consequences of the definition(Lemma 3.2), in particular, we prove that a PNO N induces a Lie algebra structure [ , ] N on h compatiblewith the initial one and that ( N + λI )( h ) is a Lie subalgebra for any λ . We also formulate some sufficientor necessary and sufficient conditions on a partial operator to be a PNO (Lemmas 3.3, 3.8, Remark3.7), discuss some sufficient conditions on a restriction of a Nijenhuis operator to a subalgebra to be aPNO (Lemma 3.9). All these results are of independent interest, however their main aim are geometricapplications in further sections.In Section 4 we recall the notions of a Lie algebroid and the related notion of linear Poisson structure.This framework is very convenient for defining the geometric version of PNOs and the construction ofthe canonical bihamiltonian structure related to a Kronecker web (cf. Section 7). We also discusscompatibility of Lie algebroid structures and the corresponding linear Poisson structures.The central notion of this article, a geometric PNO, appears in Section 5. This is a morphism ofbundles N : T F →
T M , where F is some foliation on a manifold M , such that the correspondingmapping induced on the spaces of sections Γ( T F ) → Γ( T M ) is an algebraic PNO (the Lie bracket beingthe commutator of vector fields). This notion naturally generalizes the well-known differential geometricconditions of vanishing of the Nijenhuis torsion of a (1,1)-tensor (in our terminology this last is a PNOwith T F = T M ). We further generalize some results of Section 3 to the geometric context. A newaspect with respect to the purely algebraic situation is that a PNO N together with the induced Liealgebra structure [ , ] N on Γ( T F ) form a Lie algebroid structure on T F , which, moreover, is compatiblewith the canonical one (see Lemma 5.3(4,5)). The fact that the image of N + λI is a subalgebra, i. e.is the tangent bundle to some foliation F λ , indicates that geometric PNOs are related to 1-parameterfamilies of foliations such as Veronese and Kronecker webs.These last are the main objects of Section 6. We first recall the definition of a Veronese web whichis a collection {F λ } λ ∈ RP of foliations of corank 1 on a manifold M such that the annihilating one-form ( T x F λ ) ⊥ sweeps a Veronese curve in P T ∗ x M for any x ∈ M . We than show that there is a 1–1-correspondence between Veronese webs and PNOs N : T F →
T M of generic type with F ∞ = F and T F = N T F (Theorem 6.2). Here the genericity of type means that there is a sole Kronekcer block inthe Jordan–Kronecker decomposition of the pair of operators N x , I x : T x F → T x M for any x ∈ M . Nextwe naturally generalize this result to Kronecker webs and Kronecker PNOs, the last one admitting morethan one Kronecker block in the decomposition. In Remark 6.7 we touch the problem of “integrabilityof Veronese curves of distributions” and its generalizations.3n Section 7 we use the relations between Lie algebroids and linear Poisson structures established inSection 4 to construct, given a PNO N : T F →
T M , the canonical bihamiltonian structure on T ∗ F .We then specify this construction to two particular cases: a Kronecker PNO and a Jordan PNO (i.e. aPNO with T F = T M , a Nijenhuis operator).In section 8 we discuss in full generality relations { Kronecker webs } $ $ e e { Kronecker bihamiltonianstructures } , in particular the compositions of the passages $ $ and e e in different order. Thisincludes the procedure e e of passing to the local base of a bilagrangian foliation and reconstruction ' ' • f f of a bihamiltonian structure from its Kronecker web up to a local diffeomorphism.Section 9 is devoted to the problem of local bisymplectic realizations of a Kronecker bihamiltonianstructure. More precisely it can be formulated as follows. Let η , be a Kronecker bihamiltonian structureon a small open set U ⊂ R n . Does there exist a manifold M with a Jordan bihamiltonian structure ¯ η , and a surjective submersion p : M → U such that p ∗ ¯ η , = η , ? If such bisymplectic realizations exist,how many nonequivalent ones there are? We show that this problem is reduced to the following problemof “realization of a Kronecker PNO”: (1) given a Kronecker PNO N : T F →
T M does there exist aNijenhuis operator N : T M → T M such that N | T F = N ? (2) how many locally nonequivalent onesthere are? The answer to question (1) is affirmative by a result of Turiel. The answer to question (2) israther impossible in full generality in view of great range of different nonequivalent Kronecker PNOs andNijenhuis operators. However, in the next section we give an answer to this question in the particularcase of Kronecker PNOs of generic type in 3 dimensions thus solving the problem of local bisymplecticrealizations of 5-dimensional generic bihamiltonian structures.More precisely, in Section 10 we prove that, given a Kronecker PNO N : T F →
T M of generic type(Veronese web) on a 3-dimensional manifold, in a neighborhood of every point p ∈ M there exists anextension of N to any of normal forms of a Nijenhis operator, necessarily cyclic. Such normal forms wereobtained by Turiel and Grifone–Mehdi; they are listed in Appendix. We conjecture that the same is truein any dimension: a Kronecker PNO N : T F →
T M of generic type can be extended to any normalform of a cyclic Nijenhuis operator.In Section 11 we apply a sufficient condition for the restriction N | T F of a Nijenhuis operator N : T M → T M to be a PNO (Lemma 5.5) to the case M = R , the foliation F of rank 2 and N being thesimplest Nijenhuis operator with constant distinct eigenvalues. As a result we get a nonlinear secondorder PDE on the function f defining the foliation F . This is the above mentioned dispersionless Hirotaequation. We further prove that any Veronese web in R defines a solution of this equation and, viceversa, any solution defines a Veronese web. This provides a 1–1-correspondence between Veronese websand classes of solutions with respect to a natural equivalence relation.Section 12 is devoted to generalizing these results to other types of Nijenhuis operators in R . Moreprecisely, we get a series of pairwise contactly nonequivalent nonlinear second order PDEs on a function f of three variables. For each of these equations we establish a 1–1-correspondence between classes oftheir solutions and Veronese webs. A crucial ingredient in this correspondence is the solution for therealization problem of a Veronese web obtained in Section 10.In Section 13 we discuss generalizations of the results of the two preceding sections to higher dimen-sions. In particular, we establish a 1–1-correspondence between (classes of) solutions of a certain systemof nonlinear second order PDEs and certain Kronecker webs in 4-dimensional case.Finally, in Section 14 we make a short overview of related bibliography.The notion of partial Nijenhuis operator as well as the majority of related results of this paper arebased on [PZ] and are products of discussions with Ilya Zakharevich, to whom the author would liketo express his deep gratitude. The problem of bisymplectic realization of a bihamiltonian structure was4osed to the author by Stanis law Zakrzewski shortly before this prominent mathematical physicist haspassed away in 1998. This paper is dedicated to his memory. Let ( g , [ , ]) be a Lie algebra, N : g → g a linear operator. A bilinear map T N : g × g → g given by T N ( x, y ) := [ N x, N y ] − N [ x, y ] N , x, y ∈ g , where [ x, y ] N := [ N x, y ] + [ x, N y ] − N [ x, y ] , is called the Nijenhuis torsion of the operator N . One calls N (algebraic) Nijenhuis if T N ≡ . This notion has its origin in the well known in differential geometry notion of the Nijenhuis torsion ofa (1,1)-tensor e N : T M → T M on a smooth manifold M : if g is the Lie algebra Γ( T M ) of the vector fieldson the manifold M with the usual commutator bracket and N : g → g is generated by the endomorphism e N of the tangent bundle T M , the definition above in fact defines a (2,1)-tensor which coincides with theNijenhuis torsion tensor. [KSM90] Let N : g → g be a linear operator acting on a Lie algebra ( g , [ , ]) .1. The bracket [ , ] N is a Lie algebra bracket if and only if d T N = 0 (here we regard T N as a -cochain onthe Lie algebra ( g , [ , ]) with the coefficients in the adjoint module and d stands for the correspondingcoboundary operator).2. Assume d T N = 0 . Then the Lie bracket [ , ] N is automatically compatible with [ , ] , i.e., λ [ , ] + λ [ , ] N is a Lie bracket for any λ , λ ∈ K , here K is the ground field. Pairs ([ , ] , [ , ] ) of compatible (as in the lemma) Lie brackets on a vector space will be called bi-Liestructures . The families of Lie brackets { [ , ] λ } λ ∈ K , [ , ] λ := λ [ , ] + λ [ , ] , λ := ( λ , λ ), generated bybi-Lie structures ([ , ] , [ , ] ) are called Lie pencils [Bol92]. In particular, any algebraic Nijenhuis operatoron ( g , [ , ]) generates a bi-Lie structure on g , hence also a bihamiltonian structure on g ∗ (consisting of thecorresponding Lie–Poisson structures).The following two examples are essential in our further considerations. Let g = gl ( n ), A ∈ g be a fixed matrix, N := L A be the operator of left multiplicationby A . Then N is algebraic Nijenhuis, [ x, y ] N = xAy − yAx =: [ x, A y ] is a Lie bracket, brackets [ , ] , [ , ] N are compatible. Let h = so ( n ), A be a fixed symmetric matrix. Then [ , A ] is a Lie bracket on h compatiblewith [ , ].In the second example we constructed the bracket [ , A ] “by analogy” with the first example. Itis natural to ask whether one can include this bracket into a framework similar to that of Nijenhuisoperators, i.e. whether [ , A ] = [ , ] N for some N . Note that for general symmetric A and N = L A : gl ( n ) → gl ( n ) and we have N so ( n ) so ( n ). However we observe that [ x, y ] N = [ x, A y ] for any x, y ∈ so ( n ). In5rder to understand what happens, assume for a moment that A is nondegenerate, i.e., N is invertible.Although N so ( n ) so ( n ) the subspace N so ( n ) is a Lie subalgebra in gl ( n ). From this we conclude that N − [ N x, N y ] ∈ so ( n ) for any x, y ∈ so ( n ), i.e. N − [ N · , N · ] is a new Lie algebra bracket on so ( n ). Onthe other hand, the fact that T N ≡ gl ( n ) implies that N − [ N x, N y ] = [ x, y ] N = [ x, A y ], x, y ∈ so ( n ),in particular, this new bracket is compatible with the standard one.Let us codify these considerations in a way which allows N to be not invertible. Let g be a Lie algebra and h ⊂ g a Lie subalgebra. We say that a pair ( h , N ), where N : h → g is a linear operator, is an (algebraic) partial Nijenhuis operator on g (PNO for short) if thefollowing two conditions hold:(i) [ x, y ] N ∈ h for any x, y ∈ h ;(ii) T N ( x, y ) = 0 for any x, y ∈ h .(Here [ , ] N and T N are given by the same formulas as above; note that it follows from condition (i) thatthe term N [ x, y ] N which appears in the definition of T N is correctly defined.)The examples above give the following two instances of PNOs: (1) let g = h = gl ( n ), N = L A , then( h , N ) is a PNO on g (which in fact is a Nijenhuis operator since h = g ); (2) let h = so ( n ) , g = gl ( n ), N = L A | h , where A is a symmetric matrix. Then ( h , N ) is a PNO on g .In these examples, given a PNO ( h , N ) on a Lie algebra g , we obtained a bi-Lie structure ([ , ] , [ , ] N )on h . It turns out that it is also true in general (see Lemma 3.2). Vice versa, given a bi-Lie structure([ , ] , [ , ] ) on a vector space h such that ( h , [ , ]) is a semisimple Lie algebra, one can identify h with˜ h = ad ( h ) ⊂ End( h ) and define N : ˜ h → End( h ) by N := ad ◦ ad − , where ad , ad : h → End( h ) arethe corresponding adjoint representations. Then N is a PNO; this fact was helpful in an approach tothe problem of classification of bi-Lie structures ([ , ] , [ , ] ) on semisimple Lie algebras ( h , [ , ]) [Pan14].In order to study PNOs and their relations to bihamiltonian structures and Kronecker webs we recalla classical result on normal forms of a pair of linear operators. [Gan59] Consider a pair of operators S , S : V → W between finite-dimensional vec-tor spaces over C . Then there are direct decompositions V = L nm =1 V m , W = L nm =1 W m , S = L nm =1 S ,m , S = L nm =1 S ,m , where S j,m : V m → W m , j = 1 , , m = 1 , . . . , n , such that each 4-tuple ( S ,m , S ,m , V m , W m ) is from the following list:1. [the Jordan block j λ ( j m )]: dim V m = dim W m = j m and in an appropriate bases of V m and W m thematrix of S ,m is equal to I j m (the unity j m × j m -matrix) and the matrix of S ,m is equal to J λj m (the Jordan j m × j m -block with the eigenvalue λ );2. [the Jordan block j ∞ ( j m )]: dim V m = dim W m = j m and in an appropriate bases of V m and W m thematrix of S ,m is equal to J j m and the matrix of S ,m is equal to I j m ; . [the Kronecker block k + ( k m )]: dim V m = k m , dim W m = k m + 1 and in an appropriate bases of V m , W m the matrices of S ,m , S ,m are equal to . . .
00 1 . . .
00 0 . . . . . . . . .
10 0 . . . , . . .
01 0 . . .
00 1 . . . . . . . . .
00 0 . . . , respectively ( ( k m + 1) × k m -matrices);4. [the Kronecker block k − ( k m )]: dim V m = k m + 1 , dim W m = k m and in an appropriate bases of V m , W m the matrices of S ,m , S ,m are equal to . . . . . . . . . . . . , . . . . . . . . . . . . , respectively ( k m × ( k m + 1) -matrices). The decomposition from the theorem above will be called the
Jordan–Kronecker (J–Kfor short) decomposition of the pair S , S . We will call the Kronecker blocks k + ( k m ) ( k − ( k m )) increasing (respectively decreasing ). Consider the pencil of operators S • = { S λ } , S λ := λ S + λ S , λ := ( λ , λ ),generated by the operators S , S : V → W . The set E S • := { λ ∈ C | rank S λ < max µ rank S µ } will becalled exceptional for S • .It is clear from the theorem above that the exceptional set E S • is either { } (Kronecker case: the Jordanblocks are absent) or a finite union of lines in C . In this section we consider vector spaces defined over a field K equal to R or C . We study elementaryproperties of PNOs. Let W be a vector space, V ⊂ W its subspace, and S : V → W a linear operator.We say that a pair ( V, S ) is a partial operator on W . The subspace V is called the domain of S .Recall that algebraic PNOs were introduced in Definition 1.5. If ( h , N ) is a PNO on g , then:1. N h is a Lie subalgebra in g ; . ( h , N λ ) , N λ := λ I + λ N , is a partial Nijenhuis operator on g for any λ := ( λ , λ ) ∈ K , here I : h → g is the natural embedding;3. N λ h is a Lie subalgebra in g for any λ ;4. [ , ] N λ is a Lie algebra structure on h and N λ : h → g is a homomorphism between Lie algebras ( h , [ , ] N λ ) and ( g , [ , ]) ;5. the Lie bracket [ , ] N is compatible with the Lie bracket [ , ] (see Lemma 1.2 for the definition). Indeed, Item 1 is obvious. Item 2 is due to the equality [ , ] λ I + λ N = λ [ , ] + λ [ , ] N and to the equality T λ I + λ N = λ T N . Item 3 follows from Items 1 and 2.Now Items 4 and 5 follow easily from the equality [ x, y ] λ I + λ N = ( λ I + λ N ) − [( λ I + λ N ) x, ( I + λ N ) y ], which makes sense for ( λ , λ ) E S • (see Definition 2.3), where S • is the pencil of operatorsgenerated by I, N . (cid:3) In the following lemma we give some sufficient conditions for a partial operator ( h , N ) on g to be aPNO. Let g be a Lie algebra and h ⊂ g be a Lie subalgebra. Let N : h → g be an operator suchthat N h is also a Lie subalgebra. Then, if there exist ( a k , b k ) , k = 1 , . . . , K , not proportional to (1 , and to (0 , such that h k := ( a k I + b k N ) h is a Lie subalgebra and T Kk =1 h k = { } , the pair ( h , N ) is aPNO. For such ( a k , b k ), a k = 0; put ρ k = b k /a k . By the assumption, for any x, y ∈ h there exists s = s ( x, y ) ∈ h such that [ N x, N y ] =
N s ( x, y ). Thus[ x + ρ k N x, y + ρ k N y ] = [ x, y ] + ρ k ([ N x, y ] + [ x, N y ]) + ρ k [ N x, N y ]= ( I + ρ k N )[ x, y ] + ρ k [ x, y ] N + ρ k N s ( x, y )= ( I + ρ k N )([ x, y ] + ρ k s ( x, y )) + ρ k ([ x, y ] N − s ( x, y )) . Therefore [ x, y ] N − s ( x, y ) ∈ h k for any k (since h k is a subalgebra); hence [ x, y ] N − s ( x, y ) ∈ T Kk =1 h k = { } and [ x, y ] N = s ( x, y ) ∈ h .Now T N ( x, y ) of Definition 1.1 is correctly defined and T N ( x, y ) = [ N x, N y ] − N [ x, y ] N = N s ( x, y ) − N s ( x, y ) = 0. (cid:3) The idea of this lemma and its proof is borrowed from [BD06, Theorem 4.1].
Note that the assumption of existence of ( a k , b k ), k = 1 , . . . , K , such that h k are subalge-bras and T Kk =1 h k = { } is a sufficient but not necessary condition for the Nijenhuis property of N . Say,if N is a “usual” (i.e., h = g ) nondiagonalizable Nijenhuis operator, then this condition is not satisfied.Below we study for which cases the condition mentioned is also necessary (see Remark 3.7) and giveanother necessary and sufficient conditions for the Nijenhuis property of N in terms of the “affinization” g [ α ] of g . For instance, the operator of left multiplication by a nilpotent matrix on g = gl ( n ). .6. Lemma-Definition Let ( V, N ) be a partial operator on a finite-dimensional vector space W over C and let I : V → W be the natural embedding. Consider the pencil { N λ } , N λ := λ I + λ N , λ := ( λ , λ ) ,generated by the operators I, N . Then1. The subspace V J := T λ ∈ C \ E N • im N λ lies in V and is invariant w.r.t. N (the operator N J := N | V J will be called the Jordan part of ( V, N ) ).2. the intersection T λ ∈ C \{ (0 , } im N λ ⊂ V J , is equal to the zero subspace if and only if the Jordanpart N J is diagonalizable. Since I is injective, there are no decreasing Kronecker blocks in the corresponding J–K decomposition(see Theorem 2.1). The rest of the proof is an easy consequence of the structure of this decomposition. (cid:3) Now we see that the sufficient condition of “existence of ( a k , b k ), k = 1 , . . . , K , such that h k are subalgebras and T Kk =1 h k = { } ” from Lemma 3.3 is necessary for the Nijenhuis property of thepartial operator ( h , N ) on g if and only if the Jordan part N J is diagonalizable. Let g be a Lie algebra and h ⊂ g be a Lie subalgebra. We write g [ α ] for the Lie algebra ofpolynomials with coefficients from g with the natural Lie bracket. Then a partial operator ( h , N ) on g isa PNO if and only if the image of the operator N ′ := ( I + αN ) | h + α h : ( h + α h ) → g [ α ] is a Lie subalgebra. Indeed, im N ′ is a Lie subalgebra if and only if for any x, y ∈ h there exists u = u + αu ∈ h + α h such that [ x + αN x, y + αN y ] = u + αN u . The left hand side of this equality is equal to[ x, y ] + α ([ N x, y ] + [ y, N x ]) + α [ N x, N y ]. Comparing the coefficients of different powers of α in theequality above we conclude that im N ′ is a Lie subalgebra if and only if u = [ x, y ], N u = [ N x, N y ] and u + N u = [ N x, y ] + [ y, N x ]. The last three equalities are equivalent to conditions (i), (ii) of Definition1.5. (cid:3)
We conclude this section by studying relations between partial Nijenhuis operators and Nijenhuisoperators.
Let g be a Lie algebra, h ⊂ g a Lie subalgebra, and N : g → g a Nijenhuis operator (seeDefinition 1.1). Assume that for some λ ∈ K the following two conditions hold: (1) k := ( N + λ Id g ) h isa Lie subalgebra; (2) ( N + λ Id g ) − ( k ) = h (for instance, this condition holds if − λ is not an eigenvalueof N ).Then ( h , N | h ) is a partial Nijenhuis operator on g . Put N ′ := N + λ Id g . Due to the condition T N ′ = T N ≡
0, for any x, y ∈ h we have N ′ [ x, y ] N ′ = [ N ′ x, N ′ y ],the last expression being an element of k by assumption (1). Hence, [ x, y ] N ′ = [ x, y ] N + λ [ x, y ] ∈ h byassumption (2) and also [ x, y ] N ∈ h . On the other hand, obviously T N ≡ ⇒ T N | h ≡ (cid:3) A natural question occurs: is it true that any partial Nijenhuis operator ( h , N ) on g with h g canbe extended to a Nijenhuis operator on g ? We will come back to this question in Section 5.9 Lie algebroids and linear Poisson structures
In this section we recall some notions related to Lie algebroids and linear Poisson structures, which willbe used for defining the geometric version of PNOs and establishing their connections with bihamiltonianstructures. A Lie algebroid is a vector bundle E → M endowed with a bundle morphism (called anchor ) ρ : E → T M and a Lie algebra structure [ , ] E on the space of sections Γ( E ) satisfying(i) The induced mapping ρ : Γ( E ) → Γ( T M ) is a Lie algebra homomorphism (the space of vector fieldsΓ(
T M ) is endowed with the standard bracket; we use the same letter for the morphism of bundlesand the morphism of spaces of sections).(ii) [ x, f y ] E = f [ x, y ] E + ( ρ ( x ) f ) y for any x, y ∈ Γ( E ), f ∈ F un ( M ) (here F un ( M ) denotes the spaceof functions on M in the corresponding category). If M = {∗} , then ρ is trivial, F un ( M ) = K (the corresponding ground field), Γ( E ) = E = g is a Lie algebra. Let E = T M , [ , ] E be the commutator of vector fields, ρ = Id. We say that E is the tangent Lie algebroid on M . Let F be a foliation on M . Put E = T F (the space of elements of T M tangent to F ), ρ = I : E → T M for the natural inclusion, [ , ] E for the commutator of vector fields tangent to F . Wewill call this Lie algebroid structure canonical .Given a Lie algebroid ( E, ρ, [ , ] E ), one can build a Poisson structure on E ∗ which will be linear infibers, i.e., the Poisson bracket { , } of two sections of E interpreted as (fiberwise) linear functions on E ∗ will be a linear function on E ∗ (see [dSW99]). If x , . . . , x n are local coordinates on M and e , . . . , e r local basis of sections of E and the corresponding structure functions are defined by ρ ( e i ) = b ij ∂∂x j , [ e k , e l ] E = c mkl e m , then the linear Poisson bracket on E ∗ is defined as { x i , x j } = 0 , { ξ k , ξ l } = c mkl ξ m , { ξ i , x j } = − b ij . (4.1)Globaly, we have the following properties [Mar]:(1) { X, Y } = [ X, Y ] E (here X stands for the linear function on E ∗ corresponding to X ∈ Γ( E ));(2) { X, q ∗ f } = q ∗ ( ρ ( X ) f ) (here q denotes the projection E ∗ → M );(3) { q ∗ f, q ∗ g } = 0. 10olrmulas (4.1) show that these properties completely characterize the Poisson bracket; in other words,the Poisson bracket is completely characterized by its values on linear and base functions.One can show that in fact the notions of a Lie algebroid on E and of a linear Poisson structure on E ∗ are equaivalent, i.e. they uniquely determine each other.In the context of Examples 4.2–4.4 the corresponding linear Poisson structure on E ∗ is, respectively:1. the Lie–Poisson structure on g ∗ ;2. the canonical nondegenerate Poisson structure η T ∗ M on T ∗ M ;3. the canonical Poisson structure η T ∗ F on T ∗ F (which is degenerate if dimension of leaves of F isstrictly less than dimension of M ); recall that T ∗ F is fibered into symplectic manifolds T ∗ L , where L runs over leaves of F . Note that the Poisson structure η T ∗ F is completely determined by the anchor I : T F →
T M and the canonical Poisson structure η T ∗ M ; more precisely, η T ∗ F = I t ∗ η T ∗ M , where I t : T ∗ M → T ∗ F is the transposed map to I understood as a smooth surjective submersion. Indeed, first notice that forany X ∈ Γ( T F ) we have the following equality of linear functions on T ∗ M : IX = ( I t ) ∗ X , where ( I t ) ∗ stands for the pullback. Denote the Poisson brackets corresponding to η T ∗ F and η T ∗ M by { , } ′ and { , } correspondingly and write σ : T ∗ F → M and π : T ∗ M → M for the canonical projections. Then for any X, Y ∈ Γ( T F ) and any functions f, g on M we have( I t ) ∗ { X, Y } ′ = ( I t ) ∗ [ X, Y ] = I [ X, Y ] = { IX, IY } = { ( I t ) ∗ X, ( I t ) ∗ Y } ( I t ) ∗ { X, σ ∗ f } ′ = ( I t ) ∗ σ ∗ ( IXf ) = π ∗ ( IXf ) = { IX, π ∗ f } = { ( I t ) ∗ X, ( I t ) ∗ σ ∗ f } ( I t ) ∗ { σ ∗ f, σ ∗ g } ′ = 0 = { π ∗ f, π ∗ g } = { ( I t ) ∗ σ ∗ f, ( I t ) ∗ σ ∗ g } , which proves the claim (cf. properties (1)–(3) above). Let E → M be a vector bundle with two Lie algebroid structures ([ , ] , ρ ) and([ , ] , ρ ). They are called compatible if ( λ [ , ] + λ [ , ] , λ ρ + λ ρ ) is a Lie algebroid structure for anyconstants λ , λ . Given two compatible Lie algebroid structures ([ , ] , ρ ) and ([ , ] , ρ ) on E , the family { ( λ [ , ] + λ [ , ] , λ ρ + λ ρ ) } is a pencil of Lie algebroid structures on E . Let E → M be a vector bundle with two compatible Lie algebroid structures ([ , ] , ρ ) and ([ , ] , ρ ) . Then the corresponding linear Poisson structures on the total space of E ∗ are also compatible. The proof easily follows from the definition of compatible algebroids and properties (1)–(3) which com-pletely characterize the linear Poisson structure. (cid:3)
One can also proceed in the other direction:
Let E = T ∗ M . Assume S i : E → T M , i = 1 , M . Put [ x, y ] i := L S i x y − L S i y x + d h S i x, y i , x, y ∈ Γ( T ∗ M ), i = 1 ,
2, for the corresponding Lie algebrastructures on Γ( T ∗ M ) [KSM90]. Then ([ , ] , S ) , ([ , ] , S ) are compatible Lie algebroid structures on T ∗ M .However, note that these two constructions are not inverse to each other. Starting with Lie algebroidstructures on E , one gets Poisson structures on the total space E of E ∗ . The second construction wouldgive Lie algebroid structures on T ∗ E . 11 Partial Nijenhuis operators (geometric version)
Let E = T F for some foliation F on M . We say that a pair ( E, N ), where N : E → T M is a bundle morphism, is a (geometric) partial Nijenhuis operator (PNO for short) on M if thefollowing two conditions hold:(i) [ x, y ] N := [ N x, y ]+[ x, N y ] − N [ x, y ] ∈ Γ( E ) for any x, y ∈ Γ( E ) (here [ , ] stands for the commutatorof vector fields on M );(ii) T N ( x, y ) := [ N x, N y ] − N [ x, y ] N = 0 for any x, y ∈ Γ( E ) (it follows from condition (i) that thesecond term is correctly defined).In other words, a bundle morphism N : E → T M is a geometric PNO if it is an algebraic PNO regardedas a map of Lie algebras Γ( E ) → Γ( T M ) (which will be denoted by the same letter).
This notion is very natural and probably existed in the literature earlier with no specialname. A similar notion appeared in [CGM04] under the name ”outer Nijenhuis tensor”.F. J. Turiel used equivalent notion in [Tur10, Tur11a, Tur11b] in different terms. Namely, he consid-ered a foliation F on a manifold M and a morphism N : T F →
T M such that(1) N ∗ α is closed along the leaves of F for any closed 1-form α satisfying ker α ⊃ T F .Then he proved that, given any extension N of N to a morphism from T M to T M , the restriction of T N to T F does not depend on the extension. So one can require that(2) T N | T F× T F = 0.We claim that in fact the two notions are equivalent, i.e. the following equivalences hold: ( i ) ⇐⇒ (1),and, under the assumption that ( i ) or (1) is satisfied, ( ii ) ⇐⇒ (2). Indeed, assume that condition(1) is satisfied. If α is a 1-form such that dα = 0 , α | T F = 0, then for any vector fields x, y we have α ([ x, y ]) = xα ( y ) − yα ( x ) and for X, Y ∈ Γ( T F ) we have ( N ∗ α )([ X, Y ]) = X ( N ∗ α )( Y ) − Y ( N ∗ α )( X ),i.e. α ( N [ X, Y ]) = Xα ( N Y ) − Y α ( N X ). Thus for any such 1-form we have α ([ X, Y ] N ) = α ([ N X, Y ] + [
X, N Y ] − N [ X, Y ]) =
N Xα ( Y ) − Y α ( N X ) + Xα ( N Y ) − N Y α ( X ) − Xα ( N Y ) +
Y α ( N X ) = 0 . This implies [
X, Y ] N ∈ Γ( T F ), hence condition ( i ). These considerations are reversible and ( i ) ⇐⇒ (1).Now if one of these equivalent conditions hold, T N ( X, Y ) coincides with the expression T N ( X, Y ) fromcondition ( ii ), is independent of the prolongation N , and, obviously, ( ii ) ⇐⇒ (2).Recall that the bundle E = T F has the canonical Lie algebroid structure with the canonical inclusion I : E → T M as the anchor and the commutator of vector fields tangent to F as the Lie bracket on Γ( E ). Let ( E, N ) , N : E → T M , be a PNO on M . Then:1. N Γ( E ) is a Lie subalgebra in Γ( T M ) ;2. N λ := λ I + λ N is partial Nijenhuis for any λ := ( λ , λ ) ; . N λ Γ( E ) is a Lie subalgebra in Γ( T M ) for any λ ; in particular if rank of the distribution N λ E isconstant, it is tangent to some foliation F λ ;4. [ , ] N is a Lie algebra structure on Γ( E ) which together with the anchor N : E → T M form a Liealgebroid structure on E ;5. this new Lie algebroid structure on E is compatible with the canonical Lie algebroid structure on E , i.e. the family { ([ , ] N λ , N λ ) } is a pencil of Lie algebroid structures on E (see Definition 4.6). Items 1, 2, 3 are proven as in the algebraic case (Lemma 3.2). Let us prove that ( E, [ , ] N λ , N λ ) is a Liealgebroid for any λ . The fact that [ , ] N λ is a Lie algebra and that N λ is a homomorphism of Lie algebrasis also proven as in algebraic case. It remains to check the condition of compatibility of the bracket withthe anchor; by linearity it is enough to prove it with N instead of λ I + λ N :[ x, f y ] N = [ N x, f y ] + [ x, N f y ] − N [ x, f y ]= f [ N x, y ] + ((
N x ) f ) y + [ x, f N y ] − N ( f [ x, y ] + ( xf ) y )= f [ N x, y ] + ((
N x ) f ) y + f [ x, N y ] + ( xf ) N y − N ( f [ x, y ] + ( xf ) y )= f [ x, y ] N + (( N x ) f ) y ;note that we used only the linearity of N . (cid:3) The proofs of the following two lemmas follow from the corresponding lemmas in the algebraic case(see Lemmas 3.3 and 3.9).
Let F be a foliation on M . Let N : T F →
T M be a vector bundle morphism such that
N T F is the tangent bundle to some foliation. Then, if there exist ( λ ( k )1 , λ ( k )2 ) , k = 1 , . . . , K , linearlyindependent with (1 , and with (0 , such that ( λ ( k )1 I + λ ( k )2 N ) T F = T F ( k ) for some foliation F ( k ) and T Kk =1 T x F ( k ) = { } for any x ∈ M , the pair ( T F , N ) is a PNO. Let F be a foliation on M . Let N : T M → T M be a Nijenhuis (1,1)-tensor such that forsome λ ∈ K the following two conditions hold: (1) the distribution B := ( N + λ Id T M ) T F is tangent tosome foliation; (2) ( N + λ Id T M ) − ( B ) = T F . Then the pair ( T F , N | T F ) is a PNO. Let N is a “usual” Nijenhuis operator ((1,1)-tensor). Then N : E → T M be a PNOwith E = T M .Now we provide a simplest nontrivial example of a partial Nijenhuis operator.
Let M be any manifold and let v, w ∈ Γ( T M ) be linearly independent (at each point)vector fields. Put E := h v i , N : E → T M , N v := w . Since E is a vector bundle with one-dimensionalfibers, the integrability condition on N E is trivial, and it is easy to check that the operator N is partialNijenhuis.Assume that v, w are generic. It is clear that there is no coordinate system in which N is translation-invariant. For example, if dim M >
2, then E and N E are not simultaneously tangent to any 2-dimensional foliation.The 1-parameter family of foliations of rank 1 appearing in this example via Lemma 5.3 is an exampleof the so-called Kronecker web. In more details this notion is considered in the next section.13
Veronese and Kronecker webs and PNOs
Recall the definition of a Veronese web [GZ91].
Let {F s } s ∈ RP be a collection of foliations of rank n on a manifold M n +1 of dimension n + 1 such that in a neighbourhood of any point there exists a local coframe α , . . . , α n with T F s = h α + sα + · · · + s n α n i ⊥ (here h·i ⊥ stands for the annihilator of the span h·i ) for any s ∈ RP = R ∪ {∞} (by definition T F ∞ := h α n i ⊥ ). Thus the map RP ∋ t
7→ h ( α + sα + · · · + s n α n ) | x i ∈ P T ∗ x M parametrizesa Veronese curve for any x ∈ M . The whole collection {F s } s ∈ RP is a Veronese web .It turns outs that there exists a 1-1-correspondence between Veronese webs and special PNOs. Letus say that a PNO ( T F , N ) on a manifold M n +1 is of of generic type if the pair of operators N, I : T F →
T M , where I : T F ֒ → T M is the canonical inclusion, has a unique Kronecker block in the J–Kdecomposition (see Section 2), i.e. there exist local frames v , . . . , v n ∈ Γ( T F ), w , . . . , w n ∈ Γ( T M ), inwhich I =
10 1. . . . . .0 10 , N =
01 0. . . . . .1 01 . (6.1) There exists a 1-1-correspondence between Veronese webs {F s } on M n +1 and PNOs ( T F , N ) of generic type such that F ∞ = F and T F = N T F . Let {F s } s ∈ RP be a Veronese web on M n +1 . It turns out that {F s } is determined by the foliation F ∞ and an (everywhere defined) Nijenhuis operator which is built as follows [BD06, Tur99b, Tur89]. Fix s , . . . , s n ∈ R to be pairwise distinct nonzero numbers; for i = 0 , . . . , n define a rank-1 foliation S i by T x S i := T nj =0 ,j = i T x F t j , x ∈ M . Then T x S i + T x S k is an integrable distribution for any i, k , hence putting N | T x S i := s i Id T x S i we will get a Nijenhuis operator.It is easy to see that T x F s i = ( N − s i I ) T x F ∞ , i = 0 , . . . , n , where I := Id T M (indeed ker( N − s i I ) = T S i is transversal to T F ∞ and im( N − s i I ) = P j = i T S j = T F s i ). On the other hand, one can see that themap RP ∋ s (( N − sI ) T x F ∞ ) ⊥ ∈ P T ∗ x M is a Veronese curve (a priori different from the initialone). These two curves pass through n + 2 distinct points of P T ∗ x M : n + 1 mentioned above and ∞ (since T x F ∞ = lim s →∞ ( N − sI ) T x F ∞ ). We conclude by the uniqueness property of the Veronese curve(Lagrange interpolation theorem) that they coincide. Hence T x F s = ( N − sI ) T x F ∞ for any s ∈ RP and x ∈ M .By Lemma 5.5 (put λ = 0) this gives us a partial Nijenhuis operator N = N | T F ∞ : T F ∞ → T M .Alternatively one can use Lemma 5.4 since T ni =0 T x F s i = { } .The constructed PNO ( F ∞ , N ) is independent of the choice of the numbers s i . Indeed, let ( T F s ) ⊥ = h α + sα + · · · + s n α n i =: h α s i and let X , . . . , X n be the frame dual to the coframe α , . . . , α n . Thenthe partial operator N : T F ∞ = h X , . . . , X n − i → T M satisfying α s (( N − sI ) T F ∞ ) = 0 for any s (now I : T F ∞ ֒ → T M is the canonical inclusion) is uniquely determined by
N X k = X k +1 , 0 ≤ k < n . Notealso that the pair ( N, I ) has canonical matrix form (6.1) in the frames X , . . . , X n − and X , . . . , X n .Vice versa, let ( T F , N ) be a PNO of generic type on M . Then it is easy to see that ( N − sI ) T F = h α + sα + · · · + s n α n i ⊥ , where α , . . . , α n is the coframe dual to w , . . . , w n ∈ Γ( T M ) (see (6.1)). Theintegrability of the distribution ( N − sI ) T F follows from Lemma 5.3(3). (cid:3) .3. Remark The proof above shows that for any PNO of generic type ( T F , N ) on a manifold M thereexists a Nijenhuis operator N : T M → T M such that N = N | T F . It turns out that such a Nijenhuisoperator is not unique. The related problem of realization of PNOs of generic type is considered in Section10 (which in turn is related to the problem of bisymplectic realizations of bihamiltonian structures, seeSection 9).Veronese webs are particular cases of a more general notion of a Kronecker web [Zak01]. Noticethat F. J. Turiel (and initially the author [Pan00]) uses the term Veronese web for both the notions[Tur00],[Tur10]. [Zak01] Let {F s } s ∈ RP be a collection of foliations on a manifold M . Assume thatthere is a vector bundle Φ → M and two bundle morphisms φ i : T ∗ M → Φ, i = 1 ,
2, such that forany s , s ∈ R , ( s , s ) = 0 we have ker φ ( s ,s ) = ( T F s : s ) ⊥ , here φ ( s ,s ) := s φ + s φ . We say that {F s } s ∈ RP is a Kronecker web if for any ( s , s ) ∈ C \ { (0 , } the morphism s φ + s φ : ( T ∗ M ) ⊗ C → Φ ⊗ C is fiberwise surjective, or in other words, dim ker( s φ + s φ ) does not depend on ( s , s ) ∈ C \ { (0 , } for any fixed point of M . Equivalently, the J–K decomposition of the pair of operators φ ,x , φ ,x : T ∗ x M → Φ x , x ∈ M , does not contain Jordan blocks (this explains the name “Kroneckerweb”).It turns out that the dualization of this definition gives an example of a PNO. Indeed, given aKronecker web {F s } s ∈ RP , consider the pencil of the transposed morphisms φ t ( s ,s ) : Φ ∗ → T M (whichare fiberwise injective for any s , ). Note that, im φ t ( s ,s ) = (ker φ ( s ,s ) ) ⊥ = T F s : s , in particularim φ t = T F ∞ . Hence φ t identifies Φ ∗ with T F ∞ . Consider the map ( φ t ) − : T F ∞ → Φ ∗ and the map N := φ t ◦ ( φ t ) − : T F ∞ → T M .We claim that ( T F ∞ , N ) is a PNO. Indeed, ( s I + s N ) T F ∞ = im φ t ( s ,s ) = T F s : s for any s : s ∈ RP , where I is the canonical embedding T F ∞ ֒ → T M . Moreover, one can find a finite number (whichdepends on the structure of Kronecker blocks in the J–K decomposition) of points in RP such that theintersection of the corresponding foliations is trivial. By Lemma 5.4 we conclude that ( T F ∞ , N ) is aPNO. One can immediately see that a Kronecker web is the same as a PNO N such that themorphism N λ is injective at any point of the base manifold and for any λ = 0 (provided one can takecomplex λ ). We will call such a PNO Kronecker , since for such N the pair of morphisms ( N, I ) containsonly (increasing) Kronecker blocks in the Jordan–Kronecker decomposition at any point. Veronese websare distinguished by the case of a sole Kronecker block (PNOs of generic type).
The proof of Theorem 6.2 suggests a question: is it true that any Kronecker PNO is arestriction to the tangent bundle of some foliation of some “usual” Nijenhuis operator on M as it is forthe particular case of Veronese webs, see Remark 6.3. The answer to this question is positive [Tur10,Theorem 2.1] (see also Sections 9–10 for the discussion of the realization problem). In the context of Veronese webs the following theorem is true [Pan02, BD06]. Let α , . . . , α n be a local coframe on R n +1 and let D s := h α s i ⊥ , where α s := α + sα + · · · + s n α n . Assumethat the distribution of hyperplanes D s ⊂ T R n +1 is integrable for n + 3 different values of s ∈ RP . Then D s is integrable for any s , i.e., induces a Veronese web.15ote that this statement is surprising starting from n = 3 since the condition of integrability dα s ∧ α s =0 is polynomial in s of degree 2 n , thus one would expect that a sufficient condition would be vanishingof the polynomial at 2 n + 1 different points.In [Pan02, BD06] also a generalization of this theorem was proven, considering Kronecker webs withKronecker blocks of equal dimension.The construction of PNO related to Kronecker webs and Lemma 5.4 allow to prove an analogue of this theorem for the most general Kronecker webs without any restrictions on the dimensions of theKronecker blocks (another proof of such a theorem is obtained by F. J. Turiel [Tur10, Corollary 2.1.2]). Combining the construction of a linear Poisson structure from a Lie algebroid described in Section 4 withLemmas 5.3(4-5) and 4.7 one obtains, given a PNO ( T F , N ) on a manifold M , a canonically definedpencil of (linear) Poisson structures on the total space of T ∗ F . We will say that this bihamiltonianstructure is obtained by means of “up construction” from a PNO ( T F , N ).Let us consider this bihamiltonian structure in detail. One of the linear Poisson structures from thispencil, η T ∗ F , corresponds to the canonical Lie algebroid structure on T F with the anchor I : T F →
T M (the canonical inclusion). We know (see Remark 4.5) that η T ∗ F = ( I t ) ∗ η T ∗ M , where η T ∗ M is the canonicalPoisson structure on T ∗ M . Analogous statement is true for the second generator of this pencil. Consider the transposed map N t : T ∗ M → T ∗ F as a smooth map. Then for the canonicallinear Poisson structure η N related to the Lie algebroid T ∗ F with the Lie algebra structure [ , ] N and theanchor N the following equality holds: η N = N t ∗ η T ∗ M . To prove this claim we shall proceed as in Remark 4.5. First notice that for any X ∈ Γ( T F ) we havethe following equality of linear functions on T ∗ M : N X = ( N t ) ∗ X , where ( N t ) ∗ stands for the pullback.Now the following calculations, which use this equality and the definition of the algebroid ( T ∗ F , [ , ] N , N ),prove the claim (in view of properties (1)–(3) of the linear bracket which determine it, see Section 4):( N t ) ∗ { X, Y } ′ = ( N t ) ∗ [ X, Y ] N = N [ X, Y ] N = [ N X, N Y ] = { N X, N Y } = { ( N t ) ∗ X, ( N t ) ∗ Y } ( N t ) ∗ { X, σ ∗ f } ′ = ( N t ) ∗ σ ∗ ( N Xf ) = π ∗ ( N Xf ) = { N X, π ∗ f } = { ( N t ) ∗ X, ( N t ) ∗ σ ∗ f } ( N t ) ∗ { σ ∗ f, σ ∗ g } ′ = 0 = { π ∗ f, π ∗ g } = { ( N t ) ∗ σ ∗ f, ( N t ) ∗ σ ∗ g } ;here { , } ′ and { , } are the Poisson brackets corresponding to η N and η T ∗ M correspondingly and σ : T ∗ F → M and π : T ∗ M → M are the canonical projections. (cid:3) Summarizing, the canonical bihamiltonian structure on T ∗ F related to a PNO ( T F , N ) is generatedby the linear Poisson structures η := ( I t ) ∗ η T ∗ M and η := ( N t ) ∗ η T ∗ M . Note that the fibers of thecanonical projection T ∗ F → M are lagrangian submanifolds in any symplectic leaf of any of these twoPoisson structures, i.e. the fibers form a bilagrangian foliation .Below we consider two particular cases of the “up construction”. With n + 3 values, where n + 1 is the dimension of the target space of the highest Kronecker block. .2. Example Let {F s } s ∈ RP be a Kronecker web on a manifold M and φ i : T ∗ M → Φ be the corre-sponding bundle morphisms (see Definition 6.4). In the particular case of the Kronecker PNO ( T F ∞ , N ), N = φ t ◦ ( φ t ) − , related to a Kronecker web the “up construction” gives a bihamiltonian structure η , : T ∗ M ′ → T M ′ , M ′ := T ∗ F ∞ . We can say more about this bihamiltonian structure in comparisonwith the general case.First of all, since N is fiberwise injective, N t : T ∗ M → T ∗ F ∞ is a smooth surjective submersion andby Lemma 7.1 we can define η = η N as N t ∗ η T ∗ M .Second, let x , . . . , x n be a local coordinate system on M such that ∂∂x , . . . , ∂∂x k are the basic vectorfields tangent to F ∞ and let ξ , . . . , ξ k be the corresponding linear functions on T ∗ F ∞ . Then by formulas(4.1) the symplectic foliation F s of the linear Poisson structure corresponding to the Lie algebroid( T F ∞ , N − sI ) (here I is the canonical embedding T F ∞ ֒ → T M ) is generated by the vector fields ∂∂ξ , . . . , ∂∂ξ k and ( N − sI ) ∂∂σ ∗ x , . . . , ( N − sI ) ∂∂σ ∗ x k (here σ : T ∗ F ∞ → M is the canonical projection,i.e σ ∗ x i is a base function on T ∗ F ∞ ). Due to the kroneckerity of N the rank of the distribution D s generated by these vector fields is constant even if we admit s ∈ C , which means that the correspondingbihamiltonian structure η , , is Kronecker itself, i.e. for any p ∈ M ′ the J–K decomposition of the pair ofoperators η ,p , η ,p : T ∗ p M ′ → T p M ′ does not contain Jordan blocks. Moreover, we observe the followingobvious facts: (1) T s ( D s ) p coincides with the fiber of σ passing through p ∈ M ′ , i.e. the canonicalbilagrangian foliation W of the Kronecker bihamiltonian structure η , (see Section 8) coincides withthe foliation of fibers of σ ; (2) the base of this foliation is correctly defined and coincides with M ; (3) theprojection of the symplectic foliation F s with respect to σ coincides with the initial foliation F s fromthe web for any s . Let N : E → T M be a PNO with the domain E = T M , i.e., N is a “usual” Nijenhuisoperator. Then the “up construction” gives a bihamiltonian structure η := η T ∗ M , η = η N on themanifold M ′ := T ∗ M , where η T ∗ M is the canonical Poisson structure on M ′ = T ∗ M .The (1,1)-tensor N ′ : T M ′ → T M ′ uniquely defined by N ′ = η N ◦ η − T ∗ M has zero Nijenhuis torsion dueto the compatibility of η N and η T ∗ M . In case, when N is fiberwise invertible, the Poisson structure η N is nondegenerate and ( N ′ ) − = η T ∗ M ◦ η − N coincides with the so-called cotangent lift of the operator N defined as η T ∗ M ◦ ( N t ) ∗ η − T ∗ M (see [Tur92]); here the transposed operator N t : T ∗ M → T ∗ M is regardedas a smooth map of M ′ and η − T ∗ M is the canonical symplectic form. We know from Lemma 7.1 thatthe following equality holds η N := N t ∗ η T ∗ M , which in the case of fiberwise invertible N can serve as thedefinition of the linear Poisson structure η N . The bihamiltonian structure η T ∗ M , η N on T ∗ M from Example 7.3 will be called the bisymplectic or Jordan bihamiltonian structure of type N .The last terminology is motivated by the fact that there are only Jordan blocks in the J–K decompositionof the pair of operators η T ∗ M | p , η N | p : T ∗ p M ′ → T p M ′ for any p ∈ M ′ . F. J. Turiel [Tur92] proved that under some additional assumption of regularity (whichis satisfied for generic cases) any Jordan bihamiltonian structure is locally equivalent to a bihmiltonianstructure of type N . In the next section we shall also see that any Kronecker bihamiltonian structure islocally equivalent to the one built in Example 7.2. Thus the examples above show that the notion of aPNO is a proper geometric framework for simultaneous treatment of Jordan and Kronecker bihamiltonianstructures. 17 .6. Lemma Let ( T F , N ) be a PNO on a manifold M . Assume there exists a Nijenhuis operator N : T M → T M such that N = N | T F . Write I : T F →
T M for the canonical inclusion. Let I t : T ∗ M → T ∗ F be the transposed operator regarded as a smooth surjective submersion. Then I t ∗ η T ∗ M = η T ∗ F and I t ∗ η N = η N . The first equality was already discussed (see Remark 4.5). The second equality follows from the commu-tativity of the following diagram T F I / / N ❋❋❋❋❋❋❋❋ T M N (cid:15) (cid:15) T M (which implies N t = I t ◦ N t and in view of Lemma 7.1 η N = ( N t ) ∗ η T ∗ M = I t ∗ ◦ ( N t ) ∗ η T ∗ M = I t ∗ η N ). (cid:3) There are two constructions relating Kronecker webs with bihamiltonian structures, which are mutuallyinverse in the sense that will be explained below (see [GZ91], [Pan00], [Tur00]).Let η , : T ∗ M → T M be a
Kronecker bihamiltonian structure , i.e., a bihamiltonian structure suchthat for any x ∈ M the J–K decomposition of the pair of operators η ,x , η ,x : T ∗ x M → T x M does notcontain Jordan blocks. The rank of the Poisson bivector λ η + λ η does not depend on x and λ , (when( λ , λ ) = 0); denote by F λ , λ = λ : λ , the corresponding symplectic foliation. Then {F λ } λ ∈ P is afamily of foliations of constant rank; as linear algebra shows, they contain a unique common subfoliation W such that T x W = T λ ∈ P T x F λ for any x ∈ M . Such a foliation is lagrangian in any symplectic leafof any of two Poisson structures and is called the bilagrangian foliation of the Kronecker bihamiltonianstructure. Reduce attention to a sufficiently small open subset U ⊂ M on which the foliation W has alocal base B .Finally, it turns out that B carries a rich geometric structure of a Kronecker web: a collection offoliations in general position F λ depending on λ ∈ P such that the normal spaces N m F λ ⊂ T ∗ m M dependin a particular way on parameter λ . These foliations are the “projections” of the foliations F λ w.r.t. thereduction of U to B . As we know from Section 6 such a structure is equivalent to a geometric KroneckerPNO.Note that the operators η ,x , η ,x being skew symmetric necessarily contain both increasing and de-creasing Kronecker blocks (see Definition 2.2) in the J–K decomposition, which are mutually transposedto each other. Algebraically the construction described, which will be referred to as “down construction” ,consists in cutting off the decreasing blocks.Vice versa, let {F s } s ∈ RP be a Kronecker web on a manifold M and φ i : T ∗ M → Φ be the correspondingbundle morphisms (see Definition 6.4). Then “up construction”, which was discussed in Section 7, appliedto the Kronecker PNO ( T F ∞ , N ), N = φ t ◦ ( φ t ) − , related to the Kronecker web gives a Kroneckerbihamiltonian structure η , : T ∗ M ′ → T M ′ , M ′ := T ∗ F ∞ .From Example 7.2 we see that starting from a Kronecker web and applying first “up construction”and then “down construction” results in the initial Kronecker web.Applying these constructions other way round is more subtle. Starting from any Kronecker bi-hamiltonian structure η , we can always perform locally “down construction” and get a Kronecker web {F s } s ∈ RP . Applying to it the “up construction” results in a bihamiltonian structure η ′ , which a priori18eed not coincide with the initial one. It was the initial conjecture of Gelfand and Zakharevich (for-mulated by them in the case of generic Kronecker bihamiltonian structures [GZ91], i.e. with Kroneckerwebs which are Veronese webs) that the bihamiltonian structures η , and η ′ , are locally equivalent, i.e.there exists a local diffeomorfism bringing one structure to another.This conjecture was proved by Turiel in the particular cases listed in the following theorem (see[Tur10, Theorem 3.2] and references therein). (Turiel) A Kronecker bihamiltonian structure can be locally reconstructed from itsKronecker web obtined by means of the “down construction” in the following cases: • in complex or real analytic category; • in C ∞ category for generic Kronecker bihamiltonian structures and Kronecker bihamiltonian struc-tures with flat Kronecker webs. A Kronecker web {F s } s ∈ RP is called flat if in a vicinity of every point there exists a local diffeomorphismbringing simultaneously all the foliations F s to the foliations of parallel planes on an open set in R n . It is well known [Wei83] that, given a Poisson structure η on a manifold M , for any point of M there existsan open neighbourhood of this point U and a symplectic manifold ( U , ω ) with a surjective submersion p : U → U such that p ∗ ω − = η | U ; here ω − is the Poisson structure inverse to the symplectic form ω . Inother words, any Poisson structure has a local symplectic realization . This is a first step to the problemof existence of global symplectic realization which is very important and led in particular to the theoryof symplectic groupoids.Analogous problem can be formulated in the bihamiltonian context: given a bihamiltonian structure η , on a manifold M such that λ η + λ η is degenerate for any λ , does it have a bisymplectic realization,i.e. does there exist a manifold M with a bihamiltonian structure ω − , (such bihamiltonian structuresnecessarily are Jordan , i.e. for any x ∈ M the pair of operators ω − ,x , ω − ,x : T ∗ x M → T x M containsonly Jordan blocks in the J–K decomposition) and a surjective submersion p : M → M such that p ∗ ω − , = η , ? In this section we consider the problem of local bisymplectic realization for Kroneckerbihamiltonian structures.Note that there is a crucial difference between the two realization problems above: in the Poissoncase there is only one local model of the symplectic form ω given by the Darboux theorem while there aremany local models of bisymplectic bihamiltonian structures ω − , , i.e. Jordan bihamiltonian structures.For instance, the Jordan bihamiltonian structures of type N (see Definition 7.4), which are completelydetermined by a Nijenhuis (1,1)-tensor N , are locally inequivalent for locally inequivalent N .A quite natural and desirable feature of the symplectic and bisymplectic realization is its minimality:once dimension of M is fixed, try to find M of possibly minimal dimension. Since for a Kroneckerbihamiltonian structure η , both the bivectors have the same rank, say 2 r , and corank, say l , it is easyto see that the minimal possible dimension for M we can think about is 2 r + 2 l .Now we can make our problem more precise. 19 roblem 1 (a) Given a Kronecker bihamiltonian structure η , , rank η , = 2 r , on an open set U ⊂ M , dim M = m , do there exist a Jordan bihamiltonian structure η , on an open set U ⊂ M ,dim M = 2 m − r , and a smooth surjective submersion p : U → U such that p ∗ η , = η , ?(b) List all locally inequivalent Jordan bihamiltonian structures η , on U with the property p ∗ η , = η , .Below we set some preliminary steps for solving this problem. In view of Theorem 8.1 we can assumethat the bihamiltonian structure η , is equal to the bihamiltonian structure η T ∗ F ∞ , η N on the manifold T ∗ F ∞ , T F ∞ ⊂ T B , where B is the local base of the canonical bilagrangian foliation W of η , and N : T F ∞ → T B is the Kronecker PNO corresponding to the Kronecker web obtained on B by meansof the “down construction” (see Example 7.2). Now assume that there exists a Nijenhuis operator N : T B → T B such that N = N | T F ∞ . Then by Lemma 7.6 we have I t ∗ η T ∗ B = η T ∗ F ∞ , I t ∗ η N = η N , where I : T F ∞ → T B is the canonical inclusion and I t : T ∗ B → T ∗ F ∞ is the corresponding surjectivesubmersion.We see that Problem 1 is intimately related to the following Problem 2 (a) Given a Kronecker PNO ( T F , N ), rank F = r , on an open set V ⊂ R m − r , m > r ,does there exist a Nijenhuis operator N : T V → T V such that N | T F = N ?(b) List all locally nonequivalent Nijenhuis operators N on V satisfying N | T F = N .The considerations above show that once Problem 2(a) is solved we obtain also a solution of Problem1(a). Recall (see Remark 6.6) that Problem 2(a) has a solution for any Kronecker web, hence Problem1(a) has a solution for any Kronecker bihamiltonian structure.On the other hand, a solution of Problem 2(b), which will be called the “realization problem forKronecker webs” , would imply only a particular solution of Problem 1(b), i.e. a solution in the class ofJordan bihamiltonian structures of type N on U , where N : T V → T V is a Nijenhuis (1,1)-tensor and U is an open set in T ∗ V (cf. Remark 7.5). Solutions to the realization problem will be obtained in thenext section for particular Kronecker webs.
10 Realization problem for Veronese webs
The realization problem for Kronecker webs was formulated in the previous section (Problem 2(b)).Below we discuss this problem and we start from describing a solution to this problem for 3-dimensionalVeronese webs obtained in [KP17]. We begin with the general situation, and then specify to the 3-dimensional case. For simplicity consider only complex analytic case (which excludes the normal formof a real Nijenhuis operator with complex eigenvalues, see [KP17] for this case).Recall that one of the local models of the Nijenhuis operators N , namely a semisimple operator withsimple spectrum the elements of which are constant functions, was obtained in the proof of Theorem 6.2.To get other local models we need to introduce the following notion.20 Consider a Veronese web {F λ } λ ∈ CP on a manifold M n +1 , given by T F λ = h α λ i ⊥ ,where α λ = α + λα + · · · + λ n α n and α , α , . . . , α n is a local coframe on an open set U ⊂ M . Ananalytic function φ : U → C is called self-propelled if dφ is proportional to α φ . If the coefficient ofproportionality is nonzero, we denote this by dφ ∼ α φ . However, the coefficient is allowed to be zero, soa constant function is also considered self-propelled. Let {F λ } be a Veronese web on M n +1 . Then in a vicinity of any point x ∈ M thereexist n + 1 functionally independent self-propelled functions φ ( x ) , φ ( x ) , . . . , φ n ( x ) . If X , . . . , X n is theframe dual to the coframe α , . . . , α n defining the Veronese web, the condition on the function φ to beself-propelled is the following system PDEs: φX φ = X φ, . . . , φX n − φ = X n φ. (10.1)The required relation α + · · · + φ n α n ∼ ( X φ ) α + · · · + ( X n φ ) α n is equivalent to vanishing of thedeterminants (cid:12)(cid:12)(cid:12)(cid:12) φX φ X φ (cid:12)(cid:12)(cid:12)(cid:12) , . . . , (cid:12)(cid:12)(cid:12)(cid:12) φ n − φ n X n − φ X n φ (cid:12)(cid:12)(cid:12)(cid:12) , which is equivalent to system (10.1). Let F ( x, λ ) be a λ -parametric first integral of the folitation F λ ,where x = ( x , . . . , x n ). The following formula gives a family of implicit solutions φ ( x ) of system (10.1)depending on an arbitrary smooth function of one variable f = f ( λ ) that locally satisfies f ′ ( λ ) = F λ : F ( x, φ ( x )) = f ( φ ( x )) . (10.2)Indeed, differentiating this equality along X k − φ ( x ) X k − we get d x F ( x, λ )( X k − λX k − ) | λ = φ ( x ) + ( F λ ( x, φ ( x )) − f ′ ( φ ( x ))) · ( X k φ ( x ) − φ ( x ) X k − φ ( x )) = 0 . (10.3)The first term vanishes since X k − λX k − ∈ h α λ i ⊥ , and the claim follows.Choosing n solutions φ , . . . , φ n with initial values c , . . . , c n at x ∈ M being pairwise different andwith nonzero ψ i := X φ i | x , we compute from (10.1) the Jacobian at x :Jac x ( φ , φ , . . . , φ n ) ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ c ψ . . . c n ψ ψ c ψ . . . c n ψ ... ... . . . ... ψ n c n ψ n . . . c nn ψ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ψ ψ · · · ψ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c . . . c n c . . . c n ... ... . . . ...1 c n . . . c nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since the Vandermonde determinant with the second column consisting of pairwise different entries isnonzero, we obtain n functionally independent solutions of (10.1). (cid:3) Let ( T F , N ) be a Kronecker PNO of generic type (see Theorem 6.2) on a 3-dimensionalmanifold M . Then in a neighborhood U of every point p ∈ M there exists a Nijenhuis operator N : T M → T M of any type A, B or C listed in Appendix such that N | T F = N . Consider ( T F , N ) locally near p ∈ M . The intersection D := T F ∩
N T F is a one dimensionaldistribution. Choose arbitrarily a nonvanishing vector field X ∈ Γ( D ) and put X := N − X , X := N X . Then X , X , X is a frame such that there exist functions b , b , c , c satisfying the followingcommutation relations: 21i) [ X , X ] = b X + b X and [ X , X ] = c X + c X ;(ii) [ X , X ] = c X + ( c + b ) X + b X .Item (i) is due to the integrability of the distributions T F and N T F . To prove Item (ii) let [ X , X ] = d X + d X + d X for some functions d , d , and d and use the definition of a PNO 5.1: by condition1 of this definition we have [ X , X ] N = [ N X , X ]+ [ X , N X ] − N [ X , X ] = [ X , X ] + [ X , X ] − N [ X , X ] = [ X , X ] − N [ X , X ] = d X + d X + d X − ( b X + b X ) = d X + ( d − b ) X +( d − b ) X ∈ T F , which implies d = b ; by condition 2 of this definition we have c X + c X =[ X , X ] = [ N X , N X ] = N ([ X , X ] N ) = N ( d X + ( d − b ) X ) = d X + ( d − b ) X , which implies d = c , d = c + b .If ( X , X , X ) is a frame satisfying relations (i-ii) for some functions and ( α , α , α ) is the dualcoframe, it is easy to see that the distribution h α + λα + λ α i ⊥ ⊂ T M is integrable for any λ , i.e.defines a Veronese web {F λ } . This is of course the Veronese web corresponding to N by Theorem 6.2(see its proof).The matrix of the operator N : T F →
T M with respect to the bases ( X , X ) in T F and ( X , X , X )in T M is equal to . Define N by N | T F = N and N X = f X + f X + f X , where f i are local analytic functions, i.e.putting the matrix of N in the frame ( X , X , X ) to be equal to f f f . Direct calculations taking into account relations (i), (ii) show that T N ( X , X ) = 0, if and only if thefollowing system of nonlinear first order equations is satisfied: X f = f X f , X f = X f + f X f , X f = X f + f X f , (10.4)and, analogously, the equality T N ( X , X ) = 0 is equivalent to the system X f = f X f , X f = X f + f X f , X f = X f + f X f . (10.5)Now let f = φ φ φ , f = − φ φ − φ φ − φ φ , f = φ + φ + φ for some local functions φ , φ , φ .Then it is easy to see that once the functions φ i satisfy the system of equations (10.1), the functions f i satisfy the systems of equations (10.4), 10.5). In other words, if the functions φ , φ , φ are self-propelledfor the corresponding Veronese web, the Nijenhuis torsion T N of the (1,1)-tensor N given in the frame X , X , X by the matrix F ( φ , φ , φ ) := φ φ φ − φ φ − φ φ − φ φ φ + φ + φ (10.6)vanishes (recall that T N ( X , X ) = T N ( X , X ) = 0 by the assumptions of the theorem).Now let ψ , ψ , ψ be functionally independent self-propelled functions with pairwise distinct ψ ( p ), ψ ( p ), ψ ( p ) (they exist by Lemma 10.2) and let a , a , a be pairwise distinct constants. Put22 F A := F ( ψ , ψ , ψ ); F A := F ( ψ , ψ , a ); F A := F ( ψ , a , a ); F A := F ( a , a , a ); • F B := F ( ψ , ψ , ψ ); F B := F ( ψ , ψ , a ); F B := F ( a , a , ψ ); F B := F ( a , a , a ); • F C := F ( ψ , ψ , ψ ); F C := F ( a , a , a ).We have shown above that all these matrices represent Nijenhuis (1,1)-tensors. On the other hand,we recognize in these matrices the Frobenius forms of all the Nijenhuis (1,1)-tensors listed in Appendix.Consequently, by [Tur96] for each F X there should exist local coordinates ( x , x , x ) such that thematrix of the corresponding Nijenhuis (1,1)-tensor N in the basis { ∂∂x i } has the form N X from the listof Appendix. (cid:3) We conclude this section by a conjecture that the realization problem for a Veronese web can besimilarly solved in any dimension (in fact its proof should go in the same way as above).
Let ( T F , N ) be a Kronecker PNO of generic type (see Theorem 6.2) on a n -dimensional manifold M , n > . Then in a neighborhood U of every point p ∈ M there exists acyclic Nijenhuis operator N : T M → T M of any type of [Tur96] such that N | T F = N . Note that condition of cyclicity is necessary when we speak about the extensions of PNOs of generictype.
The situation with nongeneric Kronecker PNOs, i.e. having more than one Kroneckerblock in the J–K decomposition seems to be much more involved. The extension here can have morethan one cyclic blocks, however not necessarily. The analogues of systems of equations (10.1), (10.4),(10.5) would be much more complicated.
11 The Hirota equation
In this section we assume that dim M = 3. The aim of this section is to show that there is a 1–1-correspondence between Veronese webs in 3 dimensions and solutions of the so-called dispersionlessHirota PDE a f x f x x + a f x f x x + a f x f x x = 0 , where a i are constants such that a + a + a = 0.It follows from Theorem 6.2 and its proof that, given a Veronese web, one can construct a PNOwhich, at least locally, can be extended to a Nijenhuis operator defined on the whole tangent bundle T M . In Section 10 we have shown that in fact such an extension is possible essentially to any of normalforms of Nijenhuis operators in 3 dimensions.Conversely, starting from a Nijenhuis (1,1)-tensor N we can try to construct a Veronese web by meansof constructing a PNO (cf. Theorem 6.2) ( F , N | T F ) for some foliation F . Assuming that the foliation F is given by f = const for some smooth function f , we can use Lemma 5.5 to obtain sufficient conditionsfor N | T F to be a PNO in terms of a PDE on f , the form of which essentially depends on the form of theinitial Nijenhuis operator.Let us illustrate these idea choosing the simplest normal form of a Nijenhuis operator: the diagonalone with constant pairwise distinct eigenvalues. 23 Consider M = R ( x , x , x ) and a Nijenhuis operator N : T M → T M definedby
N ∂ x i = λ i ∂ x i , (11.1)where λ , λ , λ are pairwise distinct nonzero numbers. Let f : R → R be a smooth function such that f x i = 0. Define a foliation F ∞ by f = const, i.e. by T F ∞ := h df i ⊥ . Then ( N ( T F ∞ )) ⊥ = h ω i , where ω = ( N t ) − df = λ − f x dx + λ − f x dx + λ − f x dx . The condition of integrability of the distribution N ( T F ∞ ), dω ∧ ω = 0 (which by Lemma 5.5 impliesthat N | T F ∞ is a PNO), is equivalent to( λ − λ ) f x f x x + ( λ − λ ) f x f x x + ( λ − λ ) f x f x x = 0 , (11.2)in which we recognize the Hirota equation.The following theorem is a variant of [Zak00, Theorem 3.8] (our proof is different). Let λ , λ , λ be distinct real numbers.1. For any solution f of (11.2) on a domain U ⊂ M with f x i = 0 , i = 1 , , , the 1-form α λ = ( λ − λ )( λ − λ ) f x dx + ( λ − λ )( λ − λ ) f x dx + ( λ − λ )( λ − λ ) f x dx (11.3) defines a Veronese web F λ on U by T F λ = h α λ i ⊥ such that F λ i = { x i = const } , F ∞ = { f = const } . (11.4)
2. Conversely, let {F λ } be a Veronese web on a 3-dimensional smooth manifold M . Then in a neigh-bourhood of any point on M there exist local coordinates ( x , x , x ) such that any smooth firstintegral f of the foliation F ∞ is a solution of equation (11.2) with f x i = 0 .Consequently, we obtain a 1–1-correspondence between Veronese webs {F λ } satisfying (11.4) and theclasses [ f ] of solutions f of (11.2) with f x i = 0 modulo the following equivalence relation: f ∼ g ifthere exist local diffeomorphisms ψ, φ , φ , φ of R such that f ( x , x , x ) = ψ ( g ( φ ( x ) , φ ( x ) , φ ( x )) (obviously, if f ∼ g and f solves (11.2), then g does the same). On a solution f of equation (11.2) we get dω ∧ ω = 0, hence the distribution N ( T F ∞ ) is integrable.Consequently, N | T F ∞ is a PNO by Lemma 5.5. The condition f x i = 0 implies that the pair ( N | T F ∞ , I )has generic type (one Kronecker block in the J–K decomposition) and thus defines a Veronese web F λ by Theorem 6.2. The Veronese curve α λ in T ∗ U such that ( T F λ ) ⊥ = h α λ i annihilates the distribution N λ ( T F ∞ ) = T F λ . Direct check shows that it is given by formula (11.3), in particular satisfies (11.4).Conversely, let F λ be a Veronese web and f a first integral of F ∞ . The proof of Theorem 6.2 yieldsthe coordinates ( x , x , x ) and a Nijenhuis operator by (11.1). The distribution N ( T F ∞ ) = T F isintegrable, hence dω ∧ ω = 0 and f solves (11.2). The condition f x i = 0 follows from nondegeneracy ofthe curve α λ .Finally, the last statement follows from the fact that the first integrals of the three Veronese foliationscorresponding to different λ , λ , λ determine the first integral of any other foliation up to postcompo-sition with a local diffeomorphism. (cid:3) Repeating Construction 11.1 for other types of Nijenhuis operators listed in Appendix we get anotherPDEs on the function f , which are pairwise contactly nonequivalent (see [KP17, Section 6]). Below welist these PDEs corresponding to the cases A, B, C of Appendix, and indicate the Veronese curves α λ (the the one-forms ω such that ( N ( T F ∞ )) ⊥ = h ω i are given by ω = α λ | λ =0 ). (A) ( λ ( x ) − λ ( x )) f x f x x + ( λ ( x ) − λ ( x )) f x f x x + ( λ ( x ) − λ ( x )) f x f x x = 0 α λ = ( λ ( x ) − λ )( λ ( x ) − λ ) f x dx + ( λ ( x ) − λ )( λ ( x ) − λ ) f x dx + ( λ ( x ) − λ )( λ ( x ) − λ ) f x dx . (B) f x f x x − f x f x x + ( λ ( x ) − λ ( x ))( f x f x x − f x f x x ) + λ ′ ( x ) f x f x = 0 α λ = ( λ ( x ) − λ )( λ ( x ) − λ )( f x dx + f x dx ) + ( λ ( x ) − λ ) f x dx − ( λ ( x ) − λ ) f x dx . (C) C0 ( f x f x x − f x f x x ) x + f x f x x − f x f x x + f x f x x − f x f x x + f x f x = 0 α λ = f x (( x − λ ) dx − ( x − λ ) dx ) + f x ( − ( x − λ ) dx + ( x − λ ) dx +( x ( x − λ ) + 1) dx ) + f x ( x − λ ) dx . C1 f x f x x − f x f x x + f x f x x − f x f x x = 0 α λ = f x (( a − λ ) dx − ( a − λ ) dx + dx ) + f x (( a − λ ) dx − ( a − λ ) dx ) + f x ( a − λ ) dx . Here the following specifications should be made in order to exhaust the corresponding cases ( a , a , a are arbitrary pairwise different constants): (A) A λ ( x ) = x , λ ( x ) = x , λ ( x ) = x ; A λ ( x ) = x , λ ( x ) = x , λ ( x ) = a ; A λ ( x ) = x , λ ( x ) = a , λ ( x ) = a ; A λ ( x ) = a , λ ( x ) = a , λ ( x ) = a . (B) B λ ( x ) = x , λ ( x ) = x ; B λ ( x ) = x , λ ( x ) = a ; B λ ( x ) = a , λ ( x ) = x ; B λ ( x ) = a , λ ( x ) = a .Note that case A A solution f of any of the equations A, B, C on an open set U ⊂ M with coordinates( x , x , x ) is called nondegenerate if the corresponding one-form α λ ∈ T ∗ U defines a Veronese curve atany x ∈ U (equivalently: the curve λ α λ = α + λα + λ α does not lie in any plane, i.e., the 1-forms α , α , α are linearly independent at any point). 25
1. A generic solution f of any of the equations A, B, C is nondegenerate on asmall open set U . If f is such a solution, then the corresponding one-form α λ defines a Veroneseweb F λ on U by T F λ = h α λ i ⊥ .2. Conversely, let F λ be a Veronese web on a 3-dimensional smooth manifold M . Then for any symbol S = Ai, Bi, Ci in a neighbourhood of any point on M there exist local coordinates ( x , x , x ) suchthat any smooth first integral f of the foliation F ∞ is a nondegenerate solution of the equation oftype S .Here by a generic solution we mean a solution with a generic jet in the Cauchy problem setup. We omitthe formulation of the analogue of the last part of Theorem 11.2 as it follows immediately. The proof of the second statement of Item 1 is the same as that of Theorem 11.2(1). For the explanationwhy a generic solution of the equations A–C is nondegenerate see [KP17, Theorem 5.2].The proof Item 2 goes essentially as that of Theorem 11.2(2) with the account of Theorem 10.3. (cid:3)
13 Generalizations to higher dimensions: systems of PDEs
Generalization of the correspondence between Veronese webs and PDEs to higher dimensions (and thecase of Kronecker webs) is straightforward. Let {F λ } be a Kronecker web defined on an open set U ⊂ R n and let N : T F ∞ → T U be the corresponding kronecker PNO, see Remark 6.5. By Remark 6.6 (seealso Theorem 10.3) there exists a Nijehuis operator N : T U → T U such that N | T F ∞ = N . If f , . . . , f k are functionally independent first integrals of the foliation F ∞ , the condition of the integrability of thedistribution N ( T F ∞ ) (which follows from Lemma 5.3) is equivalent to a system of nonlinear PDEs onthe functions f i (depending on the form of the extension N ).Conversely, given a Nijenhuis (1,1)-tensor N : T U → T U and foliation F on U defined by a systemof first integrals f , . . . , f k , we can try to construct a Kronecker PNO ( T F , N | T F ) (and thus a Kroneckerweb {F λ } with F ∞ = F by requiring the integrability of the distribution N ( T F ) (cf. Lemma 5.5). Thecondition of the integrability of N ( T F ) is equivalent to a system of nonlinear PDEs on the functions f i .Of course, one should impose additional algebraic conditions on the pair F , N in order to guarantee thekroneckerity of N | T F .Note that the system of PDEs mentioned is overdetermined unless rank of the foliation F is not equaltwo (a reasonable bound is rank F ≥
2, since the rank one case gives a trivial differential constraint).Say in the case of Veronese web in 4 dimensions we get 4 equations on one function (the components ofthe 3-form dω ∧ ω , where ω is the 1-form annihilating N ( T F )).Let us illustrate the simplest higher dimensional case, when the system is determined: a Kroneckerweb {F λ } with foliations F λ of rank two in M = R . If N : T F ∞ → T M is the corresponding PNO, thepair (
N, I ) has two Kronecker blocks in the J–K decomposition. It is known that such Kronecker websare related with torsionless 3-webs on M , i.e. triples of foliations of rank 2 in general position with thetorsionless Chern connection. For any 3-web ( F , F , F ) there exists a unique 1-parametric family ofdistributions { D λ } λ ∈ RP of rank two such that D ∞ = T F , D = T F , D := T F and D λ is integrablefor any λ if and only if the torsion of the canonical Chern connection vanishes [Nag01, Theorem 4.14].The corresponding family of foliations {F λ } form a Kronecker web. We shall say that such Kroneckerwebs are of -web type .Consider M = R ( x , x , x , x ) and a Nijenhuis operator N : T M → T M defined by
N ∂ x i = λ i ∂ x i , λ , λ , λ , λ are pairwise distinct nonzero numbers. Let f , : R → R be a generic pair of smoothfunctions. Define a foliation F ∞ by T F ∞ := h df , df i ⊥ . Then ( N ( T F ∞ )) ⊥ = h ω , ω i , where ω i = ( N t ) − df i = X j =1 λ − j f ix j dx j . The condition of integrability of the distribution N ( T F ∞ ), dω ∧ ω ∧ ω = 0 , dω ∧ ω ∧ ω = 0 , is equivalent to the following system of equations( λ − λ ) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) + ( λ − λ ) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) +( λ − λ ) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) + ( λ − λ ) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) +( λ − λ ) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) + ( λ − λ ) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) = 0 , i = 1 , , (13.1)where ¯1 = 2 and ¯2 = 1. Once this system is satisfied by a pair of functions f , , the distribution D λ = ( N − λ Id T M )( T F ∞ ) is integrable for any λ and generates a Kronecker web, as it follows from Lemma5.5. Note that D λ is annihilated by the pair of forms ω λ , := P j =1 ( λ − λ ) · · · \ ( λ − λ j ) · · · ( λ − λ ) f , x j dx j for any λ = λ j , where c ( · ) means that the corresponding term is omitted (for λ = λ j the 1-forms ω λ , become linearly dependent).Another system of equations is obtained if we consider a Nijenhuis operator with two double eigen-values. For instance, put λ = λ and λ = λ in the example above. Then system (13.1) becomes f ix x ( f ix f ¯ ix − f ¯ ix f ix ) − f ix x ( f ix f ¯ ix − f ¯ ix f ix ) − (13.2) f ix x ( f ix f ¯ ix − f ¯ ix f ix ) + f ix x ( f ix f ¯ ix − f ¯ ix f ix ) = 0 , i = 1 , , and the corresponding annihilating one-forms are ω λ , = ( λ − λ )( f , x dx + f , x dx ) + ( λ − λ )( f , x dx + f , x dx ) (13.3)(now they span D ⊥ λ for all λ ). We see that D λ i , i = 1 ,
3, coincide with the corresponding coordinateplanes. We can prove an analogue of Theorem 11.2.
Let λ , λ be distinct real numbers.1. For any solution f , of (13.2) on a domain U ⊂ M satisfying (cid:12)(cid:12)(cid:12)(cid:12) D ( f , f ) D ( x , x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , (cid:12)(cid:12)(cid:12)(cid:12) D ( f , f ) D ( x , x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 (13.4) the 1-forms (13.3) define a Kronecker web F λ on U of -web type by T F λ = h ω λ , ω λ i ⊥ such that F λ i = { x i = const, x i +1 = const } , F ∞ = { f = const, f = const } . (13.5)
2. Conversely, let {F λ } be a Kronecker web of -web type on a 4-dimensional smooth manifold M .Then in a neighbourhood of any point on M there exist local coordinates ( x , x , x , x ) such that anyindependent smooth first integrals f , of the foliation F ∞ are solutions of system (13.2) satisfying(13.4). onsequently, we obtain a 1–1-correspondence between Kronecker webs {F λ } of -web type satisfying(13.5) and the classes [ f , ] of solutions f , of (13.2) satisfying system (13.4) modulo the followingequivalence relation: f , ∼ g , if there exist local diffeomorphisms ψ = ( ψ , ψ ) , φ = ( φ , φ ) and ζ = ( ζ , ζ ) of R such that f , ( x , x , x , x ) = ψ , ( g ( φ ( x , x ) , φ ( x , x ) , ζ ( x , x ) , ζ ( x , x )) , g ( φ ( x , x ) , φ ( x , x ) , ζ ( x , x ) , ζ ( x , x ))) (obviously, if f , ∼ g , and f , solves (13.2), then g , does the same). Item 1 is already argued. To prove Item 2 we let {F λ } be a Kronecker web of 3-web type. Then T F λ ⊕ T F λ = T M an we can find coordinates x , . . . , x such that T F λ = h ∂∂x , ∂∂x i , F λ = h ∂∂x , ∂∂x i .Define a Nijenhuis operator N by N | T F λi = λ i Id T F λi .It turn out that the distributions T F λ and H λ := ( N − λ Id T M )( T F ∞ ) coincide for any λ . Indeed,both of them are of the form h ( λ − λ ) X + ( λ − λ ) X , ( λ − λ ) Y + ( λ − λ ) Y i , where the vectorfields X , , Y , are linearly independent everywhere. Since H λ and F λ coincide for λ = λ , , we have T H λ = h ( λ − λ ) X + ( λ − λ ) X , ( λ − λ ) Y + ( λ − λ ) Y i and T F λ = h ( λ − λ )( a X + b Y ) + ( λ − λ )( a X + b Y ) , ( λ − λ )( a X + b Y ) + ( λ − λ )( a X + b Y ) i for some vector fields X , , Y , andfunctions a ij , b ij . On the other hand, the equality H ∞ = F ∞ , h X + X , Y + Y i = h ( a X + b Y ) +( a X + b Y ) , ( a X + b Y ) + ( a X + b Y ) i , implies due to the linear independence of X , , Y , that a = a , a = a , b = b , b = b and T F λ = h a [( λ − λ ) X + ( λ − λ ) X ] + b [( λ − λ ) Y + ( λ − λ ) Y ] , a [( λ − λ ) X + ( λ − λ ) X ] + b [( λ − λ ) Y + ( λ − λ ) Y ] i , which proves the claim.In particular, H = F = N ( F ∞ ) is integrable and considerations above show that independent firstintegrals of F ∞ should satisfy (13.2) and conditions (13.4).The last part of the theorem can be argued in the same way as that of Theorem 11.2. (cid:3) Of course the construction can be repeated for other normal forms of Nijenhuis operators in R .However a natural question whether each Kronecker web leads to some solution of the correspondingsystem is more subtle (cf. Remark 10.5).
14 An overview of related results
Below list some related results that are beyond the scope of this paper.For the general theory of Veronese and Kronecker webs, including their local classification see [Tur99a],[Tur10], [Kry12]. In the last article and in [Kry16a] the relations of Kronecker webs with systems of ODEsare discussed and adapted connections are built, which allow to distinguish among flat and nonflat webs(cf. the definition of flatness after Theorem 8.1).The problem of bisymplectic realizations of generic Kronecker bihamiltonian structures is studied in[Pet00].B¨acklund transformations, contact symmetry algebras and some exact solutions of the equations oftypes A–C (see Section 12) and also of type D, which corresponds to the case of a Nijenhuis operatorwith imaginary eigenvalues and which we omit in this article, can be found in [KP17].As mentioned in Introduction in paper [DK14] one can find a description of relations of Veronesewebs in 3D with the hyper-CR Einstein–Weyl structures, in particular an explicit formula of such astructure based on a solution of the Hirota equation. Similar formulae for other equations of types A–Dare discussed in [KP17]. In recent paper [Kry17] a twistor geometric approach is used to treat on the28ame base equations of types A–D and mentioned in Introduction hyper-CR equation, which cannot beincluded in the scheme of [KP17]. In the same paper [Kry17] there appears system (13.2) and its twistorgeometric deformations.Finally, these deformations, their generalizations to higher dimensions, and relations with the Pleba´nskiequation are discussed in [Kry16b].
Appendix: Classification of cyclic Nijenhuis operators in 3D(after F. J. Turiel)
In papers [Tur96, GM97] there was obtained a local classification of complex analytic Nijenhuis (1,1)-tensors N : T M → T M (in a vicinity of a regular point [Tur96, p. 451]) under additional assumptionof existence of a complete family of the so-called conservation laws. This assumption is equivalent tovanishing of the invariant P N , which is automatically trivial in the case of cyclic N [Tur96, p. 450], i.e.when the space T x M is cyclic for N x , x ∈ M . Here we recall the normal forms obtained in this case for3-dimensional M .The results of [Tur96] imply that for any cyclic Nijenhuis (1,1)-tensor in a vicinity of a regular point x there exist a local system of coordinates ( x , x , x ) and pairwise distinct constants a , a , a suchthat the coordinates ( x , x , x ) of x are also pairwise distinct and the matrix N of the correspondingoperator in the basis ∂∂x , ∂∂x , ∂∂x is one from the following list. Besides the matrices N themselves belowwe list also their Frobenius forms F and their Jordan forms J . A0. N A = N A ( x , x , x ) := x x
00 0 x , F A = F A ( x , x , x ) = x x x − x x − x x − x x x + x + x , J A = N A . A1. N A := N A ( x , x , a ) , F A := F A ( x , x , a ) , J A = N A . A2. N A := N A ( x , a , a ) , F A := F A ( x , a , a ) , J A = N A . A3. N A := N A ( a , a , a ) , F A := F A ( a , a , a ) , J A = N A . B0. N B = N B ( x , x ) := x x
00 0 x , F B = F B ( x , x ) := x x − x − x x x + x , J B = N B . B1. N B := N B ( x , a ) , F B := F B ( x , a ) , J B := N B . B2. N B := N B ( a , x ) , F B := F B ( a , x ) , J B := N B . B3. N B := N B ( a , a ) , F B := F B ( a , a ) , J B := N B . C0. N C = N C ( x , x ) := x x − x x , F C = F C ( x ) := x − x x , J C = J C ( x ) := x x
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