REDUCTION OF BIHAMILTONIAN MANIFOLDS AND RECURSION OPERATORS. 1. Poisson manifolds and the Schouten-Nijenhuis bracket

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1 DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 { Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 523{538 REDUCTION OF BIHAMILTONIAN MANIFOLDS AND RECURSION OPERATORS Joana Margarida Nunes da Costa and Charles-Michel Marle Acknowledgements. The authors, particularly C.-M. M., are indebted to Professor Franco Magri for many helpful discussions. All the manifolds, maps, dierential forms, vector and tensor elds, considered in what follows are assumed to be dierentiable of class C Poisson manifolds and the Schouten-Nijenhuis bracket 1.1. Some general properties and notations. Let be a bivector, that means a 2-times contravariant skew-symmetric tensor eld, on a smooth manifold M. It denes a vector bundle map ] : T M! T M, such that, for any x 2 M, and 2 T x M, ; ] () = x (; ) : We can therefore associate with any dierential 1-form, a vector eld ] (). For any pair of smooth functions f and g 2 C 1 (M; R), we set ff; gg = (df; dg) = i? ] (df) dg =?i? ] (dg) df : We shall say that (f; g) 7! ff; gg is the bracket associated with. It satises, for all f, g, g 1 and g 2 2 C 1 (M; R), ff; gg =?fg; fg ; ff; g 1 g 2 g = ff; g 1 gg 2 + g 1 ff; g 2 g : 1.2. Denition. A bivector on a smooth manifold M is said to be a Poisson tensor if the bracket associated with it satises the Jacobi identity, that is, if for all f, g and h 2 C 1 (M; R), f; fg; hg + g; fh; fg + h; ff; gg = 0 : 1991 Mathematics Subject Classication. 58F05. Key words and phrases. Poisson manifolds, bihamiltonian manifolds, reduction. This paper is in nal form and no version of it will be submitted for publication elsewhere. 523

2 524 J. M. NUNES DA COSTA AND CH.{M. MARLE In that case, the manifold M equipped with is called a Poisson manifold and denoted by (M; ); the bracket associated with is called a Poisson bracket and, for any smooth function f on M, the vector eld ] (df) is called the Hamiltonian vector eld associated with f. Let us recall the well known result (Lichnerowicz [8]): 1.3. Proposition. Let be a bivector on a smooth manifold M. The following three properties are equivalent: (i) the bivector is a Poisson tensor; (ii) the Schouten-Nijenhuis bracket [; ] of with itself vanishes; (iii) for all f and g 2 C 1 (M; R), ] (df); ] (dg) = ]? dff; gg : When these equivalent properties are satised, C 1 (M; R), with the Poisson bracket as composition law, is a Lie algebra and the map f 7! ] (df) is a Lie algebra homomorphism Some properties of the Schouten-Nijenhuis bracket. Let us recall the denition and some properties of the Schouten-Nijenhuis bracket [15, 13, 14, 5]. We use in the following the conventions of Koszul [5], which may dier by signs and even numerical factors from those used in [8] or [9]. Let M be a smooth manifold. We denote by A(M) = p A p (M) and by (M) = p p (M) the graded algebras, repectively, of skew-symmetric contravariant tensor elds, and of dierential forms on M. The Schouten-Nijenhuis bracket is a bilinear composition law on A(M), denoted by (P; Q) 7! [P; Q], which satises the following properties: (i) if P 2 A p (M) and Q 2 A q (M), then [P; Q] 2 A p+q?1 (M); (ii) if X 2 A 1 (M) is a vector eld on M, then for any Q 2 A(M), [X; Q] is the Lie derivative L(X)Q of Q with respect to X; (iii) if P 2 A p (M) and Q 2 A q (M), then [P; Q] =?(?1) (p?1)(q?1) [Q; P ] ; (iv) for any P 2 A p (M), the map Q 7! [P; Q] is a derivation of degree p? 1 of the graded exterior algebra A(M) (with the wedge product as composition law); in other words, for any Q 1 2 A q1 (M) and Q 2 2 A(M), [P; Q 1 ^ Q 2 ] = [P; Q 1 ] ^ Q 2 + (?1) (p?1)q1 Q 1 ^ [P; Q 2 ] : These four properties determine the Schouten-Nijenhuis bracket in a unique way. The Schouten-Nijenhuis bracket has several other remarkable properties. In particular,

3 REDUCTION OF BIHAMILTONIAN MANIFOLDS 525 (v) it satises the graded Jacobi identity (?1) (p?1)(r?1) P; [Q; R] + (?1) (q?1)(p?1) Q; [R; P ] + (?1) (r?1)(q?1) R; [P; Q] = 0 ; where P 2 A p (M), Q 2 A q (M) and R 2 A r (M); therefore, A(M), with the Schouten-Nijenhuis bracket as composition law, is a graded Lie algebra. There is another important property of the Schouten-Nijenhuis bracket, which links it with the exterior dierentiation operator d. We must rst recall some conventions about the exterior algebras A(M) and (M). There exists a bilinear map from (M) A(M) into C 1 (M; R), called the coupling map, denoted by (; P ) 7! h; P i. We dene it rst when = 1^ ^ q 2 q (M) is a decomposable q-form (the i 2 1 (M) being 1-forms on M), and when P = P 1^ ^P p 2 A p (M) is a decomposable p-vector (the P i 2 A 1 (M) being vector elds on M). Then 1 ^ ^ q ; P 1 ^ ^ P p = 0 if p 6= q; det? h i ; P j i if p = q: We have denoted by h i ; P j i = i(p j ) i the usual coupling of a 1-form i with a vector eld P j. The coupling map, which is local, can then be extended by bilinearity to the whole of (M) A(M), since locally any dierential q-form and any p-vector are sums of decomposable elements. We dene next a bilinear map from A(M) (M) into (M), called the inner product of a form by a multivector, and denoted by (P; ) 7! i(p ). When P 2 A p (M) and 2 q (M) are homogeneous of degrees p and q, respectively, then i(p ) 2 q?p is the unique (q? p)-form such that, for any R 2 A q?p (M), i(p ); R = (?1) (p?1)p=2 h; P ^ Ri : We observe that when p > q, i(p ) vanishes. The denition of the inner product i(p ) can be readily extended by bilinearity when P and are not homogeneous. When P 2 A p (M) is a xed homogeneous p-tensor, then i(p ) : 7! i(p ) is a graded endomorphism of the graded exterior algebra (M), of degree?p. The numerical coecient (?1) (p?1)p=2 in the above formula may seem unnatural; however it is such that when X 2 A 1 (M) is a vector eld, then i(x) is the usual inner product by the vector eld X; it is also such that, for any P 2 A p (M) and Q 2 A q (M), i(p )i(q) = i(p ^ Q) : Let D 1 and D 2 be two graded endomorphisms of the exterior algebra (M), of degrees d 1 and d 2, respectively. We dene their bracket, denoted by [D 1 ; D 2 ], as the graded endomorphism of (M), of degree d 1 + d 2, [D 1 ; D 2 ] = D 1 D 2? (?1) d1 d2 D 2 D 1 :

4 526 J. M. NUNES DA COSTA AND CH.{M. MARLE We can now indicate the next property of the Schouten-Nijenhuis bracket: (vi) let P 2 A p (M), Q 2 A q (M), [P; Q] their Schouten-Nijenhuis bracket; let i(p ), i(q) and i? [P; Q] be the inner products by P, Q and [P; Q], respectively, considered as graded endomorphisms of (M); then we have i? [P; Q] = [i(p ); d]; i(q) ; where d denotes the exterior dierentiation operator. The above property determines the Schouten-Nijenhuis bracket in a unique way, and could be used as a denition Applications of the Schouten-Njenhuis bracket to Poisson manifolds. Let be a bivector on a smooth manifold M. Using the denitions of the bracket associated with, of the coupling map and of the inner product given above we have, for all f and g 2 C 1 (M; R), ff; gg = i? ] (df) dg = (df; dg) = df ^ dg; =?i()(df ^ dg) : By property 1.4.(vi), we have Therefore, i? [; f] dg = [i(); d]; i(f) dg = i()? df ^ dg : ] (df) =?[; f] =?[f; ] ; ff; gg =? [; f]; g = [; g]; f : Similarly, for all f, g and h 2 C 1 (M; R), [; ](df ^ dg ^ dh) =?i? [; ] (df ^ dg ^ dh) =? [i(); d]; i() (df ^ dg ^ dh) =?2i()di()? df ^ dg ^ dh) = 2i()? dff; gg ^ dh + dfg; hg ^ df + dfh; fg ^ dg =?2? ff; gg; h + fg; hg; f + fh; fg; g = 2? f; fg; hg + g; fh; fg + h; ff; gg = 2? ] (df); ] (dg)? ]? dff; gg :h : Proposition 1.3 follows easily from this formula. Let 0 and 1 be two bivectors on the smooth manifold M. For all f and g 2 C 1 (M; R), we denote by ff; gg 0 = 0 (df; dg) and ff; gg 1 = 1 (df; dg) the brackets dened by 0 and 1, respectively. We set X f = 0 (df) ; Y f = 1 (df) :

5 REDUCTION OF BIHAMILTONIAN MANIFOLDS 527 Then by a calculation similar to that above (or by polarizing the above formulas) we have, for all f, g and h 2 C 1 (M; R), [ 0 ; 1 ](df; dg; dh) = f; fg; hg g; fh; fg h; ff; gg 0 1 +? f; fg; hg g; fh; fg h; ff; gg 1 0 = [X f ; Y g ] + [Y f ; X g ]? X ff;gg 1? Y ff;gg 0 :h : We can also write [ 0 ; 1 ](df; dg; dh) = f; fg; hg g; fh; fg h; ff; gg f; fg; hg g; fh; fg h; ff; gg 1 0 = Y f :fg; hg 0? fg; Y f :hg 0? fy f :g; hg 0 + X f :fg; hg 1? fg; X f :hg 1? fx f :g; hg 1 =? L(Y f ) 0 + L(X f ) 1 (dg; dh) : These formulas will be used in the next section. 2. Bihamiltonian manifolds 2.1. Denition. Two linearly independent Poisson tensors 0 and 1 dened on the same smooth manifold M are said to be compatible when is again a Poisson tensor. In such a case the manifold M, equipped with the two Poisson tensors 0 and 1, is called a bihamiltonian manifold, and denoted by (M; 0 ; 1 ) Proposition. Let M be a smooth manifold equipped with two linearly independent Poisson tensors 0 and 1. With the notations indicated at the end of the preceding section, the following properties are equivalent: (i) the Poisson tensors 0 and 1 are compatible; (ii) the Schouten-Nijenhuis bracket [ 0 ; 1 ] vanishes; (iii) for all f, g and h 2 C 1 (M; R), f; fg; hg0 1 + g; fh; fg h; ff; gg f; fg; hg g; fh; fg h; ff; gg 1 0 = 0 ; (iv) for all f and g 2 C 1 (M; R), (v) for all f 2 C 1 (M; R), [X f ; Y g ] + [Y f ; X g ]? X ff;gg 1? Y ff;gg 0 = 0 ; L(Y f ) 0 + L(X f ) 1 = 0 :

6 528 J. M. NUNES DA COSTA AND CH.{M. MARLE When these properties are satised, any two linearly independent linear combinations of 0 and 1 are compatible Poisson structures. Proof. Taking into account the bilinearity of the Schouten-Nijenhuis bracket, it follows immediately from Proposition 1.3 and the formulas in 1.5. The importance of bihamiltonian manifolds arises from the fact that many completely integrable systems are vector elds on a bihamiltonian manifold, which are Hamiltonian with respect to both Poisson structures. Euler's equations, which govern the motion of a rigid body with a xed point, are of that type Example: Euler's equations for the rigid body. Let us assume for simplicity that the three principal moments of inertia I 1, I 2 and I 3 of the body, at its xed point, are strictly positive and distinct. Euler's equation are dx 1 dt dx 2 dt dx 3 dt = (I?1 2? I?1 3 )x 2x 3 ; = (I?1 3? I?1 1 )x 3x 1 ; = (I?1 1? I?1 2 )x 1x 2 : We have denoted by x 1, x 2, x 3 the components of the angular momentum of the body at its xed point, in the inertial orthogonal frame, attached to the moving body, made by the eigenvectors of the inertia operator. We set: 0 =? x 1 ^ + x 2 ^ + x 3 ^ ; x 2 x 3 x 3 x 1 x 1 x 2 1 = I?1 1 x 1 ^ + I?1 x 2 x 3 2 x 2 H = 1 2 (I?1 1 x2 1 + I?1 2 x2 2 + I?1 3 x2 3) ; L = 1 2 (x2 1 + x x 2 3) : ^ + I?1 x 3 x 1 3 x 3 ^ ; x 1 x 2 We observe that ] 0 (dh) = ] 1 (dl) ; and that the dierential equations dened by that vector eld, which is Hamiltonian with respect to both 0 and 1, are Euler's equations. In [4], Holm and Marsden have used the fact that Euler's equations are Hamiltonian with respect to any nonvanishing linear combination of 0 and 1 to derive remarkable properties of that completely integrable system.

7 REDUCTION OF BIHAMILTONIAN MANIFOLDS Poisson-Nijenhuis manifolds 3.1. The Nijenhuis torsion. Let N be a tensor eld of type (1; 1) (one time contravariant and one time covariant) on a smooth manifold M. We shall consider N : T M! T M as a vector bundle map of the tangent bundle T M into itself, and we shall denote by t N : T M! T M the transpose map. Nijenhuis [12] has shown that by setting, for any pair (X; Y ) of vector elds on M, T (N)(X; Y ) = [N X; N Y ]? N? [N X; Y ] + [X; N Y ]? N[X; Y ] ; we dene a vector bundle map T (N) : T M T M! T M or, in other words, a tensor eld T (N) of type (1; 2) (one time contravariant and two-times covariant), called the Nijenhuis torsion of N The recursion operator. Let 0 be a Poisson tensor and N a tensor eld of type (1; 1) on a smooth manifold M. With the same conventions as above, we assume that N ] 0 = ] 0 t N ; and we set ] 1 = N ] 0 = ] 0 t N : The bundle map ] : 1 T M! T M is associated with a 2-times contravariant skew-symmetric tensor eld 1. For any smooth functions f and g on M, we set and similarly X f = ] 0 (df) ; Y f = ] 1 (df) = N X f ; ff; gg 0 = 0 (df; dg) = X f :g ; ff; gg 1 = 1 (df; dg) = Y f :g : As seen in 1.5, 1 is a Poisson tensor if and only if, for any pair (f; g) of smooth functions on M, [N X f ; N X g ] = N X ff;gg 1 : Similarly, the Schouten-Nijenhuis bracket [ 1 ; 0 ] vanishes if and only if, for any pair (f; g) of smooth functions on M, [N X f ; X g ] + [X f ; N X g ]? X ff;gg 1? N X ff;gg 0 = 0 : Since 0 is a Poisson tensor, we have X ff;gg 0 = [X f ; X g ] ; therefore [ 1 ; 0 ] vanishes if and only if, for any pair (f; g) of smooth functions on M, X ff;gg 1 = [N X f ; X g ] + [X f ; N X g ]? N[X f ; X g ] :

8 530 J. M. NUNES DA COSTA AND CH.{M. MARLE Therefore, when 1 is a Poisson tensor compatible with 0 we have, for any pair (f; g) of smooth functions on M, [N X f ; N X g ]? N? [N X f ; X g ] + [X f ; N X g ]? N[X f ; X g ] = 0 : Using the expression of the Nijenhuis torsion of N, we may state: 3.3. Proposition. Let 0 be a Poisson tensor and N a tensor eld of type (1; 1) on a smooth manifold M, such that N ] = 0 ] t 0 N : We assume that the tensor eld 1 associated with the vector bundle map ] 1 = N ] 0 satises [ 1 ; 0 ] = 0 : Then 1 is a Poisson tensor if and only if the Nijenhuis torsion T (N) vanishes on the image ] 0 (T M) of ] 0. In such a case, (M; 0; 1 = N 0 ) is a bihamiltonian manifold, called a Poisson-Nijenhuis manifold, and the tensor eld N is called its recursion operator The Magri-Morosi concomitant. In what follows, M is a smooth manifold and, for any p-times contravariant skew-symmetric tensor eld P 2 A p (M), and any family ( 1 ; : : : ; p?1) of p? 1 Pfa forms on M, we will denote by P ( 1 ; : : : ; p?1) the unique vector eld on M such that, for any Pfa form p on M, p ; P ( 1 ; : : : ; p?1) = P ( 1 ; : : : ; p?1; p ) : In particular, for p = 2, we will write P () = P ] () : Let P 2 A 2 (M) be a skew-symmetric, two-times contravariant tensor eld, and N a tensor eld of type (1; 1) on the smooth manifold M. We assume that N P ] = P ] t N : Magri and Morosi [9] have shown that with P and N one can construct a new tensor eld R(P; N) of type (2; 1), by setting, for any pair of a 1-form and a vector eld X, R(P; N)(; X) =? L(P ] )N X? P ]? L(X)( t N ) + P ]? L(N X) : We will call R(P; N) the Magri-Morosi concomitant of P and N. It satises the following properties [9], which link it with the Nijenhuis torsion and the Schouten bracket:

9 REDUCTION OF BIHAMILTONIAN MANIFOLDS 531 (i) Let N P be the skew-symmetric two-times contravariant tensor eld associated with the vector bundle map N P ]. It satises N(N P ) ] = (N P ) ] t N, and for any 1-form and any vector eld X, R(N P; N)(; X) = NR(P; N)(; X) + T (N)(X; P ] ) : (ii) For any pair (; ) of 1-forms on M, [N P; N P ](; ) = N? [P; P ]( t N ; )? 2R(P; N)(; P ) + 2T (N)(P ; P ) : (iii) Let P and Q 2 A 2 (M) be two 2-times contravariant skew-symmetric tensor elds and N a tensor eld of type (1; 1) such that N P ] = P ] t N ; N Q ] = Q ] t N : Then, for any pair (; ) of Pfa forms on M, [N P; Q](; ) = N[P; Q](; )? QR (P; N)(; )? R(Q; N)(; P ) + R(Q; N)(; P ) ; where the transpose R (P; N) of R(P; N) is dened by the formula, in which and are any Pfa forms and X any vector eld, R (P; N)(; ); X = ; R(P; N)(; X) : 3.5. Remark. Let us indicate another useful formula involving the Schouten- Nijenhuis bracket and the Nijenhuis torsion, but not the Magri-Morosi concomitant. We shall use the same conventions as in Section 3.4. Let P 2 A 2 (M) be a two-times contravariant skew-symmetric tensor eld and N a tensor eld of type (1; 1) such that N P ] = P ] t N : Then [N P; N P ]? 2N[N P; P ]? N 2 [P; P ] = 2T (N) (P; P ) ; where T (N) (P; P ) is the vector bundle map such that, for any pair (; ) of Pfa forms, T (N) (P; P )(; ) = T (N)(P ; P ) : This formula can be easily proven by using results indicated in 1.5. By using the above formulas, we can easily prove the following result, which is a slight generalization of a theorem due to Magri and Morosi [9]: 3.6. Theorem. Let (M; 0 ; 1 = N 0 ) be a Poisson-Nijenhuis manifold. For any integer k 1, we set ] k = N k ] 0 ;

10 532 J. M. NUNES DA COSTA AND CH.{M. MARLE and we denote by k the two-times contravariant skew-symmetric tensor eld associated with the vector bundle map ]. Then, for all k 2 N, the k k are Poisson tensors and are pairwise compatible. In other words, for all k and l 2 N, [ k ; l ] = 0 : 4. Reduction of bihamiltonian manifolds The reduction theorem of Marsden and Ratiu for Poisson manifolds [11] can be easily extended to bihamiltonian manifolds, as was shown independently by F. Magri (personal communication) and one of the authors (J. M. N. da C.): 4.1. Theorem. Let (M; 0 ; 1 ) be a bihamiltonian manifold, S a submanifold of M, and D a vector sub-bundle of T S M, which satisfy the following conditions: (i) For any pair (F; G) of smooth functions on M whose dierentials df and dg, restricted to S, vanish on the sub-bundle D, the dierentials dff; Gg 0 and dff; Gg 1 of the Poisson brackets of F and G, for each Poisson structure 0 and 1, restricted to S, vanish on D. (ii) Let D 0 T M be the annihilator of D. Then S ] 0 (D0 ) T S + D and ] 1 (D0 ) T S + D : (iii) The distribution T S \ D on S is completely integrable; the set _ S of leaves of the foliation dened on S by that distribution is a smooth manifold, and the canonical projection : S! _ S is a submersion. Then there exists on _ S a unique pair ( _ 0 ; _ 1 ) of compatible Poisson structures, such that, for any pair (f; g) of smooth functions on _ S and any smooth extensions (to a neighbourhood of S in M) F of f and G of g whose dierentials df and dg, restricted to S, vanish on D, ff; Gg 0 S = ff; gg 0 ; ff; Gg 1 S = ff; gg 1 : We will say that ( _S; _ 0 ; _ 1 ) is the reduced bihamiltonian manifold dened by S and D Remarks. 1. When (M; 0 ; 1 = N 0 ) is a Poisson-Nijenhuis manifold, condition (ii) of Theorem 4.1 may be replaced by the slightly more restrictive condition ] 0 (D0 ) T S + D and N(T S + D) T S + D :

11 REDUCTION OF BIHAMILTONIAN MANIFOLDS 533 But the reduced bihamiltonian manifold ( _ S; _ 0 ; _ 1 ) is not in general a Poisson- Nijenhuis manifold. However, in many cases the higher order Poisson tensors k = N k 0 on M (k 1) induce on _ S higher order Poisson tensors _ k, which are pairwise compatible, although they are not deduced from _ 0 by repeated application of a recursion operator. We shall see an example of that construction in the next section. 2. Magri and Morosi [9, 10] considered the following situation: (M; 0 ; 1 ) is a bihamiltonian manifold, and S a symplectic leaf (see for example Weinstein [17], Vaisman [16] or [7]) of the Poisson manifold (M; 1 ). The sub-bundle D of T S M chosen here is D = ] 0 (ker ] 1 ) S : Assuming that D \ T S is of constant rank, one can easily prove that D \ T S is completely integrable, and that conditions (i) and (ii) of Theorem 4.1 are satised. If in addition the set of leaves of the foliation dened on S by D \ T S is a smooth manifold, and if the canonical projection : S! _ S is a submersion, Theorem 4.1 can be applied A simple case. In view of its application to the Toda lattice, we consider the following simple case. Let (M; 0 ) be an 2n-dimensional Poisson manifold whose Poisson tensor 0 is everywhere of rank 2n; in other words, M is a symplectic manifold, equipped with the Poisson structure canonically associated with its symplectic structure. Let 1 be another Poisson tensor on M, which may not be compatible with 0. Since ] 0 : T M! T M is an isomorphism, we may set N = ] 1 (] 0 )?1 : Then N is a tensor eld of type (1; 1) which satises For any integer k 1, we set ] = = 1 N] 0 ] t 0 N : ] k = N k ] 0 ; and we denote by k the 2-times contravariant, skew-symmetric tensor eld associated with the vector bundle map ]. Let Z be a nowhere vanishing vector eld k on M such that L(Z) 0 = 0 ; L(Z) 1 = 0 : It satises also L(Z)N = 0 : We assume that the set M_ of integral curves of Z is a smooth manifold and that the canonical projection : M! _M is a submersion. We can easily prove that there exists on M_ a pair ( _ 0 ; _ 1 ) of Poisson structures, such that : M! _M is a Poisson map from (M; 0 ) onto ( _M ; _ 0 ), and also from (M; 1 ) onto ( _M ; _ 1 ).

12 534 J. M. NUNES DA COSTA AND CH.{M. MARLE Similarly, for any integer k 1, there exists on M _ a 2-times contravariant, skewsymmetric tensor eld _ k, which is the direct image k of k by the map Proposition. With the same notations and assumptions as in 4.3, let T (N) be the Nijenhuis torsion of N. We assume that, for any pair (; ) of Pfa forms on M, T (N)( ] 1 ; ] 1 ) = 0 : Then for all k 1, the k are Poisson tensors on M, and are pairwise compatible; in other words, for all k 1 and l 1, [ k ; l ] = 0. Proof. Let be a Pfa form and X a vector eld on M. We use 3.4.(ii), with P = 0 and = ( ] 0 )?1 X. Since [ 0 ; 0 ] = 0 and [ 1 ; 1 ] = 0, we obtain NR( 0 ; N)(; X) = T (N)( ] 0; X) : () Now we use 3.4.(iii) with P = Q = 1. Since [ 1 ; 1 ] = 0, we obtain for any pair (; ) of Pfa forms, [ 2 ; 1 ](; ) =? 1 R ( 1 ; N)(; )? R( 1 ; N)(; ] 1 ) + R( 1; N)(; ] 1 ) : But by 3.4(i) and the formula () above, since T (N)(X; Y ) =?T (N)(Y; X), R( 1 ; N)(; X) = NR( 0 ; N)(; X) + T (N)(X; ] 0 ) = 0 : Similarly, R ( 1 ; N)(; ); X = ; R( 1 ; N)(; X) = 0 : Therefore [ 2 ; 1 ] = 0. Now we use Proposition 3.3, in which we replace 0 by 1 and 1 by 2, and we obtain the stated results Remarks. 1. Under the assumptions of Proposition 4.4, for all k 1, the images _ k = k of the Poisson tensors k by the submersion are Poisson tensors on _M, and they are pairwise compatible. This result remains true under assumptions slightly weaker than those made in that Proposition. 2. By using the formula indicated in 3.5, we can easily prove that the assumption of Proposition 4.4, that for any pair (; ) of Pfa forms on M, T (N)( ] 1 ; ] 1 ) = 0 ; is equivalent to the following assumption: for any triple (; ; ) of Pfa forms on M, [ 1 ; 0 ]( t N ; t N ; t N ) = 0 :

13 REDUCTION OF BIHAMILTONIAN MANIFOLDS 535 Proposition 4.4 does not say anything about the compatibility of the k with 0, nor with the compatibility of the _ k with _ 0. The next Proposition deals with that question Proposition. Under the assumptions of Proposition 4.4, we assume in addition that for any pair ( _; ) _ of Pfa forms on M _ and any vector eld X on M, [ 1 ; 0 ]( _; ) _ = 0 ; R( 0 ; N)( _; X) = 0 ; T (N)(X; ] 1 _) = 0 : Then for any k 1, the Poisson tensor _ k on _M is compatible with _ 0. Proof. Let ( _; _ ) be a pair of Pfa forms on _M. We have seen in the proof of 4.4 that R( 1 ; N) = 0. We can easily prove by induction on k that for all k 1, rst (by using 3.4.(i)) that R( k ; N)( _; X) = 0, second (by using 3.4.(iii)) that [ k ; 0 ]( _; _ ) = 0. This result implies that for all k 1, [ _ k ; _ 0 ] = Application to the Toda lattice 5.1. The Toda lattice and the Flaschka transformation. On R 2n (coordinates q 1 ; : : : ; q n ; p 1 ; : : : ; p n ), let 0 = nx p i ^ q i be the canonical Poisson tensor. The Toda lattice is the Hamiltonian system (relative to 0 ) with Hamiltonian H = 1 2 nx We set, for 1 i n? 1, 1 j n, X n?1 p i2 + exp(q i? q i+1 ) : a i = 1 2 exp 1 2 (qi? q i+1 ) ; b j =? 1 2 p j : The map : R 2n! R 2n?1, (q 1 ; : : : ; q n ; p 1 ; : : : ; p n ) 7! (a 1 ; : : : ; a n?1 ; b 1 ; : : : ; b n ), is the Flaschka transformation [3]. The bers of are integral curves of the vector eld! nx nx Z = q = i ] 0 d p i :

14 536 J. M. NUNES DA COSTA AND CH.{M. MARLE Since L(Z) 0 = 0, there is on the image of a unique Poisson tensor _ 0 such that is a Poisson map. This Poisson tensor, which extends to the whole R 2n?1, is _ 0 = 1 4 n?1 X a i a ^ i b i? b i+1 : The Hamiltonian H of the Toda lattice factors through : it can be expressed as H = _H, with n?1 nx X _H = 2 b i2 + 4 (a i ) 2 ; and the Toda vector eld projects by onto the Hamiltonian vector eld T = _ ] 0 (d _ H) Higher order Poisson structures for the Toda lattice. The projection T of the Toda vector eld on R 2n?1 is Hamiltonian with respect to _ 0 ; it is Hamiltonian for another Poisson tensor _ 1, obtained by Adler [1]: X n?1 _ 1 =?2 (a i ) 2 ^ b i b i n?1 X a i a ^ i a i+1 with, by convention, a n = 0. We have indeed a + 2b i+1 i+1? 2b i ; b i+1 b i T = _ ] 0 (d _H) = _ ] 1 (d _ L) ; with _ L = nx Moreover, the Poisson tensors _ 1 and _ 0 are compatible: [ _ 1 ; _ 0 ] = 0. Kupershmidt [6] has found a third Poisson tensor _ 2 on R 2n?1, whose coecients are polynomials of degree 3 in the coordinates a i and b j, compatible with _ 0 and _ 1. Damianou [2] has shown the existence of an innite sequence ( _ k ); k 2 N, of Poisson tensors on R 2n?1, the coecients of _ k being polynomials of degree k + 1 in the coordinates a i, b j, the rst terms _ 0, _ 1, _ 2 of that sequence being the Poisson tensors already obtained by Flaschka, Adler and Kupershmidt. Moreover the terms of that sequence are pairwise compatible. These Poisson tensors are not obtained from each other by repeated application of a recursion operator, and Damianou does not indicate an explicit formula for obtaining them. Such an explicit formula can be obtained by going back to R 2n and using Proposition 4.4. First we set z = 1 n nx q i : b i :

15 REDUCTION OF BIHAMILTONIAN MANIFOLDS 537 Then z; a 1 ; : : : ; a n?1 ; b 1 ; : : : ; b n can be used as (nonlinear) coordinates on R 2n. The Poisson tensors _ 0 and _ 1 on R 2n?1 can now be considered as Poisson tensors on R 2n, whose expressions, in terms of these coordinates, are those given above, which do not contain z nor. We have z 0 = _ 0 + Z ^ Y ; where Y is a vector eld and Z the vector eld already dened. Their expressions, in the new coordinates z, a i, b j, are Y = 1 2n nx b i ; Z = z : Although 0 and _ 1, considered as Poisson tensors on R 2n, are not compatible, we can dene a recursion operator N = _ ] 1 (] 0 )?1, and set, for any integer k 1, ] = N k ]. Let k 0 k be the corresponding 2-times contravariant skew-symmetric tensor elds. By using 4.5.2, we can easily prove that the torsion T (N) of the recursion operator N satises the condition of Proposition 4.4. Therefore, the k, for k 1, are pairwise compatible Poisson tensors. Their projections _ k by are Poisson tensors on R 2n?1 ; the rst terms _ 1 and _ 2 are equal to those found by Adler and Kupershmidt. We conjecture that the whole sequence ( _ k ); k 2 N, obtained by that method, is the same as that obtained by Damianou (the non uniqueness of the terms in that sequence observed by Damianou corresponding to the various possible choices of a Poisson tensor 1 on R 2n which projects by onto _ 1 ). 6. References [1] Adler, M., On a trace functional for formal pseudo-dierential operators and the symplectic structure of the Korteweg de-vries type equations, Invent. Math., 50, 1979, 219{248. [2] Damianou, P. A., Master symmetries and R-matrices for the Toda lattice, Letters in Mathematical Physics, 20, 1990, 101{112. [3] Flaschka, H., On the Toda lattice, Phys. Rev. B 9, 1974, 1924{1925. [4] Holm, D. D., and Marsden, J. E., The rotor and the pendulum, in Symplectic geometry and mathematical physics, (P. Donato et al., eds), Birkhauser, Boston, 1991, 189{203. [5] Koszul, J.-L., Crochet de Schouten-Nijenhuis et cohomologie, in Elie Cartan et les mathematiques d'aujourd'hui, Asterisque, numero hors serie, 1985, [6] Kupershmidt, B., Discrete Lax equatioons and dierential dierence calculus, Asterisque 123, 1985.

16 538 J. M. NUNES DA COSTA AND CH.{M. MARLE [7] Libermann, P., and Marle, C.-M., Symplectic geometry and analytical mechanics, D. Reidel Publishing Company, Dordrecht, [8] Lichnerowicz, A., Les varietes de Poisson et leurs algebres de Lie associees, J. Dierential Geometry, 12, 1977, 253{300. [9] Magri, F., and Morosi, C., A geometric characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S 19, 1984, Universita di Milano. [10] Magri, F., Geometry and soliton equations, in La \Mecanique analytique" de Lagrange et son heritage, Acta Academiae Scientiarum Taurinensis, Torino 1990, 181{209. [11] Marsden, J. E., and Ratiu, T., Reduction of Poisson manifolds, Letters in Mathematical Physics, 11, 1986, 161{169. [12] Nijenhuis, A., X n+1 -forming sets of eigenvectors, Proc. Kon. Ned. Akad. Wet. Amsterdam, A 54, 1951, 200{212. [13] Nijenhuis, A., Jacobi-type identities for bilinear dierential concomitants of certain tensor elds I, Indagationes Math., 17, 1955, 390{403. [14] Ouzilou, R., Hamiltonian actions on Poisson manifolds, in Symplectic geometry (A. Crumeyrolle and J. Grifone, eds), Research notes in mathematics 80, Pitman, Boston, 1983, 172{183. [15] Schouten, J. A., On the dierential operators of the rst order in tensor calculus, in Convegno Int. Geom. Di. Italia, Ed. Cremonese, Roma, 1954, 1{7. [16] Vaisman, I., Lectures on the geometry of Poisson manifolds, Birkhauser, Boston, [17] Weinstein, A., The local structure of Poisson manifolds, J. Dierential Geometry, 18, 1983, 523{557 and 22, 1985, 255. Universidade de Coimbra, Departamento de Matematica Apartado 3008, 3000 Coimbra, Portugal Universite Pierre et Marie Curie, Institut de Mathematiques 4, Place Jussieu, Paris cedex 05, France E{mails : jmcostamat.uc.pt, marlemathp6.jussieu.fr