F -manifolds and integrable systems of hydrodynamic type

Transcription

1 arxiv: v1 [math.dg] 25 May 2009 F -manifolds and integrable systems of hydrodynamic type Paolo Lorenzoni, Marco Pedroni, Andrea Raimondo * Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca Via Roberto Cozzi 53, I Milano, Italy paolo.lorenzoni@unimib.it ** Dipartimento di Ingegneria dell informazione e metodi matematici Università di Bergamo - Sede di Dalmine viale Marconi 5, I Dalmine BG, Italy marco.pedroni@unibg.it *** Department of Mathematics, Imperial College 180 Queen s Gate, London SW7 2AZ, UK a.raimondo@imperial.ac.uk Abstract We investigate the role of Hertling-Manin condition [14] on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F -manifold with compatible connection generalizing a structure introduced by Manin in [17]. 1 Introduction In their seminal papers [6, 22], Dubrovin, Novikov, and Tsarev pointed out a deep relation between the integrability properties of systems of PDEs of hydrodynamic type u i t = V i j u j x, i = 1,...,n, (1) (sum over repeated indices is understood) and geometrical in particular, Riemannian structures on the target manifold M, where (u 1,...,u n ) play the role of coordinates. Probably, the most important of such structures is the notion of Frobenius manifold, introduced by 1

2 Dubrovin (see, e.g., [4]) in order to give a coordinate-free description of the famous WDVV equations. A crucial ingredient involved in the definition of Frobenius manifolds is a (1, 2)- type tensor field c giving an associative commutative product on every tangent space: (X Y ) i := c i jk Xj Y k, where X and Y are vector fields. More recently [14], Hertling and Manin showed that this product satisfies the condition [X Y, Z W] [X Y, Z] W [X Y, W] Z X [Y, Z W] + X [Y, Z] W+ +X [Y, W] Z Y [X, Z W] + Y [X, Z] W + Y [X, W] Z = 0, (2) or, in terms of the components of c, ( s c k jl )cs im + ( jc s im )ck sl ( sc k im )cs jl ( ic s jl )ck sm ( lc s jm )ck si ( mc s li )ck js = 0. (3) They called F -manifold a manifold endowed with a multiplicative structure satisfying condition (2). The aim of this paper is to study the properties of the PDEs of hydrodynamic type associated with F -manifolds. The system (3) and its relation with integrable systems has been considered from a different point of view in [15]. Here, following the insights coming from the case of the principal hierarchy in the context of Frobenius manifolds, we will assume such PDEs to be of the form u i t = (V X) i j uj x, i = 1,..., n, (V X) i j := ci jk Xk, (4) where X is a vector field on M and c satisfies (2). These assumptions have two important consequences, spelled out respectively in Section 2 and 3: 1. For any choice of the vector field X, the Haantjes tensor associated with the (1,1) tensor field V X vanishes. 2. They allow one to write the condition of commutativity of two flows of the form (4) as a simple requirement on the corresponding vector fields on M. In Section 4, we show how to construct an integrable hierarchy of hydrodynamic type starting from an F -manifold with compatible flat connection, as considered by Manin in [17]. The costruction is divided in two steps. First using a basis of flat vector fields one defines a set of flows, known as primary flows. Then, from these flows one can define recursively the higher flows of the hierarchy. In this way, each primary flow turns out to be the starting point of a hierarchy. This construction is a straightforward generalization of the principal hierarchy defined by Dubrovin in the case of Frobenius manifolds [4]. In Section 5 we observe that a flat Egorov metric (locally) gives an F -manifold with compatible flat structure, while in Section 6 we introduce the notion of F -manifold with compatible (in general non-flat) connection and we show that the associated integrable systems of hydrodynamic type are defined by a family of vector fields satisfying the following condition: c i jm kx m = c i km jx m. (5) 2

3 In the non-flat case the existence of solutions of the above system is not guaranteed. Indeed, we prove that every solution X of (5) satisfies the condition (R k lmic n pk + R k lipc n mk + R k lpmc n ik)x l = 0, where R is the curvature tensor of. It is thus natural to introduce the following requirement on the curvature: R k lmic n pk + R k lipc n mk + R k lpmc n ik = 0. (6) If the structure constants c i jk admit canonical coordinates, condition (6) is related to the wellknown semi-hamiltonian property introduced by Tsarev [22] as compatibilty condition for the linear system providing the symmetries of a diagonal system of hydrodynamic type. Finally, in Section 7, we discuss in details an important example: the reductions of the dispersionless KP hierarchy (also known as Benney chain). 2 The Haantjes tensor An important class of systems of hydrodynamic type, widely studied in the literature, consists in those systems which admit diagonal form. We say that a system (1) is diagonalizable if there exists a set of coordinates (r 1,...,r n ) usually called Riemann invariants such that the tensor Vj i (diagonal) form is diagonal in these coordinates: V i j (r) = vi δ i j rt i = vi (r 1,...,r n )rx i, i = 1,...,n.. Then the system takes the It is important to recall that there exists an invariant criterion for the diagonalizability. One first introduces the Nijenhuis tensor of V as N V (X, Y ) = [V X, V Y ] V [X, V Y ] V [V X, Y ] + V 2 [X, Y ], where X and Y are arbitrary vector fields, and then defines the Haantjes tensor as H V (X, Y ) = N V (V X, V Y ) V N V (X, V Y ) V N V (V X, Y ) + V 2 N V (X, Y ). In the case when V has mutually distinct eigenvalues, then V is diagonalizable if and only if its Haantjes tensor is identically zero. In this section, we consider the Haantjes tensor of (V Z ) i j = ci jk Zk, (7) where c satisfies the Hertling-Manin condition (2). For a (1, 1)- type tensor field of the form (7), the Nijenhuis tensor reads N VZ (X, Y ) = [Z X, Z Y ] + Z 2 [X, Y ] Z [X, Z Y ] Z [Z X, Y ]. By using the Hertling-Manin condition (2) evaluated at X = Z, this can be written as N VZ (X, Y ) = [X Z, Z] Y [X, Y ] Z Y + [Z, Y Z] X [Z, Y ] X Z. (8) Using this identity it is easy to prove the following 3

4 Theorem 1 The Haantjes tensor associated with V Z vanishes for any choice of the vector field Z. Proof. Let us write for simplicity N in place of N VZ. Then, we have that H VZ [X, Y ] = N[Z X, Z Y ] + Z 2 N[X, Y ] Z N[X, Z Y ] Z N[Z X, Y ] = = [X Z 2, Z] Y Z [X Z, Z] Z 2 Y + [Z, Y Z 2 ] X Z + [Z, Y Z] X Z 2 + [X Z] Y Z 2 [X, Z] Z 3 Y + + [Z, Y Z] X Z 2 [Z, Y ] X Z 3 [X Z, Z] Z 2 Y + + [X, Z] Z 3 Y [Z, Y Z 2 ] X Z + [Z, Y Z] X Z 2 + [X Z 2, Z] Y Z + [X Z, Z] Z 2 Y [Z, Y Z] X Z [Z, Y ] X Z 3 = 0, where Z 2 = Z Z and Z 3 = Z Z Z. Suppose now that X is a vector field such that V X has everywhere distinct real eigenvalues (v 1,...,v n ). Since the Haantjes tensor of V X vanishes, there exist local coordinates (r 1,...,r n ) such that (V X ) i j = δi j vi. These coordinates are the Riemann invariants of the corresponding system of hydrodynamic type. Moreover, we have Proposition 2 The components of the tensor field c in the coordinates (r 1,...,r n ) are given by c k ij = f i δ k i δ k j. Moreover, if f j 0 for all j, then f i depends on the variable r i only. Proof. In diagonal coordinates we have (V X ) i j = ci jkx k = v i δ i j, hence, we get c j pq ci jk Xk = c j pq vi δ i j = ci pq vi. On the other hand, due to the associativity of the algebra, we can also write c j pq ci jk Xk = c j pk ci jq Xk = c i jq vj δ j p = ci pq vp (no sum over p), and therefore, c i pq ( v i v p) = 0. (9) Since the algebra is commutative and the eigenvalues of V X are pairwise distinct, this means that the structure constants, in the coordinates (r 1,...,r n ), take the form c i jk = f iδ i j δi k, 4

5 where the f i are arbitrary functions, depending in principle on all the variables r 1,...,r n. The requirement on the structure constants c to satisfy the Herling-Manin condition (3) implies further constraints on the functions f i. Indeed, substituting (9) into (3), we get a set of equations the f i have to satisfy; considering for instance the case m = j k = i = l, we get f j j f k = 0, which means that, in the non-degenerate case when f j 0 for all j, then f k depends on r k only. It is easy to check that conditions (3) give no further restrictions on the f i ; the proposition is proved. If the functions f i are everywhere different from zero, then it is easy to show that there exist local coordinates, called canonical coordinates, such that c k ij = δi k δj k. Moreover, in this case, the vector field n 1 e = f i r i i=1 is globally defined and is the unity of the algebra. Remark 3 If the algebra has a unity e, then the Hertling-Manin condition implies Lie e c = 0. Indeed, for X = Y = e the Hertling-Manin condition becomes [e, Z W] + [e, Z] W + [e, W] Z = 0. Remark 4 An alternative proof of the existence of canonical coordinates has been given in [14] under the assumption of semisimplicty of the algebra, that is, the existence of a basis of idempotents. 3 Commutativity of the flows As a consequence of the Hertling-Manin condition, the conditions for the commutativity of two hydrodynamical flows take a rather simple form. Proposition 5 The flows and u i t = [V X] i j uj x = ci jk Xj u k x (10) u i τ = [V Y ] i j uj x = ci jk Y j u k x (11) 5

6 commute if and only if the vector fields X and Y satisfy the condition ((Lie X c) (Y, Z) (Lie Y c) (X, Z) + [X, Y ] Z) Z = 0, (12) for any vector field Z. Equivalently, ((Lie X c) (Y, Z) (Lie Y c)(x, Z) + [X, Y ] Z) W + ((Lie X c)(y, W) (Lie Y c) (X, W) + [X, Y ] W) Z = 0 for all pairs (Z, W) of vector fields. In local coordinates this means that ] [(Lie X c) ijq Y q (Lie Y c) ijq Xq + c ijq [X, Y ]q c r is + c r ij [(Lie X c) isq Y q (Lie Y c) isq Xq + c isq[x, ] Y ] q = 0. Proof. It is well-known that the commutativity of the flows (10) and (11) is equivalent to the following requirements: 1. The (1,1) tensor fields V X and V Y (seen as endomorphism of the tangent bundle) commute. 2. For any vector field Z the following condition is satisfied: that is to say, [V X (Z), V Y (Z)] V X ([Z, V Y (Z)]) + V Y ([Z, V X (Z)]) = 0, (13) [Z X, Z Y ] X [Z, Z Y ] + Y [Z, Z X] = 0. (14) The first requirement is automatically verified due to the associativity of the algebra. Making use of identity (2), the second one becomes ([Z X, Y ] + [X, Z Y ] [X, Z] Y [X, Y ] Z X [Z, Y ]) Z = 0. (15) A simple calculation shows that the quantity in the bracket, namely is equal to [Z X, Y ] + [X, Z Y ] [X, Z] Y [X, Y ] Z X [Z, Y ], (16) (Lie X c)(y, Z) (Lie Y c)(x, Z) + [X, Y ] Z. (17) Substituting (17) into (15), we get the thesis. Corollary 6 A sufficient condition for the commutativity of the hydrodynamic flows (10) and (11) is that (Lie X c)(y, Z) (Lie Y c)(x, Z) + [X, Y ] Z = 0 (18) for all vector fields Z, that is, or, equivalently, (Lie X c) i pq Y q (Lie Y c) i pq Xq + c i pq [X, Y ]q = 0 (19) Lie X V Y Lie Y V X V [X,Y ] = 0. (20) 6

7 4 Dubrovin principal hierarchy In this section, we adapt Dubrovin s construction of the principal hierarchy [4] to the case of F -manifolds with compatible flat connection introduced by Manin in [17]. Definition 7 An F -manifold with compatible flat connection is an F -manifold endowed with a flat torsionless connection satisfying the symmetry condition meaning that c is totally symmetric: for all vector fields X, Y, and Z. l c i jk = jc i lk, (21) ( X c)(y, Z) = ( Y c)(x, Z), (22) Remark 8 Notice that in flat coordinates condition (21) reads l c i jk = jc i lk. This, together with the commutativity of the algebra, implies that c i jk = jc i k = j k C i. Therefore, condition (21) is equivalent to the local existence of a vector field C satisfying, for any pair (X, Y ) of flat vector fields, the condition X Y = [X, [Y, C]]. The above condition appears in the original definition of Manin [17]. Let us construct now the principal hierarchy. In order to do so, the first step consists in defining the primary flows. Since the connection is flat, we can consider a basis (X (1,0),...,X (n,0) ) of flat vector fields; the primary flows are thus defined as Proposition 9 The primary flows (23) commute. u i t (p,0) = c i jk Xk (p,0) uj x. (23) Proof. Since the X (p,0) are flat and the torsion vanishes, they commute and Lie X(p,0) c = X(p,0) c. Therefore the commutativity condition (18), with X = X (p,0) and Y = X (q,0), follows from the symmetry of c. 7

8 Starting from the primary flows (23) one can introduce the higher flows of the hierarchy, defined as u i t (p,α) = c i jk Xj (p,α) uk x, (24) by means of the following recursive relations: j X i (p,α) = c i jkx k (p,α 1). (25) Remark 10 The flatness of the connection, the symmetry of the tensor c (condition (21)) and the associativity of the algebra with structure constants c i jk are equivalent to the flatness of the one-parameter family of connections defined, for any pair of vector fields X and Y, by X Y = X Y + zx Y, z C. (26) The vector fields obtained by means of the recursive relations (25) are nothing but the z- coefficients of a basis of flat vector fields of the deformed connection [4]. In order to show that the higher flows (24) are well-defined, it is necessary to prove the following Proposition 11 The recursive relations (25) are compatible. Proof. We note that the recursive relations (25) can be written in the form thus, we have j X i (p,α) = Γ i jkx k (p,α) c i kjx k (p,α 1), (27) ( j m m j )X(p,α) i = [ m Γ i jl j Γ i ml Γ i jkγ k ml + Γ i mkγjl] k X l (p,α) + [ m c i jl jc i ml Γi kj ck ml Γk lm ci jk + Γi km ck jl + ] Γk lj ci mk X l (p,α 1) + [ c i jk ck ml ci mk ck jl] X l (p,α 2). The flatness of the connection, together with identity (21) and the associativity of the algebra, implies the vanishing of the quantity above. Therefore, relations (25) are compatible. Since the primary flows (23) commute and the recursive relations (25) are compatible, it only remains to prove the following Theorem 12 The flows of the principal hierarchy commute. 8

9 Proof. Let us consider the hydrodynamic flows associated with the vector fields X (p,α) and X (q,β). In order to show that these flows commute, we prove that they satisfy the sufficient condition (19). In local coordinates it reads: X m (p,α)( m c i jk)x k (q,β) X m (q,β)( m c i jk)x k (p,α)+ c l jk( l X i (p,α))x k (q,β) + c i lk( j X l (p,α))x k (q,β)+ +c i jl ( kx l (p,α) )Xk (q,β) + cl jk ( lx i (q,β) )Xk (p,α) + c i lk ( jx l (q,β) )Xk (p,α) + ci jl ( kx l (q,β) )Xk (p,α) + c i jk ( ( l X k (p,α))x l (q,β) + ( l X k (q,β))x l (p,α)) = 0. In particular, if the coordinates are flat, the first row vanishes due to the symmetry of the tensor c. Moreover, using the recursive relations (25) we obtain c l jk ci ln Xn (p,α 1) Xk (q,β) + ci lk cl jn Xn (p,α 1) Xk (q,β) + +c i jlc l knx n (p,α 1)X k (q,β) + c l jkc i lnx n (q,β 1)X k (p,α)+ c i lk cl jn Xn (q,β 1) Xk (p,α) + ci jl cl kn Xn (q,β 1) Xk (p,α) = c i jk ck mn Xn (p,α 1) Xm (q,β) ci jk ck mn Xn (q,β 1) Xm (p,α) which vanishes due to the associativity of the algebra. Remark 13 The flows of the principal hierarchy are well-defined even in the case when the torsion of does not vanish. Unfortunately, their commutativity depends crucially on this additional assumption. 5 F -manifolds with compatible flat connection and flat Egorov metrics The aim of this section is to show that a flat Egorov metric locally defines an F -manifold with compatible flat connection. The special role played by Egorov metrics in the theory of integrable systems of hydrodynamic has been pointed out by several authors (see for instance [3, 20, 18]). Definition 14 A metric is called of Egorov type if there exist coordinates (r 1,...,r n ) such that it is diagonal and potential: for a certain function F. Let us assume that the metric (28) is flat. g ij = δ i j g ii(r 1,...,r n ) = δ i j if, (28) 9

10 Proposition 15 Let g be a flat Egorov metric which is diagonal and potential in some coordinates (r 1,..., r n ), defined on an open set U. Then, the metric g and the structure constants c i jk (r) = δi j δi k, (29) endow U with the structure of semisimple F -manifold with compatible flat connection. Proof. Since the metric is flat and the structure constants (29) define in the tangent spaces T x U a structure of semisimple associative commutative algebra, it remains to show that the condition (21) is verified. In the coordinates (r 1,...,r n ) such a condition reads Γ i jkc k ml Γ k mlc i jk + Γ i mkc k jl + Γ k jlc i mk = Γ i jkδ k mδ k l Γ k mlδ i jδ i k + Γ i mkδ k j δ k l + Γ k jlδ i mδ i k = Γ i jlδ m l Γ i mlδ i j + Γ i mjδ j l + Γi jlδ i m = 0. It is trivially satisfied if all the indices i, j, l, m are distinct. If i = j m l i it becomes Γ i ml = 0 that holds true since the metric g is diagonal in the coordinates (r 1,...,r n ). If i = j m = l or i = m j = l we obtain Γ i il + Γi ll = 0 that holds true since the metric is potential in the coordinates (r 1,...,r n ). All other cases are either similar or they are trivially satisfied. 6 F -manifolds with compatible connection and related integrable systems From the point of view of the theory of integrable systems of hydrodynamic type the flat case and the associated principal hierarchy are an exceptional case. Therefore, it is quite natural to extend the notion of F -manifolds with compatible flat connection to the nonflat case. As a starting point we consider an F -manifold endowed with a connection satisfying (21). If is flat we know how to construct an integrable system of hydrodynamic type. In the construction, presented in Section 4, a crucial role is played by the vanishing of the curvature. Indeed, the starting point of this costruction is a basis of flat vector fields; moreover, the recursive procedure defining the higher vector fields and the corresponding flows is well-defined as a consequence of the vanishing of the curvature. In the non-flat case, in order to define integrable systems of hydrodynamic type one needs to find an alternative way to select the vector fields. 10

11 6.1 Integrable systems associated with F -manifolds We observe that the vector fields X defining the principal hierarchy satisfy the condition ( Z X) W = ( W X) Z (30) for all pairs (Z, W) of vector fields, that is, in local coordinates, c i jm kx m = c i km jx m. (31) Indeed, in the case of the flat vector fields X (p,0) defining the primary flows both sides of (31) vanish due to k X m (p,0) = 0, p = 1,..., n, while in the case of the vector fields defining the higher flows it follows from the associativity of the algebra: c i jm k X m (p,α) = c i jmc m klx l (p,α 1) = c i kmc m jlx l (p,α 1) = c i km j X m (p,α). A crucial remark is the following: if satisfies condition (21), then any pair of solutions of (31) defines commuting flows even if the connection is not flat. More precisely, we have the following Proposition 16 If X and Y are two vector fields satisfying condition (30), then the associated flows u i t = ci jk Xk u j x (32) and commute. u i τ = ci jk Y k u j x (33) Proof. Recall from Proposition 5 that the flows (32) and (33) commute if and only if ((Lie X c)(y, Z) (Lie Y c)(x, Z) + [X, Y ] Z) Z = 0 (34) for any vector field Z. On the other hand, the vanishing of the torsion of gives the identity (Lie X c)(y, Z) = ( X c) (Y, Z) c(y,z) X + c(y, Z X) + c( Y X, Z), and this, together with the symmetry (22) of c, can be used to write the term in the bracket of (34) as Y Z X + X Z Y + [Y, X] Z. Multiplying the above identity by Z, and using property (30) for the vector fields X and Y, we obtain ( Y Z X) Z + ( X Z Y ) Z + [Y, X] Z 2 = The proposition is proved. ( Z X) (Y Z) + ( Z Y ) (X Z) + [Y, X] Z 2 = ( Y X) Z 2 + ( X Y ) Z 2 + [Y, X] Z 2 = 0. 11

12 Remark 17 From (21) and (30) it follows that the (1,1)-tensor field (V X ) i j = ci jk Xk satisfies the condition k (V X ) i j = j (V X ) i k, which is well-known in the Hamiltonian theory of systems of hydrodynamic type [6]. 6.2 Integrability condition In the non-flat case, system (31) might not have solutions. We will discuss the additional constraints we have to impose on in order to avoid this possibility. Proposition 18 If X is a solution of (30), then the identity Z R(W, Y )(X) + W R(Y, Z)(X) + Y R(Z, W)(X) = 0, (35) holds for any choice of the vector fields (Y, W, Z). Proof. Condition (30) implies W (Z Y X Y Z X)+ Y (W Z X Z W X)+ Z (Y W X W Y X) = 0. Using the symmetry condition (21) written in the form Y (X Z) X (Y Z) + Y X Z X Y Z [Y, X] Z = 0 we obtain immediately identity (35). Condition (35) must be satisfied for any solution X of the system (31). Since we are looking for a family of vector fields satisfying (31), it is natural to require that (35) holds true for an arbitrary vector field X. In local coordinates this means that R k lmic n pk + R k lipc n mk + R k lpmc n ik = 0. (36) Remark 19 An equivalent form of condition (36) can be easily obtained using the (second) Bianchi identity for the deformed connection X Y = X Y + zx Y, z C (37) 12

13 (where X and Y are arbitrary vector fields). Indeed, due to associativity and symmetry condition (21), the Riemann tensor of this connection does not depend on z [21]. Using this fact it is easy to see that the Bianchi identity reduces to 0 = X R(Y, Z)(W) + Z R(X, Y )(W) + Y R(Z, X)(W) = X R(Y, Z)(W) + Z R(X, Y )(W) + Y R(Z, X)(W) R(Y, Z)(X W) R(X, Y )(Z W) R(Z, X)(Y W) for any choice of the vector fields (X, Y, W, Z). Hence, condition (35) is equivalent to for every (X, Y, W, Z). R(Y, Z)(X W) + R(X, Y )(Z W) + R(Z, X)(Y W) = 0, From now on we will assume the existence of canonical coordinates (r 1,...,r n ), discussing the meaning of condition (36) under this additional assumption. Proposition 20 In canonical coordinates, system (31) reduces to Γ i ki = where v i are the components of X in such coordinates. kv i v k vi, i k, (38) Proof. In canonical coordinates, (31) reads δ i j ( kv i + Γ i kl vl ) = δ i k ( jv i + Γ i jl vl ). If all the indices are equal such a system is trivially satisfied. In the case i = j k (the case i = k j is completely similar), using the identities Γ i kk = Γ i ki (39) and Γ i jk = 0, i j k i, (40) which follow from (21), we obtain the system (38). Remark 21 In canonical coordinates, the components of the vector field X coincide with the characteristic velocities of the associated system of hydrodynamic type: r i t = c i jkv k r j x = v i r i x, i = 1,...,n. (41) 13

14 Compatibility condition of system (38) is well-known in the literature [22] and is given by the following conditions: i Γ k mk mγ k ik = 0, k i m k, (42) i Γ k km Γk km Γm im + Γk ik Γk km Γk ik Γi im = 0, k i m k (43) Proposition 22 Condition (36) is equivalent to conditions (42) and (43). Proof. In canonical coordinates, condition (36) reads R k lmi cn pk + Rk lip cn mk + Rk lpm cn ik = R k lmi δn p δn k + Rk lip δn m δn k + Rk lpm δn i δn k = R n lmi δn p + Rn lip δn m + Rn lpm δn i = 0. If all the indices m, i, j, p are distinct the above condition is trivially satisfied. Let us consider the case k = p (the case k p can be treated in the same way and does not add further conditions). If k = i we obtain R k lmk + Rk lkm + δk m Rk lkk = 0, that is satisfied due to the skew-symmetry of the Riemann tensor with respect to the exchange of the second and third lower indices. The same if k = m. For k i, m we obtain if l = k and R k kmi = 0 (44) R k lmi = 0 if l k. Since, due to (39), the components of the Riemann tensor vanish if all the indices are distinct, the above condition reduces to R k mmi = 0, k m i k. (45) Condition (44) is equivalent to condition (42). Indeed, we have: R k kmi = m Γ k ik p Γ k mk Γ k ilγ l mk + Γ k mlγ l ik = m Γ k ik p Γ k mk Γ k ikγ k mk + Γ k mkγ k ik = i Γ j pj pγ j ij Using (40) it is easy to check that condition (45) is equivalent to condition (43). 14

15 Remark 23 If the compatibility conditions (42) and (43) are satisfied, the general solution of the system (38) depends on n arbitrary functions of a single variable. Moreover, due to (42), any solution (v 1,...,v n ) of (38) satisfies the condition ( ) ( ) j v i k v i k v j v i j = 0, i j k i, (46) v k v i known in literature as semi-hamiltonian property. An invariant and highly non trivial formulation of such a property has been found in [19]. The previous proposition justifies the following Definition 24 An F -manifold with compatible connection is an F -manifold endowed with a torsionless connection satisfying condition (22) and condition Z R(W, Y )(X) + W R(Y, Z)(X) + Y R(Z, W)(X) = 0 (47) for any choice of the vector fields (X, Y, W, Z). 6.3 Relations with the Hamiltonian formalism In the Hamiltonian theory of integrable hierarchies of hydrodynamic type a special role is played by the Levi-Civita connections associated with metrics g whose Riemann tensor admits a quadratic expansion in terms of the flows of the hierarchy [7, 8]: This means that u i t α = (W (α) ) i ju j x, i = 1,..., n. R sk mi = α ǫ α [(W (α) ) s m(w (α) ) k i (W (α) ) s i(w (α) ) k m], ǫ α = ±1, (48) where the index α can take value on a finite or infinite even continuous set. In our case the flows have the form and the quadratic expansion (48) reads u i t α = c i jk Xk (α) uj x, i = 1,..., n, R sk mi = (cs ml ck iq cs il ck mq ) α ǫ α X l (α) Xq (α), ǫ α = ±1. (49) Proposition 25 Suppose that is the Levi-Civita connection of a metric g, and that its curvature satisfies condition (49). In this case condition (36) is automatically satisfied. 15

16 Proof. Rmic sk n pk + Ripc sk n mk + Rpmc sk n ik = ǫ α [(c s mr ck iq cs ir ck mq )cn pk + (cs ir ck pq cs pr ck iq )cn mk + (cs pr ck mq cs mr ck pq )cn ik ]Xr (α) Xq (α) = α ǫ α [(c k iq cn pk ck pq cn ik )cs mr + (ck pq cn mk ck mq cn pk )cs ir + (ck mq cn ik ck iq cn mk )cs pr ]Xr (α) Xq (α) α which vanishes due to associativity. Remark 26 If the functions g lq := α ǫ α X l (α)x q (α) define the contravariant components of a metric satisfying the condition then the operator M ǫ α (w α ) i k uk x α=1 g lq c i qp = g pq c i ql, ( ) 1 d (w α ) j h dx uh x, (w α) i j := ci jk Xk (α) is a purely nonlocal Poisson operator (see [11] for details). 7 An example: reductions of the dispersionless KP hierarchy In Section 5 we have proved that a flat Egorov metric, satisfying condition (21), locally defines an F -manifold structure with compatible flat connection. This construction can be extended to the case where the Egorov metric is non-flat; however, in order to have a compatible structure, one has to introduce an additional hypothesis on the curvature of the metric, which is required to satisfy condition (36). In this section we will consider a class of F - manifolds associated with a well-known class of hydrodynamic type systems: the reductions of the dispersionless KP hierarchy. For a generic reduction, the metric will be non-flat. The dispersionless KP (or dkp) hierarchy can be defined by introducing the formal series A k λ = p + pk+1, (50) k=0 which has to satisfy the following dispersionless Lax equations λ tn = {λ, 1n } (λn ) +. (51) 16

17 Here {f, g} = x f p g p f x g denotes the canonical Poisson bracket, and ( ) + is the polynomial part of the argument. For simplicity, we will consider here only the second flow (n = 2); all other flows of the hierarchy can be treated in the same way. For the second flow, we have λ t2 = {λ, 12 } p2 + A 0 = pλ x A 0 x λ p, (52) or, explicitly in terms of the variables A k, A k t 2 = A k+1 x + ka k 1A 0 x, k = 0, 1, 2,... (53) This last system is also known in the literature as Benney chain [1]; its Lax representation (52) appeared for the first time in [16]. An n-component reduction of the Benney chain is a restriction of the infinite dimensional system (53) to a suitable n-dimensional submanifold, that is A k = A k (u 1,...,u n ), k = 0, 1,... The reduced systems are systems of hydrodynamic type in the variables (u 1,..., u n ) that parametrize the submanifold: u i t = v i j(u)u j x, i = 1,...,n. Reductions of the Benney system were introduced in [12], and there it was proved that such systems are diagonalizable and integrable via the generalized hodograph transformation [22]. Clearly, in the case of a reduction, the coefficients of the series (50) depend on the Riemann invariants (r 1,...,r n ) and the series can be thought as the asymptotic expansion for p of a suitable function λ(p, r 1,...,r n ) depending piecewise analytically on the parameter p. It turns out [12, 13] that such a function satisfies a system of chordal Loewner equations, λ r i = ia 0 p v i λ p, i = 1...,n, (54) describing families of conformal maps (with respect to p) in the complex upper half plane. The analytic properties of λ characterize the reduction. More precisely, in the case of an n- reduction the associated function λ possesses n distinct critical points on the real axis; these are the characteristic velocities v i of the reduced system, that is, λ p (v i ) := λ p (vi ) = 0, i = 1,...,n, and the corresponding critical values can be chosen as Riemann invariants. Compatibility conditions of the Loewner system (54) are of the form i v j = ia 0 v i v j i j, (55) 2 ija 0 = 2 ia 0 j A 0 (v i v j ) 2 17

18 and were found by Gibbons and Tsarev [13]. Thus, every reduction of the Benney chain is described by a particular solution of the Loewner system (54). Starting from the function λ, we will show now how to give to the manifold parametrized by the Riemann invariants (r 1,...,r n ), a structure of F -manifold with a compatible connection in general non-flat. In order to do this, we define a metric and structure constants g(, ) = c(,, ) = n i=1 n i=1 res p=v i res p=v i ( λ(p) λ(p) λ p ( λ(p) λ(p) λ(p) λ p ) dp, (56) ) dp, (57) where,, are arbitrary tangent vectors on the manifold. In the coordinates (r 1,...,r n ), and making use of the Loewner equations (54), the metric takes the diagonal form ( ) n ( ) λ λ dp n ( ) i A 0 j A 0 λ p dp g r i, = res = res r j p=v i r i r j λ p p=v i (p v i )(p v j ) i=1 i=1 = i A 0 j A 0 λ pp (v i ) δ ij = i A 0 δ ij, where we used the fact [10] that λ pp (v i ) = 1 i A 0. In particular, the metric is Egorov. Moreover, a similar calculation for the structure constants gives ( ) c r i, r, n ( ) ( ) λ λ λ dp n i A 0 j A 0 k A 0 (λ p ) 2 dp = res = res j r k p=v i r i r j r k λ p p=v i (p v i )(p v j )(p v k ) i=1 i=1 = i A 0 j A 0 k A 0 ( λ pp (v i ) )2 δ ij δ ik = i A 0 δ ij δ ik, and from this it follows that r i r = δ j ij r i, namely (r 1,...,r n ) are canonical coordinates for the algebra. Remark 27 The metric (56) and the structure constants (57) were introduced for the first time by Dubrovin in [4], in the particular case of the Gelfand-Dikii reductions of the dkp hierarchy, where the function λ is a polynomial in p. The same metric and constants were also used by Chang [2] and Ferguson and Strachan [9], for the study of reductions where λ is rational or logarithmic. We remark that in all these examples the metric considered turns out to be flat. 18

19 We have now to prove that the metric and the structure constants defined in this way are compatible, namely that conditions (21) and (36) are satisfied. As regard condition (21) due to the results of Section 5 it is sufficient to note that the metric (56) is Egorov. On the other hand, for condition (36), we only have to recall the result of [10], where the curvature tensor of the metric (56) has been shown to possess the following quadratic expansion: R ij ij = 1 2πi C w i (λ) w j (λ) dλ, w i (λ) = p λ (p(λ) v i ) 2, where p(λ) = λ 1 (p) is the inverse of λ with respect to p, and C is a suitable contour on the complex λ-plane. Due to Proposition 25, the existence of a quadratic expansion of the curvature implies that condition (36) is satisfied. Alternatively, such a condition follows from the well-known fact that the characteristic velocities v i which satisfy condition (38) satisfy the semi-hamiltonian condition (46). Remark 28 A similar construction can be done using instead of the metric (56), one of the metrics ( ) n ( ) λ g r i, = res ϕ r j i (r i λ dp ) (58) p=v i r i r j λ p i=1 where ϕ i are arbitrary functions of a single variable, and defining the corresponding structure constants as ( ) c r i, r, n ( ) λ = res (ϕ j r k i (r i )) 2 λ λ dp. (59) p=v i r i r j r k λ p i=1 If all the functions ϕ i are different from zero, it turns out that the structure constants (59) admit canonical coordinates. Moreover, in such coordinates, the metric (58) is potential. In this way, repeating the construction described in this section, one defines, for any choice of the functions ϕ i, a new structure of F -manifold with compatible connection on the same manifold. Notice that in case of reductions related to Frobenius manifolds, like Zakharov and Gel fand-dikii reductions, among the metrics (58), there is the intersection form and the construction above reduces to the Dubrovin s duality [5]. Acknowledgments We thank Maxim Pavlov for stimulating discussions. M.P. and A.R. would like to thank the Department Matematica e Applicazioni of the Milano-Bicocca University for the hospitality. Support of A.R. through the ESF grant MISGAM 2265 for his visit to the Milan-Bicocca department is gratefully acknowledged. References [1] D.J. Benney, Some properties of long nonlinear waves, Stud. Appl. Math. 52 (1973),

20 [2] Jen-Hsu Chang, On the waterbag model of the dispersionless KP hierarchy (II), J. Phys. A: Math. Theor. 40 (2007), [3] B.A. Dubrovin, On the differential geometry of strongly integrable systems of hydrodynamics type, (Russian) Funktsional. Anal. i Prilozhen. 24 (1990), no. 4, 25 30, 96; translation in Funct. Anal. Appl. 24 (1990), no. 4, (1991). [4] B.A. Dubrovin, Geometry of 2D topological field theories, in: Integrable Systems and Quantum Groups, Montecatini Terme, Editors: M. Francaviglia, S. Greco. Springer Lecture Notes in Math (1996), pp [5] B.A. Dubrovin, On almost duality for Frobenius manifolds, in: Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 212, Amer. Math. Soc., Providence, RI, 2004, pp [6] B.A. Dubrovin, S.P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk 44 (1989), English translation in Russ. Math. Surveys 44 (1989), [7] E.V. Ferapontov, Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funct. Anal. Appl. 25 (1991), no. 3, [8] E.V. Ferapontov, O.I. Mokhov, Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Russ. Math. Surv. 45 (1990), no. 3, [9] J.T. Ferguson, I.A.B Strachan, Logarithmic deformations of the rational superpotential/landau-ginzburg construction of solutions of the WDVV equations, Comm. Math. Phys. 280 (2008), [10] J. Gibbons, P. Lorenzoni, A. Raimondo, Hamiltonian structure of reductions of the Benney system, Comm. Math. Phys. 287 (2009), , [11] J. Gibbons, P. Lorenzoni, A. Raimondo, Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type, arxiv: [12] J. Gibbons, S.P. Tsarev, Reductions of the Benney equations, Phys. Lett. A 211 (1996), no. 1, [13] J. Gibbons, S.P. Tsarev, Conformal maps and reductions of the Benney equations, Phys. Lett. A 258 (1999), no. 4-6, [14] C. Hertling, Y. Manin, Weak Frobenius manifolds, Internat. Math. Res. Notices 1999, no. 6, [15] B.G. Konopelchenko, F. Magri, Coisotropic deformations of associative algebras and dispersionless integrable hierarchies, Comm. Math. Phys. 274 (2007),

21 [16] D. Lebedev, Y. Manin, Conservation laws and Lax representation of Benney s long wave equations, Phys. Lett. A 74 (1979), [17] Y. Manin, F -manifolds with flat structure and Dubrovin s duality, Adv. Math. 198 (2005), no. 1, [18] M.V. Pavlov, Integrability of Egorov systems of hydrodynamic type, (Russian) Teoret. Mat. Fiz. 150 (2007), no. 2, ; translation in Theoret. and Math. Phys. 150 (2007), no. 2, [19] M.V. Pavlov, S.I. Svinolupov, R.A. Sharipov, An invariant criterion for hydrodynamic integrability, (Russian) Funktsional. Anal. i Prilozhen. 30 (1996), no. 1, 18 29, 96; translation in Funct. Anal. Appl. 30 (1996), no. 1, [20] M.V. Pavlov, S.P. Tsarev, Tri-Hamiltonian structures of Egorov systems of hydrodynamic type, (Russian) Funktsional. Anal. i Prilozhen. 37 (2003), no. 1, [21] I.A.B Strachan, Frobenius manifolds: natural submanifolds and induced bi- Hamiltonian structures, Differential Geom. Appl. 20 (2004), no. 1, [22] S.P. Tsarev, The geometry of Hamiltonian systems of hydrodynamic type. The generalised hodograph transform, USSR Izv. 37 (1991) [23] V.E. Zakharov, Benney equations and quasiclassical approximation in the inverse problem, Funktional. Anal. i Prilozhen 14 (1980),