Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems

Recipro cal transfor mations and lo cal Hamiltonian st ructures of h ydro dynamic t yp e systems Simonetta Ab enda Dipartimen to di Matematica and C.I.R.A.M. Univ ersit` a degli Studi di Bologna, Italy aben da@ciram.unib o.it No v em b er 7, 2018 Abstract W e start from a hyp e rb olic DN h ydro dy namic type sy s tem of dimension n which po ssesses Riemann inv ariants and we se ttle the necessary conditions o n the conser- v ation laws in the recipr o cal tra nsformation so that, after such a transformation of the indep endent v ariables, one of the metrics asso ciated to the initial system be flat. W e prov e the follo wing statement : let n ≥ 3 in the case of recipro cal transformations of a sing le indep endent v aria ble or n ≥ 5 in the case of transformations of b oth the independent v aria ble ; then the recipro cal metric may b e flat only if the co nserv ation laws in the transformation ar e linear com binations o f the canonical densities of conser - v ation laws, i.e the Casimirs, the momen tum and the Hamiltonia n densities asso ciated to the Hamiltonian op erator for the initial metric. Then, we restrict ourselves to the case in which the initial metric is either flat or of constant curv ature and we classify the recipro cal tr ansformatio ns of one or b oth the indep endent v ariables so that the recipro ca l metric is fla t. Such c har acterizatio n has an interesting geometric interpre- tation: the hyper surfaces o f tw o diago nalizable DN sy stems of dimension n ≥ 5 are Lie equiv alen t if and only if the corresp onding local hamiltonian structures a re related by a canonica l rec ipr o cal transformation. 1 In tro du c tion Systems of h ydro dynamic t yp e are qu asilinear ev olutionary hyperb olic PDEs of the form u i t = n X k =1 v i k ( u ) u k x , u = ( u 1 , . . . , u n ) , u i x = ∂ u i ∂ x , u i t = ∂ u i ∂ t . (1) They natur ally arise in app lications such as gas dynamics, hydrod ynamics, chemical ki- netics, the Whitham a verag ing pro cedur e, differentia l geometry and top ological field the- ory [7, 9, 4, 21, 22]. Dubro vin and No vik o v [7] sho we d that (1) is a local Hamiltonian sys- tem (DN system) with Hamiltonian H [ u ] = R h ( u ) dx , if there exists a flat non-degenerate metric tensor g ( u ) in R n with Christoffel symb ols Γ i j k ( u ), suc h that the matrix v i k ( u ) can b e represent ed in the form v i k ( u ) = n X k =1 g il ( u ) ∂ 2 h ∂ u l ∂ u k ( u ) − n X s =1 g ik ( u )Γ l sk ( u ) ∂ h ∂ u l ( u ) ! . (2) 1 In this pap er w e shall consider DN systems whic h p ossess Riemann in v arian ts, i.e. they ma y b e transformed to the d iagonal form u i t = v i ( u ) u i x , i = 1 , . . . , n , (3) with n ≥ 3 and with v i ( u ) all r eal and distinct (strict h yp erb olicit y prop erty). W e also su pp ose to w ork in the space of smo oth and rapidly decreasing functions so that ( d dx ) − 1 f x = f . If n = 2, (1) can alwa y s b e put in diagonal form and are in tegrable by the ho dograph metho d. F or arbitrary n , Tsarev [21] p r o v ed that a DN s y s tem as in (1), (2) can b e in te- grated b y a generaliz ed ho dograph metho d only if it m a y b e transformed to th e d iagonal form. In the latter case, moreo ver the fl at metric is diagonal, the Hamiltonia n satisfies ∂ 2 h ∂ u i ∂ u j = Γ i ij ( u ) ∂ h ∂ u i + Γ j j i ( u ) ∂ h ∂ u j , (4) and eac h solution to (4) generates a conserv ed quan tit y for the DN system (1), (2) and all Hamiltonian flo ws generated by these conserve d d ensities pairwise commute. As a conse- quence, for n ≥ 3, DN systems wh ic h p ossess Riemann in v arian ts are alw a ys integrable. W e recall that there d o also exist DN systems with an infinite n umb er of conserv ed quan- tities w hic h do not p ossess Riemann in v arian ts (see F erap ont ov [11] for th e classificatio n of the latter wh en n = 3). Since a non-degenerate flat diagonal metric in R n is asso ciated to an orthogonal co or- dinate system u i = u i ( x 1 , . . . , x n ), there is a natural link b et wee n d iagonaliza ble Hamilto- nian systems and n –orthogonal cur vilinear co ordinates in fl at spaces. Up on introdu cing the Lam ´ e coefficients, whic h in our case tak e th e form H 2 i ( u ) = X k  ∂ x i ∂ u k  2 , the m etric tensor in the co ordin ate system u i is d iagonal ds 2 = n P i =1 H 2 i ( u )( du i ) 2 , and the zero curv ature conditions R il,im ( u ) = 0 ( i 6 = l 6 = m 6 = i ) and R il,il ( u ) = 0 ( i 6 = l ) form an o v erdetermined sy s tem: ∂ 2 H i ∂ u l ∂ u m = 1 H l ∂ H l ∂ u m ∂ H i ∂ u l + 1 H m ∂ H m ∂ u l ∂ H i ∂ u m , (5) ∂ ∂ u l ∂ H i H l ∂ u l + ∂ ∂ u i ∂ H l H i ∂ u i + X m 6 = i,l 1 H 2 m ∂ H i ∂ u m ∂ H l ∂ u m = 0 . (6) Bianc hi and Cartan show ed that a general solution to the zero curv ature e qu ations (5), (6) can b e parametrized locally b y n ( n − 1) / 2 arbitrary f u nctions of t w o v ariables. If the Lam ´ e co efficien ts H i ( u ) are known, one can fi nd x i ( u 1 , . . . , u n ) solving the linear o v erdetermined problem (em b edd ing equ ations) ∂ 2 x i ∂ u k ∂ u l = Γ k k l ( u ) ∂ x i ∂ u k + Γ l lk ( u ) ∂ x i ∂ u l , ∂ 2 x i ∂ ( u l ) 2 = X k Γ k ll ( u ) ∂ x i ∂ u k . (7) 2 Comparison of Eqs. (4) and (7) implies that the flat co ordin ates for the metric g ii ( u ) = ( H i ( u )) 2 are the Casimirs of the corresp ondin g Hamiltonian op erator. Finally , Zakharo v [24 ] sho wed that the d ressing metho d ma y b e used to determine the solutions to the zero cur- v ature equations up to Com b escur e tran s formations. It then follo ws that the classification of flat d iagonal metrics ds 2 = g ii ( u )( du i ) 2 is an imp ortan t preliminary step in the classification of in tegrable Hamiltonian systems of h ydr o dynamic t yp e. Best kno wn examples of in tegrable Hamiltonian systems of hydrod y- namic t yp e p ossess Riemann in v arian ts, a pair of compatible flat metrics an d ha v e b een obtained in the framewo rk of semisimple F rob enius manifolds (axiomatic theory of inte - grable Hamilto nian sys tems) [4 , 5, 6]; in the latter case, one of the flat metrics is also Egoro v ( i.e. its rotation coefficients are s ymmetric). Recipro cal transf ormations c hange the ind ep endent v ariables of a system and are an imp ortant c lass of nonlo cal transformations wh ic h act on hydro dynamic–t yp e systems [20, 19, 12, 13, 1, 23, 2]. Recipro cal transformations map conserv ati on la ws to conserv at ion la ws and map diagonaliz able systems to diagonali zable systems, b ut act non trivially on th e metrics and on the Hamiltonian structures: for in stance, th e flatness p rop erty and the Egoro v prop erty for metrics as well as the lo calit y of the Hamiltonian structure are n ot preserve d, in general, by su c h tr ansformations. Th en, it is n atural to in ve stigate under w hic h add itional h yp otheses the recipro cal system still p ossesses a lo cal Hamilto nian structure, our ultimate goal b eing the searc h for new examples of in tegrable Hamiltonian systems and th e geometrical charac terization of the asso ciated hypersu rfaces. With th is in m ind, in the follo wing w e start from a smo oth inte grable Hamiltonian system in Riemann inv ariant form u i t = v i ( u ) u i x , i = 1 , . . . , n , (8) with smo oth conserv ation laws B ( u ) t = A ( u ) x , N ( u ) t = M ( u ) x (9) with B ( u ) M ( u ) − A ( u ) N ( u ) 6 = 0. In the new indep endent v ariables ˆ x and ˆ t defined by d ˆ x = B ( u ) dx + A ( u ) dt, d ˆ t = N ( u ) dx + M ( u ) dt, (10) the recipro cal system is s till diagonal and tak es the form u i ˆ t = B ( u ) v i ( u ) − A ( u ) M ( u ) − N ( u ) v i ( u ) u i ˆ x = ˆ v i ( u ) u i ˆ x . (11) Moreo v er, the metric of the initial systems g ii ( u ) transforms to ˆ g ii ( u ) =  M ( u ) − N ( u ) v i ( u ) B ( u ) M ( u ) − A ( u ) N ( u )  2 g ii ( u ) , (12) and all conserv ation laws and comm uting flo ws of th e original system (8 ) ma y b e recalcu- lated in the new indep endent v ariables. If the recipro cal transf ormation is linear ( i.e. A, B , N , M are constan t functions), then the recipro cal to a flat metric is still flat and localit y and compatibilit y of the asso ciated Hamiltonian structures are p reserv ed (see Refs. [22, 19, 23]). 3 Under a general recipro cal transformation, the Hamiltonian str u cture do es not b eha v e trivially and a thorough study of recipro cal Hamiltonian structur es is still a n op en p roblem. F erap onto v and Pa vlo v [13] construct the recipro cal Riemannian curv ature tensor and the recipro cal Hamiltonian op erator when the initial metric is flat, while in [2], w e construct the recipro cal Riemannian curv ature tensor a nd the recipro cal Hamiltonian operator when the initial m etric is not flat and the initial system also p ossesses a flat metric. The classificati on of the reciprocal Hamiltonian structures is also complicated by the fact that a DN system as in (1)-(2) also p ossesses an in finite n umber of nonlo cal Hamil- tonian stru ctures [17, 12, 16, 15]. It is then p ossible that t w o DN systems are linked by a recipro cal transformation and that the flat metrics of the first system are not recipro cal to the flat metrics of the second. In [1], we constructed suc h an example: the gen us one mo dulation (Whitham-CH) equations asso ciated to Camassa-Holm in Riemann inv ariant form ( n = 3 in (8)). W e p ro v ed that the Whitham-CH equations are a DN-system and p ossess a pair of compatible flat metrics (none of the metrics is Egorov) . W e also pro v ed the conn ection via a recipro cal trans formation of the Whitham-CH equations to the mo d- ulation equations asso ciated to the fir st negativ e flow of the Kortew eg de V ries hierarc hy (Whitham-KdV − 1 ). In [1], finally w e also found the relation b et w een the Poisso n s tr uc- tures of the Whitham-KdV − 1 and the Whitham-CH equations: b oth systems p ossess a pair of compatible flat metrics, and the t wo flat metrics of the first system are resp ectiv ely recipro cal to the constan t curv ature and conformally flat metrics of the second (and vice v ersa). In view of the ab o ve results, in [2] w e ha v e started to classify th e recipro cal transfor- mations whic h transform a DN system to a DN system, under th e condition that the flat metric tensor ˆ g ( u ) of the transformed system is recipro cal to a m etric tensor g ( u ) of the initial system, which is e ither fl at or of co ns tant Riemannian curv at ur e or conformally flat. In [2], we giv e necessary an d su fficien t conditions so that a recipro cal transformation whic h change s only one indep end ent v ariable m a y pr eserv e the flatness of the metric; in particular, we sho w that the conserv ation la ws in the recipro cal transformation of the indep en d en t v ariable x (resp. t ) are linear combinatio ns of Casimirs and momentum densities (resp. Casimirs and Hamiltonia n densities). F or an easier comparison with th e results known in literature, w e recall that F erap on- to v [12] tak es a recipro cal tr ansformation wher e the conserv ation la ws in (10) are a linear com bination of th e Casimirs, momen tum and Hamiltonian densities and giv es the n eces- sary and sufficien t conditions so that starting from a flat metric g ( u ), the recipro cal metric ˆ g ( u ) b e either a flat or a constant curv ature metric. F oll owing F erap onto v [11, 12], w e call canonical a recipro cal t rans f ormation in wh ic h the integrals in (10) are linear com binations of the n + 2 canonical integral s (Casimirs, Hamiltonian and momen tum) with resp ect to the giv en Hamiltonian structure. The results in [2, 12] suggest that canonical recipro cal transform ations ha ve a priv- ileged role in preserving lo calit y of the Hamiltonian structur e. In this pap er w e show that canonical transform ations are indeed the only recipro cal transformations whic h ma y transform th e in itial metric g ii ( u ) into a recipro cal fl at metric ˆ g ii ( u ) wh en the dimension of the system is n ≥ 3 (in the case of a transf ormation of a single indep enden t v ariable) or n ≥ 5 (in the case of a transformation of b oth the ind ep endent v ariables) . First of all, in Theorems 3.2 and 3.5, w e give necessary conditions on th e in itial m etric g ii ( u ) and on the conserv ation la ws (9) in th e recipro cal transform ation, so that the 4 recipro cal metric (12) b e flat. W e su p p ose that th e initial system (8) is a DN system whic h p ossesses Riemann in v arian ts and w e let g ii ( u ) b e one of the metrics asso ciated to it. Under suc h h yp otheses, w e pro v e th at if the recipr o cal metric ˆ g ii ( u ) in (12) is flat, then the recipro cal transformation is canonical for the initial metric g ii ( u ). Then, w e restrict our s elv es to the case in whic h the initial metric is either flat or of constan t curv ature and, in Theorem 4.1, w e classify the recipro cal transformations of one or b oth the indep endent v ariables so that the recipro cal metric b e flat. Finally , when b oth the initial and th e transformed metrics are fl at, w e also discuss the ge ometric in tepretation of the latter Theorem in view of the resu lts obtained by F er ap on to v in [11]. Indeed, in Theorem 4.12 we sho w that, the h yp ersurfaces of t wo diagonalizable DN systems are Lie equiv ale nt if and only if the corresp onding lo cal h amiltonian s tructures are related by a canonical recipro cal transformation whic h satisfies Theorem 4.1. There are of course still man y op en problems conn ected to the classificatio n of local Hamiltonian stru ctur es: w hat ab out the p ossible role of other t yp es of transformations among h ydro d ynamic t yp e systems? What is the geometrical meaning of the conditions settled b y Theorems 3. 2, 3.5 and 4.1 when th e initial metric is n ot flat? Moreo v er, there do exists non–diagonalizable inte grable Hamiltonian systems; it w ould b e in teresting to chec k whether the s ame conditions on the conserv atio n la ws in the recipro cal transform ations preserving the lo calit y of th e Hamiltonian str u cture still h old also in that case. Finally , several systems of ev olutionary PDEs arising in physics ma y b e written as p ertur bations of h yp erb olic systems of PDEs and their classification in case of Hamiltonian p ertur bations h as recen tly b een started b y Dubr o vin, Liu and Zhang [10]. It w ould also b e inte resting to in ve stigate the role of r ecipr o cal transformations in this p erturbation sc heme. The plan of th e pap er is as follo ws. In the next section, we in tro d uce the necessary definitions and we recall some theorems w e pro ve d in [2] on the form of the reciprocal Riemannian cur v ature tensor and of the recipro cal Hamiltonian op erator. In section 3, w e pro ve the necessary conditions on the form of the Riemannian curv ature tensor and the conserv ati on la ws in the recipro cal transformation so that the recipro cal metric b e fl at. Finally in section 4, we classify the recipro cal transformation which preserv e the fl atness of the metric or whic h transform a constan t cu r v ature metric to a flat one and w e apply suc h conditions to some examples. 2 The recipro cal Hamiltonian stru c ture In this section w e in tro duce some u seful not ations, we d iscuss th e role of additiv e constan ts in the extended recipro cal transf orm ations and we recall some theorems w e pro v ed in [2] whic h w e shall use in the follo wing sections. W e consid er a smo oth DN Hamiltonian h ydro dynamic system in Riemann in v arian ts u i t = v i ( u ) u i x , i = 1 , . . . , n , (13) with v i ( u ) all real and distinct (strict hyp erb olicit y prop ert y). Let g ii ( u ) b e a (co v ariant) non–degenerate diagonal m etric suc h that for conv enien t f i ( u i ), i = 1 , . . . , n , g ii ( u ) f i ( u i ) is a flat metric associated to the local Hamiltonian op erator of the system (13). Let g ii ( u ) = 1 /g ii ( u ). Let H i ( u ), β ij ( u ) and Γ i j k ( u ) resp ectiv ely b e the Lam ´ e coefficien ts the 5 rotation co efficients and the Christoffel sym b ol of a diagonal non-degenerate metric g ii ( u ) asso ciated to (13), H i ( u ) = p g ii ( u ) , β ij ( u ) = ∂ i H j ( u ) H i ( u ) , i 6 = j, Γ i j k ( u ) = 1 2 g im ( u )  ∂ g mk ( u ) ∂ u j + ∂ g mj ( u ) ∂ u k − ∂ g k j ( u ) ∂ u m  . Since the m etric is diagonal, the only non–zero Christoffel sym b ols are Γ j ii ( u ) = − H i ( u ) H 2 j ( u ) ∂ j H i ( u ) , ∀ i 6 = j, Γ i ij ( u ) = ∂ j H i ( u ) H i ( u ) , ∀ i, j = 1 , . . . , n. Under our h yp otheses, the system (13) p ossesses at least one fl at metric. Then, for any other metric associated to (13), the Euler–Darb oux equations (6) still h old, ∂ k β ij ( u ) − β ik ( u ) β k j ( u ) ≡ 0 , i 6 = j 6 = k , that is R ij ik ( u ) ≡ 0, ( i 6 = j 6 = k 6 = i ). F or systems (13), F erap ont o v [12] constructs th e non–lo cal Hamiltonian op erators J ij ( u ) asso ciated to non-flat metrics g ii ( u ) whic h tak e the form J ij ( u ) = g ii ( u )  δ i j d dx − Γ j ik ( u ) u k x  + X l ǫ ( l ) w i ( l ) ( u ) u i x  d dx  − 1 w j ( l ) ( u ) u j x , (14) where ǫ l = ± 1, w i ( l ) ( u ) are affinors of the metric whic h satisfy ∂ j w i ( l ) ( u ) w j ( l ) ( u ) − w i ( l ) ( u ) = ∂ j v i ( u ) v j ( u ) − v i ( u ) = ∂ j ln H i ( u ) , (15) and the cur v ature tensor of the metric tak es the form R ik ik ( u ) = − ∆ ik ( u ) H i ( u ) H k ( u ) ≡ X ( l ) ǫ l w i ( l ) ( u ) w k ( l ) ( u ) , i 6 = k , (16) where ∆ ik ( u ) = ∂ i β ik ( u ) + ∂ k β k i ( u ) + X m 6 = i,k β mi ( u ) β mk ( u ) . Remark 2.1 In p art icular, if g ii ( u ) is flat, then J ij ( u ) = g ii ( u )  δ i j d dx − Γ j ik ( u ) u k x  [7]. If g ii ( u ) is of c onstant curvatur e c , then [17] J ij ( u ) = g ii ( u )  δ i j d dx − Γ j ik ( u ) u k x  + cu i x  d dx  − 1 u j x . (17) 6 If g ii ( u ) is c onforma l ly flat, then R ij ij ( u ) = w i ( u ) + w j ( u ) , i 6 = j, (18) and J ij ( u ) = g ii ( u )  δ i j d dx − Γ j ik ( u ) u k x  + w i ( u ) u i x  d dx  − 1 u j x + u i x  d dx  − 1 w j ( u ) u j x . (1 9) In the next se ction a sp e cial r ole is playe d by the metrics g ii ( u ) for which the Rieman nian curvatur e tensor takes the sp e cial form R ik ik ( u ) = w i (1) ( u ) + w k (1) ( u ) + w i (2) ( u ) v k ( u ) + w k (2) ( u ) v i ( u ) , i 6 = k , (20) and J ij ( u ) = g ii ( u )  δ i j d dx − Γ j ik ( u ) u k x  + w i (1) ( u ) u i x  d dx  − 1 u j x + u i x  d dx  − 1 w j (1) ( u ) u j x + w i (2) ( u ) u i x  d dx  − 1 v j ( u ) u j x + v i ( u ) u i x  d dx  − 1 w j (2) ( u ) u j x (21) Giv en smo oth conserv ation laws B ( u ) t = A ( u ) x , N ( u ) t = M ( u ) x for the system (13), a recipro cal transf ormation of the indep endent v ariables x, t is d efined b y [20] d ˆ x = B ( u ) dx + A ( u ) dt, d ˆ t = N ( u ) dx + M ( u ) dt. (22) In [13], F erap onto v and P a vlo v h a v e charact erized the tensor of the recipro cal Riemannian curv ature and th e recipro cal Hamiltonian structure wh en the initial metric g ii ( u ) is flat. In [2], we ha ve computed the Riemannian curv ature and the Hamiltonian structure of the recipro cal system u i ˆ t = ˆ v i ( u ) u i ˆ x = B ( u ) v i ( u ) − A ( u ) M ( u ) − N ( u ) v i ( u ) u i ˆ x , (23) asso ciated to the recipro cal metric ˆ g ii ( u ) =  M ( u ) − N ( u ) v i ( u ) B ( u ) M ( u ) − A ( u ) N ( u )  2 g ii ( u ) , (24) with g ii ( u ) non-flat. In the follo wing, w e use the sym b ols ˆ H i ( u ), ˆ β ij ( u ), ˆ Γ i j k ( u ), ˆ R ij k m ( u ) and ˆ J ij , resp ectiv ely , for the Lam ´ e co efficien ts, the rotatio n co efficients, the Christoffel sym b ols, the Riemannian curv ature tensor and the Hamiltonian op erator asso ciated the recipro cal metric ˆ g ii ( u ). T o simplify notations, we drop the u dep endence in the length y form ulas. 7 Theorem 2.2 [2] L et g ii ( u ) b e the c ovariant diagonal metric as ab ove for the Hamilto- nian system (13) with R iemannian curvatur e tensor as in (16) or as in (20). Then, for the r e cipr o c al metric ˆ g ii ( u ) define d in (24 ), the only p ossible non-zer o c omp onents of the r e cipr o c al Riemannia n curvatur e tensor ar e ˆ R ik ik ( u ) = H i H k ˆ H i ˆ H k R ik ik − ( ∇ B ) 2 + H k ˆ H k ∇ k ∇ k B + H i ˆ H i ∇ i ∇ i B − ˆ v k ˆ v i ( ∇ N ) 2 + ˆ v k H i ˆ H i ∇ i ∇ i N + ˆ v i H k ˆ H k ∇ k ∇ k N − ( ˆ v k + ˆ v i ) < ∇ B , ∇ N >, i 6 = k (25) wher e < ∇ B ( u ) , ∇ N ( u ) > = X m g mm ( u ) ∂ m B ( u ) ∂ m N ( u ) , ∇ i ∇ i B ( u ) = g ii ( u ) ∂ 2 i B ( u ) − X m Γ m ii ( u ) ∂ m B ( u ) ! , ∇ i ∇ j B ( u ) = g ii ( u )  ∂ i ∂ j B ( u ) − Γ i ij ( u ) ∂ i B ( u ) − Γ j ij ( u ) ∂ j B ( u )  . In [2], we computed the recipro cal affin ors and the r ecipro cal Hamiltonian op erator of a h ydro dynamic system (13) with (nonlocal) Hamiltonian op erator (14). At this aim, w e in tro d uce th e auxiliary flows u i τ = n i ( u ) u i x = J ij ( u ) ∂ j N ( u ) , u i ζ = b i ( u ) u i x = J ij ( u ) ∂ j B ( u ) , (26) u i t ( l ) = w i ( l ) ( u ) u i x = J ij ( u ) ∂ j H ( l ) ( u ) , resp ectiv ely , generated b y the d en sities of conserv ation la ws asso ciated to the recipro cal transformation (22), B ( u ), N ( u ), and b y the den sities of conserv ation la ws H ( l ) ( u ) asso- ciated to the affinors w i ( l ) ( u ) of the R iemann ian curv ature tensor (16). By construction, all the au x iliary fl ows comm ute with (13). Intro ducing the follo win g closed f orm            d ˆ x = B ( u ) dx + A ( u ) dt + P ( u ) dτ + Q ( u ) dζ + X l T ( l ) ( u ) dt ( l ) , d ˆ t = N ( u ) dx + M ( u ) dt + R ( u ) dτ + S ( u ) dζ + X l Z ( l ) ( u ) dt ( l ) , d ˆ τ = dτ , d ˆ ζ = dζ , d ˆ t ( l ) = dt ( l ) , (27) it is easy to verify that the recipro cal auxiliary flo ws u i ˆ τ = ˆ n i ( u ) u i ˆ x , u i ˆ ζ = ˆ b i ( u ) u i ˆ x , u i ˆ t ( l ) = ˆ w i ( l ) ( u ) u i ˆ x , satisfy ˆ n i ( u ) = n i B − P + ( N n i − R ) ˆ v i = H i ˆ H i n i − P − ˆ v i R, ˆ b i ( u ) = b i B − Q + ( N b i − S )ˆ v i = H i ˆ H i b i − Q − ˆ v i S, ˆ w i ( l ) ( u ) = w i ( l ) B − T ( l ) + ( N w i ( l ) − Z ( l ) ) ˆ v i = H i ˆ H i w i ( l ) − T ( l ) − ˆ v i Z ( l ) . (28) 8 Using (27), w e immediately conclud e that T ( l ) ( u ), Z ( l ) ( u ) satisfy n i ( u ) = ∇ i ∇ i N + X ( l ) ǫ ( l ) Z ( l ) w i ( l ) , b i ( u ) = ∇ i ∇ i B + X ( l ) ǫ ( l ) T ( l ) w i ( l ) . (29) Moreo v er, w e ha v e v i ( u ) = ∂ i A ( u ) ∂ i B ( u ) = ∂ i M ( u ) ∂ i N ( u ) , w i ( l ) ( u ) = ∂ i T ( l ) ( u ) ∂ i B ( u ) = ∂ i Z ( l ) ( u ) ∂ i N ( u ) , b i ( u ) = ∂ i Q ( u ) ∂ i B ( u ) = ∂ i S ( u ) ∂ i N ( u ) , n i ( u ) = ∂ i P ( u ) ∂ i B ( u ) = ∂ i R ( u ) ∂ i N ( u ) . (30) Using (29) and (30 ), Q ( u ), R ( u ) an d P ( u ) + S ( u ) are uniquely defined (u p to additiv e constan ts) by the follo wing identitie s Q ( u ) = 1 2 ( ∇ B ) 2 + 1 2 X l ǫ ( l )  T ( l )  2 , R ( u ) = 1 2 ( ∇ N ) 2 + 1 2 X l ǫ ( l )  Z ( l )  2 , P ( u ) + S ( u ) = < ∇ N , ∇ B > + X l ǫ ( l ) T ( l ) Z ( l ) . (31) If the Riemannian cur v ature tensor asso ciated to g ii ( u ) tak es the sp ecial form (20), th en (29) tak e the sp ecial form n i ( u ) = ∇ i ∇ i N + w i (1) N + Z (1) + w i (2) M + v i Z (2) , b i ( u ) = ∇ i ∇ i B + w i (1) B + T (1) + w i (2) A + v i T (2) , (32) and Q ( u ) = 1 2 ( ∇ B ) 2 ( u ) + B ( u ) T (1) ( u ) + A ( u ) T (2) ( u ) , R ( u ) = 1 2 ( ∇ N ) 2 ( u ) + N ( u ) Z (1) ( u ) + M ( u ) Z (2) ( u ) , P ( u ) + S ( u ) = < ∇ N , ∇ B > + T (1) N + T (2) M + Z (1) B + Z (2) A. (33) Remark 2.3 The addition of c onstants to the r.h.s. of (31) le ave invariant the r e ci pr o c al tr ansformation is the sense that the r e cipr o c al metric ˆ g ii ( u ) , the r e cipr o c al Rieman nian tensor ˆ R ij ij ( u ) , the r e cipr o c al Hamiltonian op er ator ˆ J ij ( u ) and the r e cipr o c al Hamiltonian velo c i ty flow ˆ v i ( u ) ar e not effe cte d by them. These c onstant s just effe ct the auxiliary flows. Inde e d, let Q ( u ) , P ( u ) , R ( u ) and S ( u ) b e as in (31) and let us c onsider the mo difie d close d form            d ˆ x = B ( u ) dx + A ( u ) dt + ( P ( u ) + α ) dτ + ( Q ( u ) + β ) dζ + X l T ( l ) ( u ) dt ( l ) , d ˆ t = N ( u ) dx + M ( u ) dt + ( R ( u ) + γ ) dτ + ( S ( u ) + δ ) dζ + X l Z ( l ) ( u ) dt ( l ) , d ˆ τ = dτ , d ˆ ζ = dζ , with α, β , γ , δ arbitr ary c onstants. ˆ n i m ( u ) = ˆ n i ( u ) − β − δ ˆ v i ( u ) , ˆ b i m ( u ) = ˆ b i ( u ) − α − γ ˆ v i ( u ) , with ˆ n i ( u ) and ˆ b i ( u ) as in(28). 9 The follo wing alternativ e expressions for the recipr o cal Riemann curv ature tensor and the recipro cal Hamiltonian structure h old. Theorem 2.4 L et g ii ( u ) b e the metric for the Hamiltonian hydr o dyna mic system (13), with Riema nnian curvatur e tensor as in (16). Then, after the r e c i pr o c al tr ansformation (22), the non zer o c om p onents of the r e cipr o c al Riemannia n curvatur e tensor ar e ˆ R ik ik ( u ) = X l ǫ ( l ) ˆ w i ( l ) ( u ) ˆ w k ( l ) ( u ) + ˆ v i ( u ) ˆ n k ( u ) + ˆ v k ( u ) ˆ n i ( u ) + ˆ b i ( u ) + ˆ b k ( u ) , i 6 = k , (34) wher e the r e cipr o c al metric ˆ g ii ( u ) and the r e cipr o c al affinors ˆ n i ( u ) , ˆ b i ( u ) and ˆ w i ( l ) ( u ) ar e as in (24) and (28), r esp e ctiv ely, with Q ( u ) , P ( u ) , R ( u ) and S ( u ) as in (31). L et g ii ( u ) b e the metric for the Hamiltonian hydr o dynamic system (13), with Rie- mannian curvatur e tensor as in (20), then the nonzer o c omp onents of the tr ansforme d curvatur e tensor take the form ˆ R ik ik ( u ) = ˆ n i ( u ) ˆ v k ( u ) + ˆ n k ( u ) ˆ v i ( u ) + ˆ b i ( u ) + ˆ b k ( u ) , i 6 = k , (35) wher e the r e cipr o c al metric ˆ g ii ( u ) and the r e cipr o c al affinors ˆ n i ( u ) , ˆ b i ( u ) and ˆ w i ( l ) ( u ) ar e as in (24) and (28), r esp e ctiv ely, with Q ( u ) , P ( u ) , R ( u ) and S ( u ) as in (33). F ormula (34 ) h as already b een pr ov en in [2]. T o p ro ve (35), it is su fficien t to in sert (32) and (33) into (25). Corollary 2.5 L et the r e cipr o c al tr ansformation c hanges only x ( N ( u ) = 0 and M ( u ) = 1 in (22)), then the nonzer o c omp onents of the tr ansforme d c u rvatur e tensor take the form ˆ R ik ik ( u ) = B 2 ( u ) R ik ik ( u ) + B ( u )( ∇ i ∇ i B ( u ) + ∇ k ∇ k B ( u )) − ( ∇ B ( u )) 2 (36) Mor e over, if the Riema nnian curvatur e tensor of g ii ( u ) takes the form as i n (16), then ˆ R ik ik ( u ) = X l ǫ ( l ) ˆ w i ( l ) ( u ) ˆ w k ( l ) ( u ) + ˆ b i ( u ) + ˆ b k ( u ); if R iemannian cu rvatur e tensor asso ciate d to g ii ( u ) takes the form (20) then the nonzer o c omp onents of the tr ansforme d curvatur e tensor take the form ˆ R ik ik ( u ) = ˆ w i (2) ( u ) ˆ v k ( l ) ( u ) + ˆ w k (2) ( u ) ˆ v i ( l ) ( u ) + ˆ b i ( u ) + ˆ b k ( u ) . (37) If the r e cipr o c al tr ansformat ion changes only t ( B ( u ) = 1 and A ( u ) = 0 in (22)), then the nonzer o c omp onents of the tr ansfo rme d curvatur e tensor satisfy ˆ R ik ik ( u ) = M 2 R ik ik + M ( v k ∇ i ∇ i N + v i ∇ k ∇ k N ) − v i v k ( ∇ N ) 2 ( M − N v i )( M − N v k ) (38) Mor e over, if the Riema nnian curvatur e tensor of g ii ( u ) takes the form as i n (16), then ˆ R ik ik ( u ) = X l ǫ ( l ) ˆ w i ( l ) ( u ) ˆ w k ( l ) ( u ) + ˆ v i ( u ) ˆ n k ( u ) + ˆ v k ( u ) ˆ n i ( u ); if R iemannian curvatur e tensor asso ciate d to g ii ( u ) takes the form (20), then the nonzer o c omp onents of the tr ansforme d curvatur e tensor take the form ˆ R ik ik ( u ) = ˆ w i (1) ( u ) + ˆ w k (1) ( u ) + ˆ v i ( u ) ˆ n k ( u ) + ˆ v k ( u ) ˆ n i ( u ) . (39) F ormulas (36), (38) and their expressions when R ik ik ( u ) is as in (16) ha v e already b een pro ven in [2]. T o p ro ve (37 ) (resp. (39)) it is sufficien t to inser t (32) an d (33) in to (36) (resp. (38)). 10 3 Necessary conditions for recipro cal flat metrics In this s ection, w e start from an integrable Hamilto nian sys tem u i t = v i ( u ) u i x , i = 1 , . . . , n and w e in vest igate the necessary conditions on the initial metric and on the conserv ation la ws in the recipro cal transformation so that the recipro cal metric b e flat. The conditions settled b y T h eorem 3.5 on the conserv atio n la ws in the r ecipro cal transformations are ve ry strict: if n ≥ 5, they must b e lin ear com binations with constan t c o efficients of the Casimirs, the momen tum and th e Hamiltonian densities with resp ect to the initial Hamiltonian structure. The same Theorem settles also very strict conditions on th e admissible form of the Riemann ian cu r v ature tensor asso ciated to the initial metric g ii ( u ). In the case of recipro cal transform ations of a single indep enden t v ariable the necessary conditions are ev en more restrictiv e: if n ≥ 3, the conserv ation la w is a linear com bination of Casimirs and momen tum densities (resp ectiv ely of Casimirs and Hamiltonian densities) if ju st the x v ariable (resp. the t v ariable) c han ges. Definition 3.1 F ol lowing F e r ap ontov [11, 12], we c al l c anonic al a r e cipr o c al tr ansforma- tion as in (22), in which the inte g r al s, up to additive c onst ants, ar e line ar c ombinations of the c anonic al inte gr als (Casimirs, Hamiltonian and momentum) with r esp e ct to the given Hamiltonian structur e. Remark 3.1 If the initial metric g ii ( u ) is not flat, a Casimir density (r esp. a momen- tum density, a Hamiltonian density) asso ciate d to the c orr esp onding non-lo c al H amilto- nian op er ator J ij ( u ) in (14) is a c onservation law h ( u ) such that J ij ∂ j h ( u ) ≡ 0 (r esp. J ij ∂ j h ( u ) ≡ u i x , J ij ∂ j h ( u ) ≡ v i ( u ) u i x ). We r e mark that, under the hyp otheses of the fol lowing The or em, for e ach Hamiltonian structur e with k non–lo c alities in the Hamilto- nian op er ator, ther e do exist ( n + k + 2) c ano nic al inte gr als as pr oven by M altsev and Novikov[18]. In the follo wing Theorem we settle the necessary conditions for recipro cal flat metrics in the case of a transformation of a single v ariable. Theorem 3.2 L et u i t = v i ( u ) u i x , i = 1 , . . . , n , n ≥ 3 , b e an inte gr able strictly hyp erb olic DN hydr o dyna mic typ e system as in (13), let g ii ( u ) b e one of its metrics with Hamiltonian op er ator J ij ( u ) as in (14). i) L et d ˆ x = B ( u ) dx + A ( u ) dt , d ˆ t = dt , b e a r e cipr o c al tr ansfo rmation such that the r e cipr o c al metric ˆ g ii ( u ) define d in (24) b e flat. Then B ( u ) is a line ar c ombination of the Casimirs and the momentum densities (up to an additive c onstant), and g ii ( u ) is either a flat or a c onstant curvatur e or a c onformal ly flat metric. ii) L et d ˆ x = dx , d ˆ t = N ( u ) dx + M ( u ) dt , b e a r e cipr o c al tr ansforma tion su c h that the r e cipr o c al metric ˆ g ii ( u ) define d in (24) b e flat. In the c ase n = 3 , let mor e over v i ( u ) 6≡ 0 , i = 1 , . . . , 3 . Then N ( u ) is a line ar c ombination of the Casimirs and the Hamiltonia n densities (up to an additive c onstant), and the R iemannian curvatur e tensor asso ci ate d to the initial metric g ii ( u ) takes the form R ij ij ( u ) = w i ( u ) v j ( u ) + w j ( u ) v i ( u ) , i 6 = j, (40) 11 with w i ( u ) (p ossibly nul l) affino rs. Pro of of the theorem T o compu te the form of the Riemann ian curv at ur e tens or asso ci- ated to the initial metric g ii ( u ) it is sufficien t to in vert the recipro cal transformation (42) and to app ly Theorem 2.4 to the recipro cal flat metric ˆ g ii ( u ). i) If the reciprocal transformation c hanges only x ( N ( u ) ≡ 0, M ( u ) ≡ 1) and the recipro cal metric ˆ g ii ( u ) is fl at, the Riemann curv ature tensor associated to the initial metric g ii ( u ) tak es the form R ik ik ( u ) = w i (1) ( u ) + w k (1) ( u ), ( i 6 = k ), with p ossibly constan t or n ull affinors w i (1) ( u ) (see [13]). According to Corollary (2.5), the zero cur v ature equations ˆ R ik ik ( u ) = ˆ b i ( u ) + ˆ b k ( u ) ≡ 0, ( i 6 = k ), for the r ecipro cal metric ˆ g ii ( u ) are th en equiv alent to 0 ≡ ˆ b i ( u ) = B ( u ) b i ( u ) − Q ( u ) , i = 1 , . . . , n, as follo ws from (37 ) with Q ( u ) as in (33). Sin ce b i ( u ) = ∂ i Q ( u ) ∂ i B ( u ) , ( i = 1 , . . . , n ), w e immediately conclude that there exists a constan t κ suc h that u i ˆ ζ ≡ b i ( u ) u i x ≡ J ij ( u ) ∂ j B ( u ) = κu i x , i = 1 , . . . , n , that is B ( u ) is a linear com bination of the Casimirs and the momen tum densities u p to an ad d itiv e constant . ii) Sim ilarly , if the recipro cal transformation changes only t ( B ( u ) ≡ 1, A ( u ) ≡ 0) and the recipro cal metric ˆ g ii ( u ) is fl at, the Riemann curv ature tensor asso ciated to th e initial metric g ii ( u ) tak es the form R ik ik ( u ) = w i (2) ( u ) v k ( u ) + w k (2) ( u ) v i ( u ), ( i 6 = k ), with p ossibly constan t or null affinors w i (2) ( u ) (see [13]). According to Corollary (2.5), the zero curv ature equations for the recipro cal metric, ˆ R ik ik ( u ) = ˆ v i ( u ) ˆ n k ( u ) + ˆ v k ( u ) ˆ n i ( u ) ≡ 0, ( i 6 = k ), are equiv alen t to 0 ≡ ˆ n i ( u ) = M ( u ) n i ( u ) − R ( u ) v i ( u ) M ( u ) − N ( u ) v i ( u ) , i = 1 , . . . , n . (41) Since v i ( u ) = ∂ i M ( u ) ∂ i N ( u ) , n i ( u ) = ∂ i R ( u ) ∂ i N ( u ) , ( i = 1 , . . . , n ), we immediately conclude that there exists a constan t κ su c h that u i ˆ τ ≡ n i ( u ) u i x ≡ J ij ( u ) ∂ j N ( u ) = κv i ( u ) u i x , i = 1 , . . . , n , that is the densit y of conserv ation la w asso ciated to the inv erse tr ansformation is a linear com bination of the Casimirs an d th e Hamiltonian densities u p to an additiv e constant .  Remark 3.3 The The or em 3.2 is not applic able in the c ase n = 2 . F or instanc e, in the c ase of a tr ansformat ion of the single variable x , we get the ze r o curvatur e c ondition ˆ b 1 ( u ) = − ˆ b 2 ( u ) and it is p ossible to c onstruct non-c anonic al r e cipr o c al tr ansformations which pr eserve the flatness of the metric. H er e is a c ounter e xample suggeste d by the se c ond r e fer e e: let us take a line ar 2-c omp onent system u 1 t = pu 1 x , u 2 t = q u 2 x , 12 wher e p, q ar e c onstant. It has infinitely many Hamiltonian structur es, let’s take the one c orr esp onding to the metric g = ( du 1 ) 2 + ( du 2 ) 2 . L et us c onsider a r e cipr o c al tr ans forma- tion of x only, d ˆ x = B ( u 1 , u 2 ) dx + A ( u 1 , u 2 ) dt , ˆ t = t . F or the ab ove system, the gener al form of a density of c onservation law is B ( u 1 , u 2 ) = f 1 ( u 1 ) + f 2 ( u 2 ) . L et us r e qui r e that the tr ansfo rme d metric b e flat: this gives a functional-differ ential e quation for f 1 and f 2 which c an b e solve d explicitly. In p articular, if B ( u 1 , u 2 ) = a + bu 1 + cu 2 + d 2 (( u 1 ) 2 + ( u 2 ) 2 ) , then the flatness c ond ition gives b 2 + c 2 = 2 ad . This is the c ase of c anonic al inte gr als discusse d i n The or em 3.2. However, ther e is another solution: B ( u 1 , u 2 ) = a exp( u 1 ) + b exp( − u 1 ) + c sin ( u 2 ) + d cos( u 2 ) with c 2 + d 2 = 4 ab . Thus, the r e cipr o c al metric is flat, altho ugh the density B is not a line ar c ombination of c ano nic al inte gr als. Remark 3.4 In the c ase of time tr ansform ations and n = 3 , the hyp othesis v i ( u ) 6≡ 0 ensur es ˆ v i ( u ) 6≡ 0 . If n = 3 and v 3 ( u ) = 0 , then The or em 3.2 is not applic able for tr ansformations of the indep endent variable t . Inde e d, the zer o cu rvatur e e quations for the tr ansforme d metric take the form ˆ v 3 ( u ) = 0 , ˆ n 3 ( u ) = 0 , ˆ n 2 ( u ) ˆ v 1 ( u ) + ˆ n 1 ( u ) ˆ v 2 ( u ) ≡ 0 , inste ad of (41). The c ondition ˆ n 3 ( u ) ≡ 0 implies n 3 ( u ) ≡ 0 , bu t we c an ’t c onclude that ˆ n 1 ( u ) = 0 = ˆ n 2 ( u ) and in gener al we may get a tr ansforme d flat metric with non-c anonic al tr ansformations. Inde e d, let u 1 t = 2 u 1 x , u 2 t = u 2 x , u 3 t = 0 . The ab ove system i s inte gr able and p ossesses a lo c al Hamilto nian structur e asso ciate d to the flat metric g = ( du 1 ) 2 + ( du 2 ) 2 + ( du 3 ) 2 . L et the r e cipr o c al tr ansformation b e d ˆ x = dx , d ˆ t = N ( u ) dx + M ( u ) dt , with N ( u ) = exp( u 1 ) + exp( − u 1 ) + 2 √ 2 cos( u 2 2 ) + 2 sin( u 2 2 ) + u 3 , M ( u ) = exp( u 1 ) + exp( − u 1 ) + 4 √ 2 cos( u 2 2 ) + 4 sin( u 2 2 ) . Then the zer o curvatur e e q uations for the tr ansfo rme d metric ˆ g ii ( u ) ar e identic al ly satisfie d and n 1 ( u ) = exp( u 1 ) + exp( − u 1 ) , n 2 ( u ) = − √ 2 2 cos( u 2 2 ) − 1 2 sin( u 2 2 ) , n 3 ( u ) = 0 . In the follo wing Theorem we settle the necessary conditions for recipro cal flat metrics in the case of a recipro cal transformation of b oth the ind ep end en t v ariables. 13 Theorem 3.5 L et u i t = v i ( u ) u i x , i = 1 , . . . , n , n ≥ 5 , b e an inte gr able strictly hyp erb olic DN hydr o dyna mic typ e system as in (13), let g ii ( u ) b e one of its metrics with Hamiltonian op er ator J ij ( u ) as in (14). L e t d ˆ x = B ( u ) d x + A ( u ) dt, d ˆ t = N ( u ) dx + M ( u ) dt, ( 42) b e a r e cipr o c al tr ansformat ion such that the r e cipr o c al metric ˆ g ii ( u ) define d in (24) b e flat. Then i) Ther e exist (p ossibly nul l) affinors w i ( l ) ( u ) , i = 1 , . . . , n , l = 1 , 2 , such that the R iemannian curvatur e tensor of the initial metric g ii ( u ) takes the form R ij ij ( u ) = w i (1) ( u ) + w j (1) ( u ) + w i (2) ( u ) v j ( u ) + w j (2) ( u ) v i ( u ) , i 6 = j ; (43) ii) the r e cipr o c al tr ansformat ion (42) is c anonic al with r esp e ct to J ij ( u ) , the Hamilto- nian op er ator asso ciate d to the initial metric g ii ( u ) . In p ar ticular, the auxiliary flows u i ζ = b i ( u ) u i x = J ij ( u ) ∂ j B ( u ) , u i τ = n i ( u ) u i x = J ij ( u ) ∂ j N ( u ) , asso ciate d to such tr ans formations ar e line ar c ombinations of the x and t flows. Pro of of the theorem T o v erify prop erty i) it is sufficien t to in v ert the recipro cal transformation (42) and to app ly T heorem 2.4 to the recipro cal flat metric ˆ g ii ( u ). W e now prov e statemen t ii) in th e case of a general recipro cal transformation (42) and let the in itial metric g ii ( u ) ha v e Riemann curv ature tensor as in (43). Let n = 5. The zero curv ature equations asso ciated to the recipro cal flat metric ˆ g ii ( u ) are ˆ b i ( u ) + ˆ b j ( u ) + ˆ n i ( u ) ˆ v j ( u ) + ˆ n j ( u ) ˆ v i ( u ) = 0 , i 6 = j. Using the strict h yp erb olicit y h yp othesis, it is element ary to sho w that they ma y b e equiv ale ntly expressed as ˆ b i ( u ) = − ˆ n 1 ( u ) ˆ v i ( u ) , ˆ n i ( u ) = ˆ n 1 ( u ) , i = 1 , . . . , 5 . F or n ≥ 5, it is also easy to pro v e b y induction that th e sys tem of zero curv ature equations in the 2 n v ariables ˆ b i ( u ), ˆ n i ( u ) has r ank 2 n − 1 and that ˆ b i ( u ) = − ˆ n 1 ( u ) ˆ v i ( u ) , ˆ n i ( u ) = ˆ n 1 ( u ) , i = 1 , . . . , n. (44) Since ˆ n j ( u ) are affin ors of the transformed metric ˆ g ii ( u ), using (15) for the transform ed metric and (44), w e ha v e ∂ k ˆ n j ( u ) ≡ 0, k 6 = j . Using again (44), w e then conclude that there exists a (p ossibly null) constan t κ 0 suc h that ˆ b i ( u ) = − κ 0 ˆ v i ( u ) , ˆ n i ( u ) = κ 0 , i = 1 , . . . , n . (45) F or the inv erse recipro cal transformation, w e ha v e dx = ˆ B ( u ) d ˆ x + ˆ A ( u ) d ˆ t + ˆ Q ( u ) d ˆ ζ + ˆ P ( u ) d ˆ τ , dt = ˆ N ( u ) d ˆ x + ˆ M ( u ) d ˆ t + ˆ S ( u ) d ˆ ζ + ˆ R ( u ) d ˆ τ , ζ = ˆ ζ , τ = ˆ τ , 14 with ˆ B ( u ) = M ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ A ( u ) = − A ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ N ( u ) = − N ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ M ( u ) = B ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ Q ( u ) = S ( u ) A ( u ) − Q ( u ) M ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ S ( u ) = Q ( u ) N ( u ) − S ( u ) B ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ P ( u ) = R ( u ) A ( u ) − P ( u ) M ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) , ˆ R ( u ) = P ( u ) N ( u ) − R ( u ) B ( u ) B ( u ) M ( u ) − A ( u ) N ( u ) . (46) Since ˆ v i ( u ) = B ( u ) v i ( u ) − A ( u ) M ( u ) − N ( u ) v i ( u ) = ∂ i ˆ A ( u ) ∂ i ˆ B ( u ) = ∂ i ˆ M ( u ) ∂ i ˆ N ( u ) , i = 1 , . . . , n , and the recipro cal affinors satisfy ( i = 1 , . . . , n ) ˆ n i ( u ) = B ( u ) n i ( u ) − P ( u ) + ( N ( u ) n i ( u ) − R ( u )) ˆ v i ( u ) = ∂ i ˆ P ( u ) ∂ i ˆ B ( u ) = ∂ i ˆ R ( u ) ∂ i ˆ N ( u ) , ˆ b i ( u ) = B ( u ) n i ( u ) − Q ( u ) + ( N ( u ) n i ( u ) − S ( u )) ˆ v i ( u ) = ∂ i ˆ Q ( u ) ∂ i ˆ B ( u ) = ∂ i ˆ S ( u ) ∂ i ˆ N ( u ) , w e immediately conclude that there exist constants κ 1 , . . . , κ 4 suc h that ˆ S ( u ) = − κ 0 ˆ M ( u ) − κ 1 , ˆ Q ( u ) = − κ 0 ˆ A ( u ) − κ 2 , ˆ R ( u ) = κ 0 ˆ N ( u ) − κ 3 , ˆ P ( u ) = κ 0 ˆ B ( u ) − κ 4 . Inserting (46), into the ab o v e equations, we then get Q ( u ) = κ 2 B ( u ) + κ 1 A ( u ) S ( u ) = κ 2 N ( u ) + κ 1 M ( u ) + κ 0 , P ( u ) = κ 4 B ( u ) + κ 3 A ( u ) − κ 0 , R ( u ) = κ 4 N ( u ) + κ 3 M ( u ) , from whic h the assertion follo ws.  Remark 3.6 If n = 4 , the system of the six zer o c urvatur e e quations for the tr ansforme d metric ˆ g ii ( u ) has maximal r ank 6 in the unknowns ˆ b i , ˆ n i , and it is p ossible to expr ess, say ˆ b i ( u ) , ˆ n i ( u ) , i = 2 , 3 , 4 in function of ˆ b 1 ( u ) and ˆ n 1 ( u ) . Mor e over the c ondition ˆ n i ( u ) = ˆ n 1 ( u ) , i = 2 , 3 , 4 is satisfie d if and only if ˆ b 1 ( u ) = − ˆ v 1 ( u ) ˆ n 1 ( u ) . The ab ove observat ion implies that, for n = 4 , ther e exist non-c anonic al tr ansforma- tions which pr ese rve the flatness of the metric when ˆ n i ( u ) 6 = ˆ n 1 ( u ) for i ∈ { 2 , 3 , 4 } . 15 4 Classification of the recipro c al transformations whic h pre- serv e the flatness of the metric or transform constan t cur- v ature metrics to fl at metrics Theorems 3.2 and 3.5 state that only the recipro cal trans formations w hic h are canonical with resp ect to the initial Hamiltonian structure ma y trans form the initial metric to a flat one, resp ectiv ely for n ≥ 3 (recipro cal tr ansformations of a single indep endent v ariable) or n ≥ 5 (recipro cal transf ormations of b oth the indep en d en t v ariables) . In view of the ab o ve , in this section we restrict ourselv es to the case in whic h the initial metric g ii ( u ) is either flat ( w i (1) ≡ 0 ≡ w i (2) , i = 1 , . . . , n , in (43)) or of constant curv ature 2 c ( w i (1) ≡ c , w i (2) ≡ 0, i = 1 , . . . , n , in (43)). T hen, in T heorem 4.1, w e completely c haracterize whic h recipro cal transformations map g ii ( u ) to fl at metric ˆ g ii ( u ). Finally , the case in which b oth the initial and the transformed Hamiltonian structure are lo cal has also a nice geometric in terpretation in view of the resu lts by F erap on to v [11], whic h w e pr esent in Theorem 4.12. Theorem 4.1 L et n ≥ 5 and let u i t = v i ( u ) u i x = J ij ( u ) ∂ j H ( u ) , i = 1 , . . . , n , b e a DN inte gr able strictly hyp erb olic hydr o dynamic typ e system, with J ij ( u ) the Hamilto- nian op er ator asso ciate d to the metric g ii ( u ) and H ( u ) the Hamiltonian density. L et d ˆ x = B ( u ) dx + A ( u ) dt , d ˆ t = N ( u ) dx + M ( u ) dt b e a r e cipr o c al tr ansformation with A ( u ) , B ( u ) , M ( u ) and N ( u ) not al l c onstant f u nctions. A) L et the metric g ii ( u ) b e flat. Then the r e cipr o c al metric ˆ g ii ( u ) define d in (24) is flat i f and only if one of the fol lowing alterna tives hold: A.i) ther e exist c onstants κ 1 6 = 0 , κ 2 , κ 3 such that M ( u ) = κ 1 , N ( u ) = κ 2 , ( ∇ B ) 2 ( u ) = κ 3 ( κ 1 B ( u ) − κ 2 A ( u )) ; A.ii) ther e exist c onstants κ 1 6 = 0 , κ 2 , κ 3 such that B ( u ) = κ 1 , A ( u ) = κ 2 , ( ∇ N ) 2 ( u ) = κ 3 ( κ 1 M ( u ) − κ 2 N ( u )) ; A.iii) ther e exist c onstants κ 1 , κ 2 , κ 3 , κ 4 such that ( ∇ B ) 2 ( u ) = 2 κ 1 A ( u ) + 2 κ 2 B ( u ) , ( ∇ N ) 2 ( u ) = 2 κ 3 M ( u ) + 2 κ 4 N ( u ) , < ∇ B ( u ) , ∇ N ( u ) > = κ 1 M ( u ) + κ 2 N ( u ) + κ 3 A ( u ) + κ 4 B ( u ) . B) L et the metric g ii ( u ) b e of c onst ant cu rvatur e 2 c . Then the r e ci pr o c al metric ˆ g ii ( u ) define d in (24) is flat if and only if one of the fol lowing alternatives hold: B.i) ther e exist c onstants κ 1 6 = 0 , κ 3 , suc h that M ( u ) = κ 1 , N ( u ) ≡ 0 , ( ∇ B ) 2 ( u ) + 2 cB 2 ( u ) = 2 κ 3 B ( u ); B.ii) ther e exist c onstants κ 1 , κ 2 , κ 3 , κ 4 such that ( ∇ B ) 2 ( u )+2 cB 2 ( u ) = 2 κ 1 A ( u )+2 κ 2 B ( u ) , ( ∇ N ) 2 ( u )+2 cN 2 ( u ) = 2 κ 3 M ( u )+2 κ 4 N ( u ) , < ∇ B ( u ) , ∇ N ( u ) > +2 cB ( u ) N ( u ) = κ 1 N ( u ) + κ 2 M ( u ) + κ 3 A ( u ) + κ 4 B ( u ) . 16 Remark 4.2 Case A.i) (r esp. A.ii) ) includes the r e cipr o c al tr ansformations of the single variable x (r esp. the single variable t ) when κ 1 = 1 , κ 2 = 0 . Case B.i) c orr esp onds to r e cipr o c al tr ansformations of the single variable x (notic e that only N ( u ) ≡ 0 is admissible if c 6 = 0 ). Final ly, it is not p ossible to tr ansform a c onstant curvatur e metric to a flat one by a tr ansformation of the single variable t . Pro of: Let g ii ( u ) b e either a flat ( c = 0) or a constan t curv ature metric ( c 6 = 0). W e p ro ve first A.i) and B.i). Let κ 1 6 = 0 , κ 2 b e constants suc h that M ( u ) ≡ κ 1 , N ( u ) ≡ κ 2 . Then, the only p ossibly non-zero elemen ts of the recipro cal Riemannian curv at ur e tensor tak e the form, ˆ R ik ik ( u ) = 2 c H i ( u ) H k ( u ) ˆ H i ( u ) ˆ H k ( u ) − ( ∇ B ) 2 ( u ) + H i ( u ) ˆ H i ( u ) ∇ i ∇ i B ( u ) + H k ( u ) ˆ H k ( u ) ∇ k ∇ k B ( u ) , where ˆ H i ( u ) = κ 1 − κ 2 v i ( u ) B ( u ) κ 1 − A ( u ) κ 2 H i ( u ) , ˆ v i ( u ) = B ( u ) v i ( u ) − A ( u ) κ 1 − κ 2 v i ( u ) , i = 1 , . . . , n . F rom the necessary condition found in T heorems 3.2 and 3.5, b i ( u ) ≡ ∇ i ∇ i B ( u ) + 2 cB ( u ) = κ 3 + κ 4 v i ( u ) , i = 1 , . . . , n, (47) w e infer ( ∇ B ) 2 ( u ) + 2 cB 2 ( u ) = 2 κ 3 B ( u ) + 2 κ 4 A ( u ) + κ 5 . (48) If w e insert (47) and (48) inside the expression of th e transformed Riemann ian curv ature tensor, w e immediately get ˆ R ik ik ( u ) = − κ 5 + ( κ 1 κ 4 + κ 3 κ 2 )  ˆ v i ( u ) + ˆ v k ( u )  + 2 cκ 2 2 ˆ v i ( u ) ˆ v k ( u ) . Then th e condition ˆ R ik ik ( u ) ≡ 0, is equiv alen t to either c = κ 5 = κ 1 κ 4 + κ 3 κ 2 = 0 , or to c 6 = 0 , and κ 5 = κ 2 = κ 4 = 0 , from whic h cases A.i) and B.i) immediately follo w. W e n o w pro v e A.ii). Let κ 1 6 = 0 , κ 2 b e constan ts suc h that B ( u ) ≡ κ 1 , A ( u ) ≡ κ 2 and let the initial metric g ii ( u ) b e fl at. Then, the only p ossibly non-zero elemen ts of the recipro cal Riemannian curv ature tensor tak e the form, ˆ R ik ik ( u ) = H i ( u ) ˆ H i ( u ) ∇ i ∇ i N ( u ) ˆ v k ( u )+ H k ( u ) ˆ H k ( u ) ∇ k ∇ k N ( u ) ˆ v i ( u ) − ˆ v i ( u ) ˆ v k ( u ) ( ∇ N ) 2 ( u ) , i 6 = k . 17 Inserting in to the ab o ve equation ( i = 1 , . . . , n ) ˆ H i ( u ) = M ( u ) − N ( u ) v i ( u ) κ 1 M ( u ) − κ 2 ( u ) N ( u ) H i ( u ) , ˆ v i ( u ) = κ 1 v i ( u ) − κ 2 M ( u ) − N ( u ) v i ( u ) , w e get ˆ R ik ik ( u ) = ˆ v i ( u ) ˆ v k ( u )  ( κ 1 M ( u ) − κ 2 N ( u ) n i ( u ) κ 1 v i ( u ) − κ 2 + ( κ 1 M ( u ) − κ 2 N ( u ) n k ( u ) κ 1 v k ( u ) − κ 2 −  ∇ N  2  . Since ˆ v i ( u ) 6≡ 0, the conditions ˆ R ik ( u ) ≡ 0, ( i 6 = k ) are equiv alen t to either ( ∇ N ) 2 ( u ) ≡ 0 or ∂ i ( ∇ N ) 2 ( u ) ( ∇ N ) 2 ( u ) = ∂ i ( κ 1 M ( u ) − κ 2 N ( u )) ( κ 1 M ( u ) − κ 2 N ( u )) , i = 1 , . . . , n , from which case ii) immediately follo ws. In particular, under the same h yp otheses, w e also h a v e n i ( u ) = κ 3 ( κ 1 v i ( u )+ κ 2 ) , ˆ n i ( u ) = − 1 2 ( ∇ N ) 2 ( u ) ˆ v i ( u )+ n i ( u ) κ 1 M ( u ) − κ 2 N ( u ) M ( u − N ( u ) v i ( u ) ≡ 0 . If κ 1 6 = 0 , κ 2 are constants suc h that B ( u ) ≡ κ 1 , A ( u ) ≡ κ 2 and the initial metric g ii ( u ) is of constant curv ature c 6 = 0, then it is easy to sho w that the transform ed metric ˆ g ii cannot b e flat. T o pr o v e A.iii ) and B.ii) , w e use the closed form    d ˆ x = B ( u ) dx + A ( u ) dt + P ( u ) dτ + Q ( u ) dζ , d ˆ t = N ( u ) dx + M ( u ) dt + R ( u ) dτ + S ( u ) dζ , d ˆ τ = dτ , d ˆ ζ = dζ , (49) asso ciated to the auxiliary flo ws u i τ = n i ( u ) u i x =  ∇ i ∇ i N ( u ) + 2 cN ( u )  u i x , u i ζ = b i ( u ) u i x =  ∇ i ∇ i B ( u ) + 2 cB ( u )  u i x . (50) In view of the results of the previous section, the auxiliary flo ws (50) are n ecessarily linear com binations of the x and t flows. W e imp ose that the conserv ation la ws in the r ecipro cal transformation satisfy the n ecessary conditions settled in Theorem 3.5. Th en there exist constan ts κ j , j = 1 , . . . , 8 suc h that b i ( u ) = κ 1 v i ( u ) + κ 2 , ( ∇ B ) 2 ( u ) + 2 cB ( u ) = 2 κ 1 A ( u ) + 2 κ 2 B ( u ) + 2 κ 6 , n i ( u ) = κ 3 v i ( u ) + κ 4 , ( ∇ N ) 2 ( u ) + 2 cN ( u ) = 2 κ 3 M ( u ) + 2 κ 4 N ( u ) + 2 κ 5 , P ( u ) = κ 3 A ( u ) + κ 4 B ( u ) + κ 7 , S ( u ) = κ 1 M ( u ) + κ 2 N ( u ) + κ 8 , < ∇ B , ∇ N > + 2 cB N ≡ P + S = κ 1 M ( u ) + κ 2 N ( u ) + κ 3 A ( u ) + κ 4 B ( u ) + κ 7 + κ 8 . If w e insert the ab o v e expressions int o th e right hand side of (28 ) w e get ˆ n i ( u ) = − κ 7 − κ 5 ˆ v i ( u ) , ˆ b i ( u ) = − κ 6 − κ 8 ˆ v i ( u ) . Finally , the elements of the Riemannian curv ature tensor are ˆ R ik ik ( u ) = ˆ n i ( u ) ˆ v i ( u ) + ˆ n k ( u ) ˆ v k ( u ) + ˆ b i ( u ) + ˆ b k ( u ) = − 2 κ 6 − ( κ 7 + κ 8 )( ˆ v i ( u ) + ˆ v k ( u )) − 2 κ 5 ˆ v i ( u ) ˆ v k ( u ) , i 6 = k , 18 so that ˆ R ik ik ( u ) ≡ 0 if and only if κ 5 = κ 6 = κ 7 + κ 8 = 0, and the assertions A.iii ) and B.ii) easily follo w.  Example 4.3 If B ( u ) and N ( u ) ar e non trivial indep endent Casimirs of the flat metric g ii ( u ) and ( ∇ B ( u )) 2 6 = 0 , then ther e exist a c onstant α and A ( u ) such that , under the r e cipr o c al tr ansformation d ˆ x = ( αB ( u ) + N ( u )) dx + A ( u ) dt , the r e cipr o c al metric ˆ g ii ( u ) = g ii ( u ) / ( αB ( u ) + N ( u )) 2 is flat. Example 4.4 If B ( u ) is a density of mo mentum for the flat metric g ii ( u ) and ( ∇ B ( u )) 2 − 2 B ( u ) = 2 α , then under the r e cipr o c al tr ansformation d ˆ x = ( B ( u ) + α ) d x + A ( u ) dt , the r e cipr o c al metric ˆ g ii ( u ) = g ii ( u ) / ( B ( u ) + α ) 2 is flat. Example 4.5 If B ( u ) and N ( u ) ar e non trivial indep endent Casimirs of the flat metric g ii ( u ) and ( ∇ N ( u )) 2 6 = 0 , then ther e e xi st a c onstant α and M ( u ) such that, under the r e cipr o c al tr ansformatio n d ˆ t = ( αN ( u ) + B ( u )) dx + M ( u ) dt , the r e cipr o c al metric ˆ g ii ( u ) is flat. Example 4.6 If N ( u ) is a density o f Hamiltonian for the flat metric g ii ( u ) and ( ∇ N ( u )) 2 − 2 M ( u ) = 2 α , then u nder the r e ci pr o c al tr ansformation d ˆ t = N ( u ) dx +  M ( u ) + α  dt , the r e cipr o c al metric ˆ g ii ( u ) is flat. Example 4.7 L et N ( u ) b e a density of momentum and let B ( u ) b e a density of Hamil- tonian for the flat metric g ii ( u ) . Then under the r e cipr o c al tr ansform ation d ˆ x = B ( u ) d x + 1 2 ( ∇ B ) 2 ( u ) dt, d ˆ t = N ( u ) dx + M ( u ) dt, such that ( ∇ N ) 2 ( u ) = 2 N ( u ) , < ∇ N ( u ) , ∇ B ( u ) > = N ( u ) + B ( u ) , the r e ci pr o c al metric ˆ g ii ( u ) is flat. Example 4.8 L et N ( u ) = M ( u ) = 1 and let B ( u ) b e a density of Hamiltonian for the flat metric g ii ( u ) . Then under the r e ci pr o c al tr ansformatio n d ˆ x = B ( u ) d x + 1 2 ( ∇ B ) 2 ( u ) dt, d ˆ t = dx + dt, the r e cipr o c al metric ˆ g ii ( u ) is flat. Example 4.9 L et N ( u ) b e a density of momentum and let B ( u ) b e a density of Hamil- tonian for the metric g ii ( u ) with c onstant curvatur e 2 c . Then under the r e cipr o c al tr ans- formation d ˆ x = B ( u ) dx +  1 2 ( ∇ B ) 2 ( u ) + cB 2 ( u )  dt, d ˆ t = N ( u ) dx + M ( u ) dt, such that ( ∇ N ) 2 ( u ) + 2 cN 2 ( u ) − 2 N ( u ) ≡ 0 , < ∇ N ( u ) , ∇ B ( u ) > +2 cN ( u ) B ( u ) − N ( u ) − B ( u ) ≡ 0 , the r e cipr o c al metric ˆ g ii ( u ) is flat. 19 Example 4.10 L et N ( u ) b e a density of Hamiltonian and let B ( u ) b e a Casimir for the metric g ii ( u ) with c onstant curvatur e 2 c . Then under the r e cipr o c al tr ansformation d ˆ x = B ( u ) d x + A ( u ) dt, d ˆ t = N ( u ) dx +  1 2 ( ∇ N ) 2 ( u ) + cN 2 ( u )  dt, such that ( ∇ B ) 2 ( u ) + 2 cB 2 ( u ) ≡ 0 , < ∇ N ( u ) , ∇ B ( u ) > +2 cN ( u ) B ( u ) − B ( u ) ≡ 0 , the r e cipr o c al metric ˆ g ii ( u ) is flat. 4.1 Reciproc al transformations whic h preserv e the flatness prop erty of the metric and Lie–equi v alen t systems W e end th e pap er gi ving the ge ometrical inte rp retation o f Theorem 4. 1 in the case in whic h b oth the initial and the transformed metrics are flat. In deed, local Hamiltonian systems connected b y canonical recipro cal transformations h a v e nice geometrical prop erties as first observ ed by F erap on to v [11]. Using the theorems pro v en by F er ap on to v in [11] and Theorem 4.1, in Theorem 4.12 w e sh o w that the lo cal Hamilto nian stru ctures of tw o DN Hamiltonian systems in Riemann inv ariants are connected b y a canonical recipro cal transformation if and only if the asso ciated h yp ersurfaces are Lie equiv alen t. A DN h ydro d ynamic type system as in (1) in flat co ordinates tak es the form u i t = v i j ( u ) u i x = ǫ i δ ij d dx  δ H δ u j  , ( 51) with ǫ i = ± 1 and th e Hamiltonian H = R h ( u ) dx . T o eac h system as in (51), there corresp onds a h yp ersurface M n in a pseudo eu clidean space E n +1 in such a wa y that equations (51) ma y b e transformed into the form n t = r x , (52) where n and r are resp ectiv ely the un it n ormal and the radius v ector of M n (see [11]). Let u 1 , . . . , u n b e an y syste m of curvilinear c o ord inates on M n . S ince the tangen t bu ndle T M n is spann ed b y ∂ r ∂ u i , i = 1 , . . . , n and ∂ n ∂ u i ∈ T M n , i = 1 , . . . , n , it is p ossible to in tro d u ce the so-calle d W eingarten (or shap e) op erator w i j ( u ), b y the form ulas ∂ n ∂ u j = w i j ( u ) ∂ r ∂ u j , and (52) m a y b e rewritten in the f orm (51), with v i j = ( w i j ) − 1 . T hen the eigen v alues of the ve lo cities v i j ( u ) are the radii of th e p rincipal curv atures of M n and the corresp onding eigenfoliat ions are the curv atur e su rfaces of M n (see [11]). In particular, the h yp ersur- face M n is called Dup in if its pr incipal curv atures are constan t along the corresp onding curv ature hyp ersurfaces and such h yp ersurfaces corresp ond to w eakly–nonlinear hydro dy- namic typ e systems ( i.e. eac h eigenv alue of th e matrix v ij ( u ) in (51) is constan t along the corresp ondin g eigenfolation) as pro ven in [11]. F ollo wing [11], let us call the h yp ersurfaces associated to t w o DN systems as in (51) Lie–equiv alent if th ey are connected by a Lie sphere transformation (see [14], [3]). 20 The n + 2–canonical in tegrals (the n Casimirs, the momentum and the Hamiltonian) tak e the follo wing form in the flat co ord inates u 1 , . . . , u n (see [11]), H = hdx + 1 2  n X m =1 ǫ m ( ∂ m h ) 2 + 1  dt, P = 1 2  n X m =1 ǫ m u 2 m + 1  dx −  h − n X m =1 u m ∂ m h  dt, U i = u i dx + ǫ i ∂ i hdt, i = 1 , . . . , n. Then the follo win g T heorem settles th e follo wing imp ortant relatio n b et w een equiv ale nt h yp ersurfaces and recipro cal tr an s formations. Theorem 4.11 [11] A) Supp ose that the asso ciate d hyp ersurfac es of two DN systems as i n (51) ar e Lie– e quivalent. Then the lo c al Hamiltonian structur es of the systems themselves ar e c onne cte d by a r e cipr o c al tr ansfo rmation. B) Su pp ose that the lo c al Hamiltonian structur es of two DN systems ar e c onne cte d by the c anonic al r e cipr o c al tr ansfo rmation d ˆ x = αH + β P + n X m =1 γ i U i + η 1 dx + η 2 dt, d ˆ t = ˜ αH + β P + n X m =1 ˜ γ i U i + ˜ η 1 dx + ˜ η 2 dt, with α, β , γ m , η j , ˜ α, ˜ β , ˜ γ m , ˜ η j , ( m = 1 , . . . , n , j = 1 , 2 ) c onstants such that ( α + η 1 ) 2 + ( β + η 2 ) 2 − n X m =1 ǫ m γ 2 m − η 2 1 − η 2 2 = 0 , ( ˜ α + ˜ η 1 ) 2 + ( ˜ β + ˜ η 2 ) 2 − n X m =1 ǫ m ˜ γ 2 m − ˜ η 2 1 − ˜ η 2 2 = 0 , ( ˜ α + ˜ η 1 )( α + η 1 ) + ( ˜ β + ˜ η 2 )( β + η 2 ) − n X m =1 ǫ m ˜ γ m γ m − ˜ η 1 η 1 − ˜ η 2 η 2 = 0 . (53) Then the hyp ersurfac es asso ci ate d to the two DN systems ar e Lie–e qu ivalent. W e recall th at any n × n DN t yp e sy s tem as in (51) admits the n + 2 –canonical in tegrals, so that Theorem 4.11 applies also to the case in which Riemann in v arian ts do not exist. If w e restrict ourselv es to the case of DN systems whic h p ossess Riemann in v arian ts, then the compatibilit y conditions (53) in the flat coord inates h av e th eir corresp ondence in the conditions A.i)-A .iii) expressed in the Riemann in v arian ts in Theorem 4.1. Moreo v er, the same t heorem give s the complete c haracteriza tion of the recipro cal trans- formations w hic h pr eserv e local Hamiltonian structur e when Riemann in v arian ts exist, so that the follo wing stronger geomet rical c haracterizatio n h olds in the presen t case. Theorem 4.12 L e t n ≥ 5 . The hyp ersurfac es asso ciate d to two diagonalizable strictly hyp erb olic DN systems ar e c onne cte d b y a Lie spher e tr ansformation if and only if the c or- r e sp onding lo c al Hamiltonian structur es of the two DN systems ar e c onne cte d by c anonic al r e cipr o c al tr ansform ation satisfying The or em 4.1 . 21 Finally , w e lik e to p oint out that there is no geo metrical inte rp retation of the recipro cal transformations wh en the lo calit y of the Hamiltonian structure is not preserv ed by the transformation and b oth the initial and the transformed systems are of DN t yp e. The most in teresting example in this class are the gen us g mo du lated Camassa-Holm equations already ment ioned in the int ro duction: suc h system p ossesses t wo compatible flat metrics whic h are mapp ed to t w o non–flat metrics asso ciated to the g mo dulated equations of the first negativ e Kortewe g–de V ries fl o w b y a recipro cal trans f ormation as pro v en in [2 ]. Then, from Th eorem 4.11, it follo ws that the hyp ersurfaces asso ciated to the t w o systems are not Lie–equiv alen t. Ac kno wledgemen ts This researc h has b een partially su pp orted b y ESF Programme MISGAM, by R TN ENIGMA and by PRIN2006 ”Metodi geometrici nella teoria delle onde non lineari ed applicazioni”. I warmly thank F.V. F erap on to v, M. Pa vlov and Y. Zhang for their inte rest in the present researc h. I am also grateful to to the referees for man y usefu l remarks and esp ecially to the second referee for many v aluable observ ations wh ic h help ed me in improving the man uscrip t. References [1] S. Ab enda, T. Gra v a, Mo du lation of Camassa-Holm equ ation and recipro cal tran s for- mations. A nn. Inst. F ourier 55 , 1803– 1834 (200 5). [2] S. Ab enda, T. Gra v a, Recipro cal transformations and flat metrics on Hurwitz spaces. J. Phys. A: Math. The or. 40 , 10769– 10790 (2007). [3] T. E. Cecil, Lie spher e ge ometry. With applic atio ns to submanifolds. 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