On compatible metrics and diagonalizability of non-locally bi-Hamiltonian systems of hydrodynamic type

On compatible metrics and diagonalizabili ty of non-lo cally bi-Hamiltonian systems of h ydro dynamic t yp e 1 O. I. Mokho v Abstract W e study bi-Hamiltonian systems of h ydro dynamic t yp e with non-singular (semi- simple) non-lo cal bi-Hamiltonian structures and pro v e that suc h systems of hy dro dy- namic t ype are diagonalizable. Moreo v er, w e pro v e that for an arbitra ry non-singular (semisimple) non-lo cally bi-Hamiltonian system of h ydro dynamic type, there exist lo cal co ordinates (Riemann in v arian ts) suc h that a ll the related matrix differential-geometric ob jects, namely , the mat r ix V i j ( u ) of this system of h ydro dynamic t yp e, the metrics g ij 1 ( u ) and g ij 2 ( u ) and the affinors ( w 1 ,n ) i j ( u ) and ( w 2 ,n ) i j ( u ) of the non-singular non-lo cal bi-Hamiltonian structure of this system, are diagonal in t hese lo cal co ordina t es. The pro of is a natural consequence of the general results of the theory of compatible metrics and the theory of non-lo cal bi-Hamilto nia n structures dev elop ed earlier b y the presen t author in [21]–[33]. In tro duc tion In this pap er we consider (1+1)-dimensional non- singular (semisimple) non-lo cally bi- Hamiltonian s ystems of h ydro dynamic t yp e and prov e their diag o nalizabilit y . Moreov er, w e pro ve tha t for an a r bit r a ry non-singular (semisimple) non-lo cally bi-Hamiltonian system of h ydro dynamic ty p e, there exist lo cal co ordinates ( R iemann in v arian ts) suc h that all the r elat ed matrix differential-geometric o b jects, namely , the matrix V i j ( u ) o f this system of h ydro dynamic ty p e, the metrics g ij 1 ( u ) and g ij 2 ( u ) and the affinors ( w 1 ,n ) i j ( u ) and ( w 2 ,n ) i j ( u ) of the non- singular non-lo cal bi-Hamiltonian structure of this system, are diagonal in these lo cal co ordinates. Let us giv e here v ery briefly basic well-kno wn notions and results necessary f or us. Recall that (1 + 1)- dimensional systems of hydr o dynamic typ e [1] are arbitrary (1 + 1)-dimensional ev olutio n quasilinear systems of first-o rder partial differen tial equations, i.e., equations of the form u i t = V i j ( u ) u j x , 1 ≤ i, j ≤ N , (1) 1 The work was supp or ted by the Ma x-Planck-Institut f ¨ ur Mathematik (Bo nn, Ger many), by the Russian F oundation for Basic Resea rch (pro ject no. 08-01 -0046 4) and by the prog ramme “Leading Scient ific Sc ho ols ” (pro ject no . NSh-1 824.20 08.1). 1 where u = ( u 1 , . . . , u N ) are lo cal co ordinates on a certain smo o th N -dimensional mani- fold or in a domain of R N (or C N ); u i ( x ) are functions (fields) of one v ariable x that are ev olving with resp ect to t ; V i j ( u ) is an arbitrary ( N × N )-matrix dep ending on u (this matrix is a mixed tensor of the type (1, 1), i.e., a n affinor, with resp ect to lo cal c hanges of co ordinates u ). Non-lo cally Hamiltonian systems of h ydro dynamic t yp e W e will consider systems of the form (1) that are Hamiltonian with respect to arbi- trary non-degenerate non-lo cal P oisson brack ets of h ydro dynamic t yp e (the F e rap on tov brac k ets [2], see also [3] a nd [1] fo r t he Mokho v–F erap o n tov brac k ets and the Dubrovin– No vik ov brack ets in partial non-lo cal and lo cal cases, resp ectiv ely), i.e., u i t = V i j ( u ) u j x = { u i ( x ) , H } , 1 ≤ i, j ≤ N , (2) where the functional H = Z h ( u ( x )) dx (3) is the Hamiltonian of the system (2) (the function h ( u ) is the densit y of the Hamiltonian) and the Poiss on brac ke t has the form { u i ( x ) , u j ( y ) } = P ij δ ( x − y ) , 1 ≤ i, j ≤ N , (4) P ij = g ij ( u ( x )) d dx + b ij k ( u ( x )) u k x + + L X m,n =1 µ mn ( w m ) i k ( u ( x )) u k x  d dx  − 1 ◦ ( w n ) j s ( u ( x )) u s x , (5) where the co efficien ts g ij ( u ) , b ij k ( u ) , and ( w n ) i j ( u ) , 1 ≤ i, j, k ≤ N , 1 ≤ n ≤ L, are smo oth functions of lo cal co ordinates, det( g ij ( u )) 6 = 0 , µ mn is an arbitrary non-degenerate sym- metric constan t matrix, µ mn = µ nm , µ mn = const , det( µ mn ) 6 = 0 . F or t w o arbitrary functionals I and J t he P oisson brac ke t (4), (5) has the form { I , J } = Z δ I δ u i ( x ) P ij δ J δ u j ( x ) dx. (6) P oisson brack ets of the form (5), (6 ) were intro duced and studied b y F erap on tov in [2]; these br a c k ets are a non-lo cal g eneralization of the Dubrovin–No vik o v brac k ets (lo cal P oisson brac k ets of h ydro dynamic t yp e generated by flat metrics g ij ( u ); there are no non- lo cal terms in this case, L = 0, or ( w n ) i j ( u ) = 0) [1] and the Mokhov –F erap on tov brac ke ts (non-lo cal P oisson brac k ets of h ydro dynamic type generated b y metrics of constan t cur- v ature K ; in this case L = 1, µ 11 = K , ( w 1 ) i j ( u ) = δ i j ) [3]. F erap onto v prov ed that a 2 non-lo cal op erator P ij of the for m ( 5 ) give s a Pois son brack et (6) if and only if there is an N -dimensional submanifold with flat no r mal bundle in an ( N + L )-dimen sional pseudo-Euclidean space suc h that g ij ( u ) is the contra v ariant first fundamental fo rm; b ij k ( u ) = − g is ( u )Γ j sk ( u ); Γ j sk ( u ) are the Christoffel sym b ols of the Levi-Civita connection of the metric g ij ( u ); ( w n ) i j ( u ) , 1 ≤ n ≤ L, are the W e inga r t en op erators (the W e ingart en affinors) of the submanifold; and µ mn is the Gr am matrix of t he corresp onding pa r allel bases in the no rmal spaces of the submanifold (a ll to rsion fo rms of the submanifold with flat normal bundle v anish in these bases in the normal spaces). In o t her w ords, the non-lo cal op erator (5) giv es a P oisson brack et ( 6 ) if and only if its co efficien ts satisfy the relations (see also [4]) g ij = g j i , (7) ∂ g ij ∂ u k = b ij k + b j i k , (8) g is b j k s = g j s b ik s , (9) g is ( w n ) j s = g j s ( w n ) i s , (10) ( w n ) i s ( w m ) s j = ( w m ) i s ( w n ) s j , (11) g is g j r ∂ ( w n ) k r ∂ u s − g j r b ik s ( w n ) s r = g j s g ir ∂ ( w n ) k r ∂ u s − g ir b j k s ( w n ) s r , (12) g is  ∂ b j k s ∂ u r − ∂ b j k r ∂ u s  + b ij s b sk r − b ik s b sj r = L X m =1 L X n =1 µ mn g is  ( w m ) j r ( w n ) k s − ( w m ) j s ( w n ) k r  . (13) Non-lo cally Hamiltonian affinors The Hamiltonian H of the system (2)–(6) mus t also b e a first in tegral of all the systems of hydrodynamic type that are giv en b y the affinors ( w n ) i j ( u ) , 1 ≤ n ≤ L, of the non-lo cal op erator (5) (these systems are called the structur al flows o f the non-lo cal P oisson brack et (4 ) – (6)) [2]: u i t n = ( w n ) i j ( u ) u j x , H t n = 0 , 1 ≤ n ≤ L. (14) F or eac h n , 1 ≤ n ≤ L , there exist a function f n ( u ) suc h that ∂ h ∂ u j ( w n ) j s ( u ) = ∂ f n ∂ u s , 1 ≤ n ≤ L. (15) In this case the affinor V i j ( u ) of the system of hy dro dynamic type (2)–(6) has the form V i j ( u ) = g is ( u ) ∂ 2 h ∂ u s ∂ u j − g is ( u )Γ p sj ( u ) ∂ h ∂ u p + L X m,n =1 µ mn ( w m ) i j ( u ) f n ( u ) , (16) 3 i.e., V i j ( u ) = g is ( u ) ∇ s ∇ j h ( u ) + L X m,n =1 µ mn ( w m ) i j ( u ) f n ( u ) = = ∇ i ∇ j h ( u ) + L X m,n =1 µ mn ( w m ) i j ( u ) f n ( u ) , (17) where ∇ k is the cov ar ian t differen tiation generated by the Levi-Civita c onnection Γ j sk ( u ) of the metric g ij ( u ) . W e will call an affinor V i j ( u ) Hamiltonian (or non-lo c al ly Hamiltonian ) if there ex- ist an N - dimensional submanifold with flat normal bundle in an ( N + L )-dimensional pseudo-Euclidean space and functions h ( u ) and f n ( u ), 1 ≤ n ≤ L , suc h that the affinor V i j ( u ) has the form (17), where g ij ( u ) is the contra v ariant first fundamental form of the submanifold; Γ j sk ( u ) are the Christoffel sym b ols o f the Levi-Civita connection o f the metric g ij ( u ); ( w n ) i j ( u ) , 1 ≤ n ≤ L, are the W eingarten op erators o f the submanifold; and µ mn is the Gram matrix of the corresp onding parallel bases in the normal spaces of the submanifold (suc h t ha t all t o rsion forms of the submanifold with flat normal bundle v anish in these bases in the normal spaces), and the functions h ( u ) and f n ( u ), 1 ≤ n ≤ L , satisfy relations (15): ( w n ) j s ( u ) ∇ j h ( u ) = ∇ s f n ( u ) , 1 ≤ n ≤ L. (18) In this case w e will also sp eak that the affinor V i j ( u ) (17), (18) is Hamilto nian with resp ect to the corresp onding non-lo cal P oisson brac k et of h ydro dynamic ty p e (4)–(6) . Obviously , this definition is inv ariant. W e not e that one can also consider it as a definition of a non-lo cally Hamiltonian system of hy dro dynamic t yp e (1). Affinors that a re Hamiltonian with resp ect to the Dubro vin–Novik ov brac k ets ( lo c al ly Hamiltonian affinors ) w ere studied in detail b y Tsarev in the remark able work [5]. Affi- nors that are Hamilto nian with resp ect to the non-lo cal Mokhov–F erapo n tov brac k ets (the Mokho v–F erap onto v affinors) w ere studied in detail in the pap er [3]. Using relations (7)–(13) it is easy to prov e that the follow ing relations alw a ys hold for non-lo cally Hamiltonian affinors V i j ( u ): g is ( u ) V s j ( u ) = g j s ( u ) V s i ( u ) , (19) ∇ j V i k ( u ) = ∇ k V i j ( u ) . (20) Here g ij ( u ) is the in v erse of the matrix g ij ( u ), g is ( u ) g sj ( u ) = δ j i (the co v ariant metric). F or lo cally Ha milto nian affinors these imp ortant relations are very simple in flat lo cal co ordinates of the metric g ij ( u ) and Tsarev prov ed that in this flat case relations (1 9) and (20) a re no t only necessary but also sufficien t f o r an affinor to b e lo cally Hamiltonian 4 (an affinor V i j ( u ) is lo cally Hamiltonian if and only if there exists a flat metric g ij ( u ) suc h tha t relations (19) and (20) hold) [5]. This result w as generalized to the case of the Mokho v–F erap on tov affinor s in [3]: a n affinor V i j ( u ) is Hamiltonia n with resp ect to a non-lo cal Mokhov –F erap onto v brack et if and only if there exists a metric g ij ( u ) of constan t curv ature suc h that relations (19) and (20) hold. Let the affinor V i j ( u ) of a system of h ydro dynamic type (1) satisfy relations (19) a nd (20), i.e., there exists a metric g ij ( u ) such that relatio ns (1 9) and (20) hold. If this system of h ydro dynamic ty p e is diagona lizable, i.e., there exist lo cal co ordinates such that the affinor V i j ( u ) is a diagonal matrix V i j ( u ) = V i ( u ) δ i j in these sp ecial lo cal co ordinat es (suc h lo cal co ordinates a r e called Riemann inva ri a nts ), and strictly hyperb olic, i.e., all the eigen v alues V i ( u ), 1 ≤ i ≤ N , are distinct ( V i ( u ) 6 = V j ( u ) when i 6 = j ), then it can b e inte gr a ted by the generalized ho dograph metho d (Tsarev, see [5 ]). In this case, relation (19) is eq uiv a lent to the condition that the me tric g ij ( u ) is also diagonal in these sp ecial lo cal co ordinates, g ij ( u ) = g i ( u ) δ ij , i.e., the Riemann in v ariants are orthogonal curvilinear co o rdinates in the cor r espo nding pseudo-Riemannian space, and relatio n (20) is equiv alent to t he condition ∂ V i ∂ u j = ∂ ln p g i ( u ) ∂ u j ( V j ( u ) − V i ( u )) , i 6 = j. (21) Hence, the follo wing relation holds fo r the eigen v a lues V i ( u ): ∂ ∂ u k  1 ( V j ( u ) − V i ( u )) ∂ V i ∂ u j  = ∂ ∂ u j  1 ( V k ( u ) − V i ( u )) ∂ V i ∂ u k  . (22) A strictly h yp erb olic diago nal system o f h ydro dynamic type is called semi-Hamiltonian if relations (22) hold (Tsarev, see [5]). In [5 ] Tsarev pro v ed that an y strictly h yp erb olic diagonal semi-Hamiltonian system of h ydro dynamic type is in tegrable b y the generalized ho dograph metho d. Diagonalizable affinors Recall that the ve ry imp ortant problem of diagonalizability for a n affinor, whic h had b een p osed, in f a ct, by Riemann, w as completely solv ed b y Haan tjes in [6 ] on the base of the previous Nijenh uis’ results [7]. An affinor V i j ( u ) is diag o nalizable b y a lo cal c hange of co ordinat es in a domain if a nd only if it is diagonalizable at an y p oint and its Haan tjes tensor v anishes. The Haantjes tensor of an affinor V ( u ) = V i j ( u ) is the follo wing tensor of t he t yp e (1 , 2) (a sk ew-symmetric v ector- v alued 2-form) generated b y the affinor V i j ( u ): H ( X , Y ) = N ( V ( X ) , V ( Y )) + V 2 ( N ( X , Y )) − V ( N ( X , V ( Y ))) − V ( N ( V ( X ) , Y )) , (23) 5 where X ( u ) and Y ( u ) are arbitrary ve ctor fields, V ( X ) is the vec tor field V i j ( u ) X j ( u ), N ( X , Y ) is the Nijenhuis tensor of the affinor V i j ( u ), i.e., the follo wing tensor of the t yp e (1 , 2) (a sk ew-symmetric ve ctor- v alued 2- form) generated b y the affinor V i j ( u ): N ( X , Y ) = [ V ( X ) , V ( Y )] + V 2 ([ X , Y ]) − V ([ X , V ( Y )]) − V ([ V ( X ) , Y ]) , (24) where [ X, Y ] is the comm uta tor of the v ector fields X ( u ) and Y ( u ) . In comp onents, the Nijenh uis tensor of the affinor V i j ( u ) has the form N k ij ( u ) = V s i ( u ) ∂ V k j ∂ u s − V s j ( u ) ∂ V k i ∂ u s + V k s ( u ) ∂ V s i ∂ u j − V k s ( u ) ∂ V s j ∂ u i (25) and the Haa n tjes tensor of the affinor V i j ( u ) has the form H i j k ( u ) = V i s ( u ) V s r ( u ) N r j k ( u ) − V i s ( u ) N s r k ( u ) V r j ( u ) − − V i s ( u ) N s j r ( u ) V r k ( u ) + N i sr ( u ) V s j ( u ) V r k ( u ) . (26) Recall also that in v ariant tensor conditions that a strictly h yp erb olic system of h ydro- dynamic t yp e is semi-Hamiltonian w ere found in [8]. Non-lo cally b i-Hamiltonian systems of hydro dynamic t yp e W e will consider bi-Hamiltonian systems o f h ydro dynamic type. Recall that tw o P ois- son brack ets are called c omp atible if an y linear com bination of these P oisson brack ets is also a P oisson brac k et [9], and a sys tem of equations that is Hamiltonian w ith respect to t w o linearly indep enden t compatible P oisson bra c k ets is called bi-Hamiltonia n . In this pap er we will consider systems of hy dro dynamic ty p e that are bi-Hamiltonian with re- sp ect to t wo line arly indep enden t compatible non-degenerate non-lo cal P oisson brac ke ts of h ydro dynamic type (4 ) –(6), u i t = V i j ( u ) u j x = { u i ( x ) , H 1 } 1 = { u i ( x ) , H 2 } 2 , 1 ≤ i, j ≤ N , (27) H 1 = Z h 1 ( u ( x )) dx, H 2 = Z h 2 ( u ( x )) dx, (28) { I , J } 1 = Z δ I δ u i ( x ) P ij 1 δ J δ u j ( x ) dx, { I , J } 2 = Z δ I δ u i ( x ) P ij 2 δ J δ u j ( x ) dx, (29) P ij 1 = g ij 1 ( u ( x )) d dx + b ij 1 ,k ( u ( x )) u k x + + L X m,n =1 µ mn 1 ( w 1 ,m ) i k ( u ( x )) u k x  d dx  − 1 ◦ ( w 1 ,n ) j s ( u ( x )) u s x , (30) 6 P ij 2 = g ij 2 ( u ( x )) d dx + b ij 2 ,k ( u ( x )) u k x + + L X m,n =1 µ mn 2 ( w 2 ,m ) i k ( u ( x )) u k x  d dx  − 1 ◦ ( w 2 ,n ) j s ( u ( x )) u s x . (31) W e will call an affinor V i j ( u ) bi-Hamiltonian (or non-lo c a l ly bi-Hamiltonian ) if this affinor is Hamiltonian with resp ect to t wo linearly indep enden t compatible no n-degene- rate no n-lo cal P oisson brac k ets o f hydrodynamic ty p e (4 ) –(6). This definition is in v ari- an t. In this paper w e prov e that (1 + 1)-dimensional n on- singular (semisimple) non-lo cally bi-Hamiltonian systems o f hyd ro dynamic type are diagonalizable. Recall tha t a pair of pseudo-Riemannian metrics g ij 1 ( u ) and g ij 2 ( u ) is called non-sing ular (or semisimple ) if the eigen v alues of this pair of metrics, i.e., the ro ots o f t he equation det( g ij 1 ( u ) − λg ij 2 ( u )) = 0 , (32) are distinct. In this case the non-lo cally bi-Hamiltonian system of hy dro dynamic t yp e (27)–(31) and the corresp onding bi-Hamiltonian affinor are also called non-singular (or semisimple ). It is imp or tan t t o note that, generally sp eaking, in tegrable bi-Hamiltonian systems of hy dro dynamic type are not necess arily diagonalizable if w e consider an other class of compatible P oisson brack ets ( even if b oth the compatible Poiss on brac k ets are lo cal). This is a no ntrivial f a ct and we g ive here a v ery imp ortan t example in detail. Example ( a non-diagonaliza b le inte gr able bi-Hamiltonian system of hydr o dynamic typ e [10]-[12]). Let us consider the asso ciativity equations of tw o-dimensional t o p ological quan tum field t heories (the Witten–Dijkgraaf–V erlinde–V erlinde equations, see [13]–[16]) for a f unction ( a p otential ) Φ = Φ( u 1 , . . . , u N ), N X k =1 N X l =1 ∂ 3 Φ ∂ u i ∂ u j ∂ u k η k l ∂ 3 Φ ∂ u l ∂ u m ∂ u n = N X k =1 N X l =1 ∂ 3 Φ ∂ u i ∂ u m ∂ u k η k l ∂ 3 Φ ∂ u l ∂ u j ∂ u n , (33) where η ij is an arbitrary constant nondegenerate symmetric matrix, η ij = η j i , η ij = const , det( η ij ) 6 = 0. W e recall that the asso ciativity equations (33) are consisten t and in tegrable by the in ve rse scattering metho d, they p ossess a ric h set of nontrivial solutions, and eac h solution Φ( u 1 , . . . , u N ) of the asso ciativit y equations (33) giv es N -par ameter deformations of sp ecial F r o b enius algebras (some sp ecial comm utativ e a sso ciat ive alge- bras equipp ed with nondegenerate inv ariant symmetric bilinear forms) (see [13 ]). Indeed, consider algebras A ( u ) in an N -dimensional v ector space with the basis e 1 , . . . , e N and the m ultiplication (see [13]) e i ◦ e j = c k ij ( u ) e k , c k ij ( u ) = η k s ∂ 3 Φ ∂ u s ∂ u i ∂ u j . (34) 7 F or all v alues of the parameters u = ( u 1 , . . . , u N ) the algebras A ( u ) are comm utativ e, e i ◦ e j = e j ◦ e i , and the asso ciativit y conditio n ( e i ◦ e j ) ◦ e k = e i ◦ ( e j ◦ e k ) (35) in the algebras A ( u ) is equiv alen t to equations (33). The matrix η ij in v erse to the matrix η ij , η is η sj = δ i j , define s a no ndegenerate inv ariant symmetric bilinear form on the algebras A ( u ), h e i , e j i = η ij , h e i ◦ e j , e k i = h e i , e j ◦ e k i . (36) Recall that lo cally the tangen t space at ev ery p oint of an y F rob enius manifold (see [13 ]) p ossesses the struc ture of F rob enius algebra (34)–(36), wh ich is determined by a solutio n of the asso ciativity equations ( 3 3) and smo othly dep ends on the p oin t. Let N = 3 and the metric η ij b e an tidiago nal ( η ij ) =   0 0 1 0 1 0 1 0 0   , (37) and the function Φ( u ) has the form Φ( u ) = 1 2 ( u 1 ) 2 u 3 + 1 2 u 1 ( u 2 ) 2 + f ( u 2 , u 3 ) . In this case e 1 is the unit in the F ro b enius a lg ebra (34)– (36), and the asso ciativity equations (33) for the function Φ( u ) ar e equiv alent to the followin g remark able inte gra ble Dubro vin equation for the function f ( u 2 , u 3 ): ∂ 3 f ∂ ( u 3 ) 3 =  ∂ 3 f ∂ ( u 2 ) 2 ∂ u 3  2 − ∂ 3 f ∂ ( u 2 ) 3 ∂ 3 f ∂ u 2 ∂ ( u 3 ) 2 . (38) W e in tro duce here new indep enden t v ariables: x = u 2 , t = u 3 . The equation (38) tak es the form f ttt = ( f xxt ) 2 − f xxx f xtt . (39) This equation is connected to quan tum cohomology of pro jectiv e pla ne and classical problems of en umerativ e geometry (see [17]). It w as pro v ed b y the pres en t author in [18] (see also [1 9 ], [10]–[12]) that the equation (39) is equiv alent to t he in tegrable non- dia gonalizable system of h ydro dynamic t yp e   a 1 a 2 a 3   t =   0 1 0 0 0 1 − a 3 2 a 2 − a 1     a 1 a 2 a 3   x , (40) a 1 = f xxx , a 2 = f xxt , a 3 = f xtt . (41) 8 The first Hamiltonian structure of system (40) giv en b y a Dubrov in–Novik ov bra ck et w as found in [20]: { I , J } 1 = Z δ I δ a i ( x ) M ij 1 δ J δ a j ( x ) dx, (42) M 1 = ( M ij 1 ) =   − 3 2 1 2 a 1 a 2 1 2 a 1 a 2 3 2 a 3 a 2 3 2 a 3 2(( a 2 ) 2 − a 1 a 3 )   d dx + +   0 1 2 a 1 x a 2 x 0 1 2 a 2 x a 3 x 0 1 2 a 3 x (( a 2 ) 2 − a 1 a 3 ) x   . (43) The metric ( g ij 1 ( a )) =   − 3 2 1 2 a 1 a 2 1 2 a 1 a 2 3 2 a 3 a 2 3 2 a 3 2(( a 2 ) 2 − a 1 a 3 )   (44) is flat and the Poiss on brack et o f h ydro dynamic type (42), (43) is lo cal (a Dubrovin– No vik ov brack et). The functional H 1 = Z a 3 dx (45) is the correspo nding Hamiltonian of system (40). The bi- Hamiltonian structure of system ( 4 0) w as found in [10] (see also [11], [12]). The second Hamiltonian structure of system (40) is given by a homogeneous third-order Dubro vin–Novik ov brac k et: { I , J } 2 = Z δ I δ a i ( x ) M ij 2 δ J δ a j ( x ) dx, (46) M 2 = ( M ij 2 ) =   0 0 1 0 1 − a 1 1 − a 1 ( a 1 ) 2 + 2 a 2    d dx  3 + +   0 0 0 0 0 − 2 a 1 x 0 − a 1 x 3( a 2 x + a 1 a 1 x )    d dx  2 + +   0 0 0 0 0 0 0 0 a 2 xx + ( a 1 x ) 2 + a 1 a 1 xx   d dx . (47) The second P oisson brac k et (46), (47) is compatible with the first P oisson brac k et (42) , (43). 9 The metric ( g ij 1 ( a )) =   0 0 1 0 1 − a 1 1 − a 1 ( a 1 ) 2 + 2 a 2   (48) is flat. The non-lo cal functional H 2 = − Z   1 2 a 1  d dx  − 1 a 2 ! 2 +  d dx  − 1 a 2 !  d dx  − 1 a 3 !   dx (49) is the correspo nding Hamiltonian of system (40). First of all, we note that the class of non-lo cally bi-Hamiltonian systems of h ydro dy- namic t yp e (27)–(31) is v ery ric h, there a re man y w ell-kno wn imp o r t an t examples arising in v arious applications. An explicit general construction of lo cally a nd no n-lo cally bi- Hamiltonian systems of h ydro dynamic type and corresp onding in tegrable hierarc hies that are generated by pairs of compatible P oisson brac k ets of hydrodynamic type and in tegrable description o f lo cal and non-lo cal compatible P oisson brack ets of hydrody- namic t yp e w ere found and studied b y the presen t author in [21]–[33], [4] (see also [34 ], [35], [13]). Compatible metrics and non-lo cally bi-Hamiltonian systems of h ydro d ynamic t yp e No w recall some necessary basic facts of the general theory of compatible metrics [2 7]– [32]. Tw o Riemannian or pseudo-Riemannian c ontra v ariant metrics g ij 1 ( u ) and g ij 2 ( u ) are called c omp atible if fo r a n y linear com bination of these metrics g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) , (50) where λ 1 and λ 2 are arbitrary constan ts suc h that det( g ij ( u )) 6 = 0, the co efficien ts of the corresp onding Levi–Civita connections and the comp onen ts of the corresp onding Riemannian curv ature tensors are related by the same linear formula [27]–[29]: Γ ij k ( u ) = λ 1 Γ ij 1 ,k ( u ) + λ 2 Γ ij 2 ,k ( u ) , (51) R ij k l ( u ) = λ 1 R ij 1 ,k l ( u ) + λ 2 R ij 2 ,k l ( u ) . (52) The indices of the co efficien ts of the Levi–Civita connections Γ i j k ( u ) and the indices of the R iemannian curv ature tensors R i j kl ( u ) are raised and low ered b y the metrics corresp onding to them: Γ ij k ( u ) = g is ( u )Γ j sk ( u ) , Γ i j k ( u ) = 1 2 g is ( u )  ∂ g sk ∂ u j + ∂ g j s ∂ u k − ∂ g j k ∂ u s  , R ij k l ( u ) = g is ( u ) R j sk l ( u ) , R i j kl ( u ) = ∂ Γ i j l ∂ u k − ∂ Γ i j k ∂ u l + Γ i pk ( u )Γ p j l ( u ) − Γ i pl ( u )Γ p j k ( u ) . 10 Tw o Riemannian or pseudo-Riemannian con tra v a rian t metrics g ij 1 ( u ) a nd g ij 2 ( u ) a re called almost c omp atible if for an y linear comb inatio n of these metrics (50 ) r elat io n (51) holds [27]–[29]. Let us in tro duce the a ffinor v i j ( u ) = g is 1 ( u ) g 2 ,sj ( u ) (53) and consider the Nijenh uis tensor of this affinor N k ij ( u ) = v s i ( u ) ∂ v k j ∂ u s − v s j ( u ) ∂ v k i ∂ u s + v k s ( u ) ∂ v s i ∂ u j − v k s ( u ) ∂ v s j ∂ u i . (54) Theorem 1 [27]–[29]. Any two metrics g ij 1 ( u ) an d g ij 2 ( u ) ar e almost c omp atible if and on l y if the c orr esp onding Nijenhuis tensor N k ij ( u ) (54) vanishes. Assume that a pair o f metrics g ij 1 ( u ) and g ij 2 ( u ) is non-singular, i.e., the eigenv alues of this pair of metrics are dis tinct. F urthermore, a ssume that the metrics g ij 1 ( u ) and g ij 2 ( u ) are almost compatible, i.e., the corresp onding Nijenhuis tensor N k ij ( u ) (54) v anishes. It w as pro ve d in our pap ers [27]–[29] that, in this case, the metrics g ij 1 ( u ) and g ij 2 ( u ) are compatible, i.e., relation (52) holds. It is ob vious that the eigenv alues of the pair of metrics g ij 1 ( u ) and g ij 2 ( u ) coincide with the eigenv alues of the affinor v i j ( u ) (53). But it is w ell kno wn that if all eigen v alues of a n affinor are distinct, then it a lw a ys fo llows from the v anishing of the Nijenh uis tensor of t his affinor that there exist s p ecial local co ordinates ( Riemann invariants ) suc h that, in these co ordinates, the affinor reduces to a diagona l form in the corresp onding neigh b ourho o d [7] (see also [6]). Hence, w e can consider tha t the affinor v i j ( u ) is diagonal in the lo cal co ordinates (Riemann in v ariants) u 1 , ..., u N , i.e., v i j ( u ) = f i ( u ) δ i j , (55) where is no summation o v er the index i . By our assumption, the eigen v a lues f i ( u ) , i = 1 , ..., N , coinciding with the eigen v alues of the pair of metrics g ij 1 ( u ) and g ij 2 ( u ) are distinct: f i ( u ) 6 = f j ( u ) if i 6 = j. (56) Lemma 1. If the affinor v i j ( u ) (53) is diagonal in c ertain lo c al c o or dina tes ( Riemann invariants ) and al l its eigenvalues ar e distinct, then , in these c o or dina tes, the metrics g ij 1 ( u ) an d g ij 2 ( u ) ar e also ne c essarily diago n al, i.e., in this c ase b oth the metrics g ij 1 ( u ) and g ij 2 ( u ) a r e diagonal in the Riemann invariants. Actually , w e hav e g ij 1 ( u ) = f i ( u ) g ij 2 ( u ) . 11 It fo llo ws from the symmetry of the metrics g ij 1 ( u ) and g ij 2 ( u ) that for any indices i a nd j ( f i ( u ) − f j ( u )) g ij 2 ( u ) = 0 , (57) where is no summation o ve r indices, i.e., g ij 2 ( u ) = g ij 1 ( u ) = 0 if i 6 = j. Lemma 2. L et an affinor w i j ( u ) b e diago nal in c ertain lo c al c o or dinates ( Riemann invariants ) u = ( u 1 , ..., u N ) , i.e., w i j ( u ) = µ i ( u ) δ i j . 1) I f a l l the e i g e nvalues µ i ( u ) , i = 1 , ..., N , of the di a gonal affin or a r e distinct, i.e., µ i ( u ) 6 = µ j ( u ) for i 6 = j , then the Nijenhuis tensor of this affi nor vanishes if and only if the i th eigenvalue µ i ( u ) d e p ends only o n the c o or dinate u i . 2) I f al l the eigenvalues c o i n cide, then the Nijenhuis tensor vanishes. 3) I n the gener al c ase of an arbitr ary diag o nal affinor w i j ( u ) = µ i ( u ) δ i j , the Nijenhuis tensor vani s h es if and only if ∂ µ i ∂ u j = 0 (58) for al l in dic es i and j such that µ i ( u ) 6 = µ j ( u ) . It fo llo ws f r o m Lemmas 1 and 2 that fo r an y non-singular pa ir of almost compatible metrics there alwa ys exist lo cal co ordinates (R iemann in v aria n ts) in whic h the metrics ha v e the form g ij 2 ( u ) = g i ( u ) δ ij , g ij 1 ( u ) = f i ( u i ) g i ( u ) δ ij . Moreo v er, an y pair of diagonal metrics of the form g ij 2 ( u ) = g i ( u ) δ ij and g ij 1 ( u ) = f i ( u i ) g i ( u ) δ ij for an y nonzero functions f i ( u i ) , i = 1 , ..., N , (here they can be, for exam- ple, coinciding nonzero constan ts, i.e., the pair of metrics ma y b e “ singular”) is almost compatible, since the corresp onding Nijenhuis tensor a lw a ys v anishes for any pair of metrics of this f orm. It was prov ed in our pap ers [28], [29] that an arbitra r y pair of diagonal metrics of suc h the form, g ij 2 ( u ) = g i ( u ) δ ij and g ij 1 ( u ) = f i ( u i ) g i ( u ) δ ij , for ar- bitrary nonzero functions f i ( u i ) , i = 1 , ..., N , (the pair of metrics may b e “singular”), is a lwa ys compatible, i.e., in this case, relation (52) holds. W e no te that, as it w as sho wn in [27]–[2 9], in general almost compatible metrics are not necessarily compatible ev en in the case of flat metrics or metrics of constant curv ature, i.e., in the case of the Dubro vin–Novik ov or the Mokho v–F erap onto v brac k ets, but if a pair o f almost compat- ible me trics is not compatible, then this pair of metrics m ust b e singular. Th us, we ha ve the follo wing imp or t an t statemen ts. Theorem 2 [27]–[29]. If a p air of metrics g ij 1 ( u ) and g ij 2 ( u ) is non-sin gular, i.e., the r o o ts of the e quation det( g ij 1 ( u ) − λg ij 2 ( u )) = 0 (59) 12 ar e distinct, then it fol lows fr o m the van ishing of the Nijen huis tensor of the affin o r v i j ( u ) = g is 1 ( u ) g 2 ,sj ( u ) that the metrics g ij 1 ( u ) and g ij 2 ( u ) ar e c omp atible. Thus, a non- singular p air of metrics is c omp atible if an d only if the metrics ar e almost c omp a tible. Theorem 3 [27]–[29]. A n arbitr ary non- s ingular p ai r of metrics is c omp a tible if a nd only if ther e ex ist lo c al c o or din a tes ( Riemann invariants ) u = ( u 1 , ..., u N ) s uch that b oth the metrics ar e diagonal in these c o or dinates and have the fol lowing sp e cia l form: g ij 2 ( u ) = g i ( u ) δ ij and g ij 1 ( u ) = f i ( u i ) g i ( u ) δ ij , wher e one of the me trics, her e g ij 2 ( u ) , is an arb i tr ary diago nal metric and f i ( u i ) , i = 1 , ..., N , ar e arbitr ary ( gener a l ly sp e aking, c omplex ) nonzer o functions o f single varia b les. If some of the functions f i ( u i ) , i = 1 , ..., N , ar e c oinciding nonzer o c onstants, then the p air of metrics of this form is singular but, neverthele s s, c omp a tible . Theorem 4 [28]. I f non-lo c al Poisson br ackets of hydr o dynamic typ e (29)–(31) ar e c omp atible, then their metrics ar e c omp atible. In [2] F erap onto v pro v ed that a brac ke t (4)–(6) is a P oisson brack et, i.e., it is sk ew- symmetric and satisfies the Jacobi iden t it y , if and only if (1) b ij k ( u ) = − g is ( u )Γ j sk ( u ) , where Γ j sk ( u ) is the Riemannian connection generated by the con trav ariant metric g ij ( u ) (the Levi–Civita connection), (2) the pseudo-Riemannian metric g ij ( u ) and the set of affinors ( w n ) i j ( u ) satisfy the relations: g ik ( u )( w n ) k j ( u ) = g j k ( u )( w n ) k i ( u ) , n = 1 , ..., L, (60) ∇ k ( w n ) i j ( u ) = ∇ j ( w n ) i k ( u ) , n = 1 , ..., L, (61) R ij k l ( u ) = L X m =1 L X n =1 µ mn  ( w m ) i l ( u )( w n ) j k ( u ) − ( w m ) j l ( u )( w n ) i k ( u )  . (62) Moreo v er, t he family of a ffinors w n ( u ) is comm utativ e: [ w m , w n ] = 0 . If no n- lo cal Poisson brac k ets of h ydro dynamic ty p e (29)–(31) are compatible, then it follows from the conditions o f compatibility and fro m F erap on tov ’s theorem that, first, relat io n (51) holds, i.e., the metrics g ij 1 ( u ) a nd g ij 2 ( u ) a re almost compatible, and, secondly , the curv ature tensor for the metric g ij ( u ) = λ 1 g ij 1 ( u ) + λ 2 g ij 2 ( u ) has the form R ij k l ( u ) = L 1 X m =1 L 1 X n =1 λ 1 µ mn 1  ( w 1 ,m ) i l ( u )( w 1 ,n ) j k ( u ) − ( w 1 ,m ) j l ( u )( w 1 ,n ) i k ( u )  + + L 2 X m =1 L 2 X n =1 λ 2 µ mn 2  ( w 2 ,m ) i l ( u )( w 2 ,n ) j k ( u ) − ( w 2 ,m ) j l ( u )( w 2 ,n ) i k ( u )  = = λ 1 R ij 1 ,k l ( u ) + λ 2 R ij 2 ,k l ( u ) , 13 i.e., relation (52) holds and hence the metrics g ij 1 ( u ) and g ij 2 ( u ) are compatible. Theorem 5 [28]. L et two non-lo c al Poisson br ackets of hydr o dynamic typ e (29)– (31) c orr esp ond to submanifolds with holon o mic net of curvatur e lines and b e giv en in c o or dinates of curvatur e lines. I n this c ase, if the c orr esp ond i n g p air of metrics is non- singular, then the non -lo c al Poisson b r ackets of hydr o dynamic typ e ar e c omp a tible i f and only if their metrics ar e c omp atible. In this case the metrics g ij 1 ( u ) = g i 1 ( u ) δ ij and g ij 2 ( u ) = g i 2 ( u ) δ ij , a nd also the W ein- garten op erators ( w 1 ,n ) i j ( u ) = ( w 1 ,n ) i ( u ) δ i j and ( w 2 ,n ) i j ( u ) = ( w 2 ,n ) i ( u ) δ i j are diagonal in the co ordinates under consideration. F or any suc h “diagonal” case, condition (60) is automatically fulfilled, a ll the W eingarten op erators commute, conditions (61) and (62) ha v e the follo wing form, resp ectiv ely: 2 g i ( u ) ∂ ( w n ) i ∂ u k = ( ( w n ) i − ( w n ) k ) ∂ g i ∂ u k for all i 6 = k , (63) R ij j i ( u ) = L X m =1 L X n =1 µ mn ( w m ) i ( u )( w n ) j ( u ) , R ij k l ( u ) = 0 if i 6 = k, i 6 = l , or if j 6 = k , j 6 = l . (64) It f o llo ws f rom non-singularity of the pair o f the metrics and from compatibility of the metrics that the corresp onding Nijenh uis tensor v anishes and there exist functions f i ( u i ) , i = 1 , ..., N , suc h t ha t: g i 1 ( u ) = f i ( u i ) g i 2 ( u ) . Using relations (63) a nd (6 4), it is easy to pro v e that in this c ase it follows fr o m compat- ibilit y of the metrics tha t an ar bit r a ry linear com bination of non- lo cal P oisson brack ets under consideration is also a P oisson brack et. Theorem 5 [33] . If the p air of metrics g ij 1 ( u ) and g ij 2 ( u ) is non-sing ular, then the non-lo c al Poisson br a c kets of hydr o dynamic typ e { I , J } 1 and { I , J } 2 (29)–(31) ar e c omp atible if and only if the metrics ar e c omp atible and b o th the metrics g ij 1 ( u ) , g ij 2 ( u ) and the affinors ( w 1 ,n ) i j ( u ) , ( w 2 ,n ) i j ( u ) c an b e sam ultane ously di a gonalize d i n a domain of lo c al c o o r dinates. It is s ufficien t to pro ve here that if the pair of metrics is non-singular and the Poiss on brac k ets are compatible, then b oth the metrics g ij 1 ( u ) , g ij 2 ( u ) and the affinors ( w 1 ,n ) i j ( u ) , ( w 2 ,n ) i j ( u ) can b e sam ultaneously diagonalized in a domain of lo cal co ordinates. All the r est w as alr eady pro ve d ab ov e. First of all, it w as prov ed that in this case the metrics g ij 1 ( u ) and g ij 2 ( u ) are compatible. Since the pair of metrics is non-singular, there exist lo cal co ordinates suc h that the metrics are diagonal and ha ve the follo wing sp ecial form in these co ordinates: g ij 2 ( u ) = g i ( u ) δ ij and g ij 1 ( u ) = f i ( u i ) g i ( u ) δ ij , where f i ( u i ) , 1 ≤ i ≤ N , are functions o f single v aria ble. The functions f i ( u i ) are the eigen v alues of 14 the pair of metrics g ij 1 ( u ) a nd g ij 2 ( u ), therefore they are distinct b y assumption of the theorem ev en in the case if the y are constan ts (they can no t be coinciding constan ts). It follo ws from the compatibilit y of the P oisson brack ets { I , J } 1 and { I , J } 2 (it is necessary to consider relation (10) for the p encil { I , J } 1 + λ { I , J } 2 ) tha t g is 1 ( w 2 ,n ) j s = g j s 1 ( w 2 ,n ) i s , (65) g is 2 ( w 1 ,n ) j s = g j s 2 ( w 1 ,n ) i s . (66) Besides, f rom relation (10) for the P oisson brac ke ts { I , J } 1 and { I , J } 2 w e hav e g is 1 ( w 1 ,n ) j s = g j s 1 ( w 1 ,n ) i s , (67) g is 2 ( w 2 ,n ) j s = g j s 2 ( w 2 ,n ) i s . (68) F rom (65) and (6 8) in our sp ecial lo cal co o r dina t es we obtain g i ( w 2 ,n ) j i = g j ( w 2 ,n ) i j , (69) f i ( u i ) g i ( w 2 ,n ) j i = f j ( u j ) g j ( w 2 ,n ) i j . (70) Therefore ( w 2 ,n ) i j = g i g j ( w 2 ,n ) j i = f i ( u i ) g i f j ( u j ) g j ( w 2 ,n ) j i , (71) i.e.,  1 − f i ( u i ) f j ( u j )  ( w 2 ,n ) j i = 0 . (72) Consequen tly , since a ll the functions f i ( u i ) are distinct, w e get ( w 2 ,n ) j i = 0 for i 6 = j. (7 3 ) Similarly , from (66) and (67) w e hav e ( w 1 ,n ) j i = 0 for i 6 = j. (7 4 ) Th us, b oth the metrics g ij 1 ( u ) , g ij 2 ( u ) and the affinors ( w 1 ,n ) i j ( u ) , ( w 2 ,n ) i j ( u ) are diagonal in our special lo cal co ordinates. No w w e can prov e the main theorem of the pap er. Theorem 6. F or an a rb itr ary non-si ngular ( semisimple ) non-lo c al ly bi-Hamiltonian system of h ydr o dyna m ic typ e (27)–(31) , ther e exist lo c al c o or dinates ( Riemann invari- ants ) such that al l the r elate d ma trix differ ential-ge ometric obje cts, nam e ly, the matrix V i j ( u ) of this system of hydr o dynam ic typ e, the metrics g ij 1 ( u ) and g ij 2 ( u ) and the affinors ( w 1 ,n ) i j ( u ) and ( w 2 ,n ) i j ( u ) of the n on-lo c al bi-Hamiltonian structur e of this system, ar e diagonal in these lo c al c o or dinates. 15 If w e hav e a non- singular (semisimple) non-lo cally bi-Hamiltonian system of h ydro- dynamic type (27)–(3 1 ), then it w as pro ve d ab o v e that the metrics g ij 1 ( u ) and g ij 2 ( u ) of the non-lo cal bi-Hamilto nian structure of this system are compatible and there exist lo cal co ordinates suc h that g ij 2 ( u ) = g i ( u ) δ ij and g ij 1 ( u ) = f i ( u i ) g i ( u ) δ ij , where f i ( u i ) , i = 1 , ..., N , a re distinct nonzero functions of single v ariable (generally sp eaking, com- plex), f i ( u i ) 6 = f j ( u j ), i 6 = j . It w a s also prov ed ab ov e t ha t the a ffinor s ( w 1 ,n ) i j ( u ) and ( w 2 ,n ) i j ( u ) of the non- lo cal bi-Hamilto nia n structure of this system, are diagonal in the se lo cal co or dinates. Let us pro v e that the matrix V i j ( u ) of this system is also diago nal in these sp ecial lo cal co or dinates. Indeed, in these lo cal co ordinates, w e ha v e f r om relatio ns (19): g i ( u ) V i j ( u ) = g j ( u ) V j i ( u ) , f i ( u i ) g i ( u ) V i j ( u ) = f j ( u j ) g j ( u ) V j i ( u ) . (75) Hence, V i j ( u ) = g j ( u ) g i ( u ) V j i ( u ) = f j ( u j ) g j ( u ) f i ( u i ) g i ( u ) V j i ( u ) , (76) i.e., g j ( u ) g i ( u ) V j i ( u ) = f j ( u j ) g j ( u ) f i ( u i ) g i ( u ) V j i ( u ) . (77) Th us, ( f i ( u i ) − f j ( u j )) V j i ( u ) = 0 , (78) i.e., V i j ( u ) = 0 , i 6 = j, (79) and the diago na lizabilit y of an arbitrary non-singular (semisimple) non-lo cally bi- Ha- miltonian system of h ydro dynamic t yp e is pro ved : V i j ( u ) = V i ( u ) δ i j . (80) The diagonalizabilit y o f non-singular (semisimp le) lo cally bi-Hamiltonian systems of h ydro dynamic t yp e (27)–( 3 1) w as notice d in [35]; it fo llows immediately from the theory of non-singular pairs of compatible flat metrics [27]–[34]. W e note that it do es not follow from the pro o f that V i ( u ) 6 = V j ( u ) if i 6 = j , i.e., an arbitrary non-singular (semisimple) non-lo cally bi-Hamiltonian system of hy dro dy- namic type m ust not b e necess arily strictly h yp erb olic but we do no t kno w examples o f suc h systems with some coinciding eigen v alues (v elo cities) V i ( u ). W e conjecture that there exist suc h systems a nd this is a ve ry interes ting problem to find non- singular (semisimple) non-lo cally bi-Hamiltonian systems of h ydro dynamic type that are not strictly h yp erb olic (i.e., they ha v e some coinciding eigen v alues V i ( u )). W e also note that the non-singularity condition of the pair of metrics is very es- sen tial and we conjecture t hat there exist non-diagonalizable singular non-lo cally bi- Hamiltonian systems o f h ydro dynamic t yp e. It is also an in teresting pro blem to find 16 examples of non-diagona lizable singular non-lo cally bi-Hamiltonian systems of hyd ro dy- namic t yp e. Ac knowled gemen ts. The w ork w as supp orted b y the Max-Planc k-Institut f ¨ ur Mathematik (Bonn, Germany ), by the Russian F oundation for Basic Researc h (pro ject no. 08-0 1-00464 ) a nd b y a gran t of the Presiden t of the Russian F ederatio n (pro ject no. NSh-1824.200 8 .1). References [1] B. A. D ubro vin and S. P . No vik ov, “The Hamiltonian formalism of one-dimensi- onal systems of h ydro dynamic t yp e and the Bogolyub ov –Whitham a ve ra g ing metho d,” Dokl. Ak ad. Nauk SSSR, V ol. 270, No. 4, 1983, pp. 7 8 1–785; English translation in So viet Math. D okl., V ol. 27, 1983, pp. 6 65–669. [2] E. V. F erap onto v, “D ifferen tial geometry of nonlo cal Ha milto nian o p erators of h ydro dynamic type,” F unkts . Analiz i Ego Prilozh., V ol. 25, No. 3, 19 91, 37–49; English translation in F unctional Analysis and its Applications, V ol. 25, No. 3, 1991, pp. 195– 204. [3] O. I. Mokho v a nd E. V. F erap on tov , “Non-lo cal Hamiltonian op era t ors of h ydro- dynamic t yp e related to metrics o f constan t curv ature,” Usp ekhi Matem. Nauk, V ol. 45, No. 3, 1990, pp. 191–1 92; English t r a nslation in Russian Mathematical Surv eys, V ol. 45, No. 3 , 19 90, pp. 218–2 19. [4] O. I. Mokho v, “ Nonlo cal Hamiltonian op erators of hyd ro dynamic type with flat metrics, inte gra ble hierarc hies, and the asso ciativit y equations,” F unkts. Analiz i Ego Prilozh., V ol. 40, No. 1, 200 6 , pp. 14–2 9; English translation in F unctional Analysis and its Applications, V ol. 40, No. 1, 20 06, pp. 11–23; http://arXiv.org/math.DG/04 0 6292 (2004). [5] S. P . Tsarev, “Geometry of Hamiltonian systems of h ydro dynamic t yp e. T he generalized ho dograph metho d,” Izv estiy a Ak ad. Nauk SSSR, Ser. Matem., V ol. 5 4, No. 5 , 1990, pp. 1048–1068; English translation in Math. USSR – Izv estiy a , V ol. 54, No. 5, 1990, pp. 397 –419. [6] A. Haan tjes, “On X n − 1 -forming sets of eigen v ectors,” Indagationes Mathematicae, V ol. 17, No. 2 , 19 55, pp. 158–1 62. [7] A. Nijenh uis, “ X n − 1 -forming sets o f eigen ve ctors,” Indagationes Mathematicae, V ol. 13, No. 2 , 19 51, pp. 200–2 12. [8] M. V. P a vlov, R. A. Sharip ov and S. I. Svinolupov, “In v a r ia n t integrabilit y crite- rion for equations o f h ydro dynamic t yp e,” F unkts. Analiz i Ego Prilozh., V ol. 30, No. 1, 1996, pp. 18–29; English translation in F unctional Ana lysis and its Applications, V ol. 30, No. 1, 1996, pp. 15–2 2. [9] F. Magr i, “A simple mo del of the inte gra ble Hamiltonian equation,” J. Math. Ph ys., V ol. 19, No. 5, 1978, pp. 115 6 –1162. 17 [10] E. V. F erap on tov , C. A. P . Galv˜ ao, O. I. Mokho v and Y. Nutku, “Bi-Hamiltonian structure o f equations of asso ciativit y in 2D top o logical field theory ,” Comm. Math. Ph ys., V ol. 186, 1997, pp. 649–66 9. [11] O . I. Mokho v, “Symplectic and P oisson structures on lo o p spaces of smo oth manifolds, and integrable systems”, Usp ekhi Matematiches kikh Nauk, V ol. 53, No. 3, 1998, pp. 85–192; English translation in Russian Mathematical Surv eys, V ol. 53, No. 3, 1 998, pp. 515– 622. [12] O. I. M okhov, “Symplectic and Poiss on geometry on lo op space s of smo ot h man- ifolds and integrable equations”, Mosco w–Izhevsk , Institute of Computer Studies, 2 004 (In Russian); English v ersion: Reviews in Mathematics and Mathematical Ph ysics, V ol. 11, P art 2, Harw o o d Academic Publishers, 2001; Second Edition: R eviews in Math- ematics and Mathematical Ph ysics, V o l. 13, P art 2, Cam bridge Scien tific Publishers, 2009. [13] B. Dubrovin, “G eometry of 2D to p ological field theories,” In: In tegrable Systems and Q uan tum Gr o ups, Lecture Notes in Math., V ol. 1620 , Springer- V erlag, Berlin, 1996, pp. 120– 348; http://arXiv.org/hep-th/940701 8 (1994). [14] E. Witten, “On the structure of the top ological phase of tw o-dimensional grav - it y ,” Nuclear Ph ysics B, V o l. 340, 1990, pp. 28 1–332. [15] E. Witten, “Tw o -dimensional gravit y and interse ction theory on mo duli space,” Surv eys in Diff. Geometry , V ol. 1, 1991, pp. 2 4 3–310. [16] R . Dijkgraa f, H. V erlinde and E. V erlinde, “T op o logical strings in d < 1,” Nuclear Ph ysics B, V ol. 35 2, 1 991, pp. 59–8 6. [17] M. Konts evic h and Y u. Manin, “Gromov–Witten classe s, quan tum cohomology , and enume rat ive geometry ,” Comm. Math. Phy s., V ol. 16 4, 1 994, pp. 525–5 6 2. [18] O. I. Mokho v, “ Symplectic and P oisson geometry on lo op spaces of mani- folds and nonlinear equations,” In: T opics in T op ology and Mathematical Ph ysics, Ed. S.P .No vik o v, Amer. Math. So c., Providenc e, RI, 1995, pp. 12 1–151; h ttp://arXiv.org/ hep-th/9503076 (1995). [19] O. I. M okhov, “ P oisson and symplectic geometry o n loo p spaces of smo o t h man- ifolds,” In: Geometry fro m the P acific Rim, Pro ceedings of the P acific Rim Geometry Conference held at National Univ ersity of Singa p ore, R epublic of Singap ore, Decem- b er 12–17, Eds. A.J.Berric k, B.Lo o, H.-Y.W ang, W alter de Gruyter, Berlin, 1997 , pp. 285–309. [20] O. I. Mokho v and E. V. F erap on tov, “Equations of asso ciativity in tw o- dimen- sional top o logical field theory as in tegrable Hamiltonian non-diago nalizable systems of h ydro dynamic type,” F unkts. Analiz i Ego Prilozh., V ol. 30, No. 3, 1996, pp. 62–72; English translation in F unctional Analysis and its Applications, V ol. 30, No. 3, 1996, pp. 195– 203; http://arXiv.org/hep-th/950518 0 (1995). [21] O. I. Mokho v, “Compatible nonlo cal P oisson bra c k ets of hy dro dynamic ty p e and in tegrable hierarc hies related to them,” T eoret. Matem. Fizik a, V ol. 132, No. 1, 2002, 18 pp. 60–7 3 ; English tra nslation in Theoretical and Mathematical Phy sics, V ol. 132, No . 1, 2 002, pp. 942– 954; http://arXiv.org/math.DG/02 01242 (200 2). [22] O. I. Mokhov , “In tegrable bi-Ha milto nian systems o f hy dro dynamic ty p e,” Us- p ekhi Matematic heskikh Nauk, V ol. 57, No. 1, 2 002, pp. 15 7–158; English tra nslatio n in Russian Mathematical Surv eys, V o l. 57, No. 1, 2002 , pp. 153 –154. [23] O. I. Mokho v, “In tegrable bi-Ha miltonian hierarc hies generated b y compatible metrics of constant Riemannian curv ature,” Usp ekhi Matematiches kikh Nauk, V ol. 5 7, No. 5, 2002, pp. 157–158; English translation in Russian Mathematical Surv eys, V ol. 57, No. 5, 2002, pp. 999– 1001. [24] O. I. Mokho v, “The Liouville canonical form for compatible nonlo cal Pois- son bra c k ets o f hydrodynamic type a nd integrable hierar chies ,” F unkts. Analiz i Ego Prilozh., V ol. 37, No. 2, 200 3 , pp. 28–4 0 ; English translation in F unctional Analysis and its Applications, V ol. 3 7 , No. 2, 2003, pp. 103– 113; h ttp://arXiv.org/ math.DG/020 1223(200 2). [25] O. I. Mokho v, “F rob enius manifo lds as a sp ecial class of submanifolds in pseudo- Euclidean spaces,” In: Geometry , T op ology , and Mathematical Ph ysics (S.P .Novik ov ’s Seminar: 200 6 -2007), Amer. Mat h. So c., Pro vidence, R I, 20 08, pp. 21 3–246; ar Xiv: 0710.5860 (2007). [26] O. I. Mokho v, “Compatible Dubrov in–Novik ov Hamiltonian op erato r s, Lie deriv a- tiv e, and in tegrable systems o f Hydro dynamic t yp e,” T eoret. Matem. Fizik a, V ol. 133, No. 2 , 2 0 02, pp. 279– 2 88; Englis h translation in Theoretical and Mathematical Ph ysics, V ol. 133, No. 2, 20 02, pp. 1557–156 4 ; h ttp:// a rXiv.org/math.DG/ 0 201281 (20 02). [27] O. I. Mokho v, “Compatible and almost compatible metrics,” Usp ekhi Matem- atic heskikh Nauk, V ol. 55 , No. 4, 2 000, pp. 217–21 8; English translation in Russian Mathematical Surv eys, V ol. 55, No. 4, 200 0, pp. 819–821 . [28] O. I. Mokho v, “Compatible and a lmo st compatible pseudo-Riemannian met- rics,” F unkts. Analiz i Ego Prilozh., V o l. 35, No . 2, 200 1, pp. 24–36; English trans- lation in F unctional Analysis and its Applications, V ol. 35, No. 2, 2001, pp. 100–110 ; h ttp://arXiv.org/ math.DG/000 5051 (2000 ). [29] O. I. Mokho v, “Compatible flat metrics,” Journal of Applied Mathematics, V o l. 2, No . 7, 2002, pp. 337– 370; h ttp://arXiv.org/ math.DG/020 1224 (2002 ). [30] O. I. Mokhov , “F lat p encils of metrics and integrable reductions of the Lam ´ e equations,” Usp ekhi Matematic heskikh Nauk, V ol. 56, No. 2, 2001, pp. 221–222; English translation in Russian Mathematical Surv eys, V o l. 56, No. 2, 20 0 1, pp. 416– 418. [31] O. I. Mokho v, “In tegrabilit y of the equations for nonsingular pairs of compatible flat metrics,” T eoret. Matem. F izik a, V ol. 130 , No. 2 , 2002, pp. 2 33–250; English translation in Theoretical and Mathematical Ph ysics, V ol. 130, No. 2, 2002 , pp. 198– 212; h ttp://arXiv.org/ math.DG/000 5081 (2000 ). [32] O. I. Mokho v, “Compatible metrics of constant Riemannian curv a t ure: lo cal 19 geometry , nonlinear equations, and in tegra bility ,” F unkts. Analiz i Ego Prilozh., V o l. 36, No. 3 , 2002, pp. 36– 4 7; English translation in F unctional Analysis and its Applications, V ol. 36, No. 3 , 20 02, pp. 196–2 04; http://arXiv.org/math.DG/020 1280 (2002 ) . [33] O. I. Mokhov , “Lax pairs for equations describing compatible nonlo cal P oisson brac k ets of h ydro dynamic t yp e and in tegrable reductions of the La me equations,” T eoret. Matem. F izik a, V ol. 138, No. 2 , 2004, pp. 283– 296; English translation in Theoretical and Mathematical Ph ysics, V ol. 138, No. 2, 2004, pp. 238–249 ; h ttp://arXiv.org/ math.DG/020 2036 (2002 ). [34] E. V. F erap o nto v, “Compatible Poiss on brack ets of h ydro dynamic t yp e,” J. Phy s. A: Math. Gen., V ol. 34, 2001, pp. 2377–3388; http://arXiv.org/math.DG/00 05221 (2000). [35] B. Dubro vin, S.-Q. Liu and Y. Z hang, “On Hamiltonian perturbations of h yp er- b olic syste ms of conserv a tion la ws, I: Qua si- t r ivialit y of bi- Hamiltonian p erturbations,” Comm. Pure Appl. Math., V ol. 59, 2006, 559 - 615; arXiv:math/0410 027 (20 04). O. I . Mokhov Cen tre for Nonlinear Studies, L.D.Landau Institute for Theoretical Ph ysics, Russian Academ y of Sciences, Kosygina str., 2, Mosco w, 11794 0, Russia; Departmen t of Geometry and T op olo gy , F acult y of Mec hanics a nd Mat hematics, M.V.Lomonoso v Moscow State Unive rsity , Mosco w, 11999 2, Russia E-mail : mokhov@mi.ras.ru; mokho v@landau.ac.ru; mokho v@bk.ru 20